Free Printable Math Worksheets for Algebra 1

Created with infinite algebra 1, stop searching. create the worksheets you need with infinite algebra 1..

  • Fast and easy to use
  • Multiple-choice & free-response
  • Never runs out of questions
  • Multiple-version printing

Free 14-Day Trial

  • Writing variable expressions
  • Order of operations
  • Evaluating expressions
  • Number sets
  • Adding rational numbers
  • Adding and subtracting rational numbers
  • Multiplying and dividing rational numbers
  • The distributive property
  • Combining like terms
  • Percent of change
  • One-step equations
  • Two-step equations
  • Multi-step equations
  • Absolute value equations
  • Solving proportions
  • Percent problems
  • Distance-rate-time word problems
  • Mixture word problems
  • Work word problems
  • Literal Equations
  • Graphing one-variable inequalities
  • One-step inequalities
  • Two-step inequalities
  • Multi-step inequalities
  • Compound inequalities
  • Absolute value inequalities
  • Discrete relations
  • Continuous relations
  • Evaluating and graphing functions
  • Finding slope from a graph
  • Finding slope from two points
  • Finding slope from an equation
  • Graphing lines using slope-intercept form
  • Graphing lines using standard form
  • Writing linear equations
  • Graphing linear inequalities
  • Graphing absolute value equations
  • Direct variation
  • Solving systems of equations by graphing
  • Solving systems of equations by elimination
  • Solving systems of equations by substitution
  • Systems of equations word problems
  • Graphing systems of inequalities
  • Discrete exponential growth and decay word problems
  • Exponential functions and graphs
  • Writing numbers in scientific notation
  • Operations with scientific notation
  • Addition and subtraction with scientific notation
  • Naming polynomials
  • Adding and subtracting polynomials
  • Multiplying polynomials
  • Multiplying special case polynomials
  • Factoring special case polynomials
  • Factoring by grouping
  • Dividing polynomials
  • Graphing quadratic inequalities
  • Completing the square
  • By taking square roots
  • By factoring
  • With the quadratic formula
  • By completing the square
  • Simplifying radicals
  • Adding and subtracting radical expressions
  • Multiplying radicals
  • Dividing radicals
  • Using the distance formula
  • Using the midpoint formula
  • Simplifying rational expressions
  • Finding excluded values / restricted values
  • Multiplying rational expressions
  • Dividing rational expressions
  • Adding and subtracting rational expressions
  • Finding trig. ratios
  • Finding angles of triangles
  • Finding side lengths of triangles
  • Visualizing data
  • Center and spread of data
  • Scatter plots
  • Using statistical models

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Mathematics LibreTexts

10.5: Graphing Quadratic Equations

  • Last updated
  • Save as PDF
  • Page ID 15196

Learning Objectives

By the end of this section, you will be able to:

  • Recognize the graph of a quadratic equation in two variables
  • Find the axis of symmetry and vertex of a parabola
  • Find the intercepts of a parabola
  • Graph quadratic equations in two variables
  • Solve maximum and minimum applications

Be Prepared

Before you get started, take this readiness quiz.

  • Graph the equation \(y=3x−5\) by plotting points. If you missed this problem, review [link] .
  • Evaluate \(2x^2+4x−1\)when \(x=−3\) If you missed this problem, review [link] .
  • Evaluate \(−\frac{b}{2a}\) when \(a=13\) and b=\(\frac{5}{6}\) If you missed this problem, review [link] .

Recognize the Graph of a Quadratic Equation in Two Variables

We have graphed equations of the form \(Ax+By=C\). We called equations like this linear equations because their graphs are straight lines.

Now, we will graph equations of the form \(y=ax^2+bx+c\). We call this kind of equation a quadratic equation in two variables .

definition: QUADRATIC EQUATION IN TWO VARIABLES

A quadratic equation in two variables , where a,b,and c are real numbers and \(a\neq 0\), is an equation of the form \[y=ax^2+bx+c \nonumber\]

Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.

Let’s look first at graphing the quadratic equation \(y=x^2\). We will choose integer values of x between −2 and 2 and find their y values. See Table .

Notice when we let \(x=1\) and \(x=−1\), we got the same value for y.

\[\begin{array} {ll} {y=x^2} &{y=x^2} \\ {y=1^2} &{y=(−1)^2} \\ {y=1} &{y=1} \\ \nonumber \end{array}\]

The same thing happened when we let \(x=2\) and \(x=−2\).

Now, we will plot the points to show the graph of \(y=x^2\). See Figure .

This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, 4), (-1, 1), (1, 1) and (2, 4).

The graph is not a line. This figure is called a parabola . Every quadratic equation has a graph that looks like this.

In Example you will practice graphing a parabola by plotting a few points.

Example \(\PageIndex{1}\)

\(y=x^2-1\)

We will graph the equation by plotting points.

Example \(\PageIndex{2}\)

Graph \(y=−x^2\).

This figure shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, -4), (-1, -1), (1, -1) and (2, -4).

Example \(\PageIndex{3}\)

Graph \(y=x^2+1\).

This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 1). Other points on the curve are located at (-2, 5), (-1, 2), (1, 2) and (2, 5).

How do the equations \(y=x^2\) and \(y=x^2−1\) differ? What is the difference between their graphs? How are their graphs the same?

All parabolas of the form \(y=ax^2+bx+c\) open upwards or downwards. See Figure .

This figure shows two graphs side by side. The graph on the left side shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the inequality a greater than 0 which means the parabola opens upwards. The graph on the right side shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the inequality a less than 0 which means the parabola opens downwards.

Notice that the only difference in the two equations is the negative sign before the \(x^2\) in the equation of the second graph in Figure . When the \(x^2\) term is positive, the parabola opens upward, and when the \(x^2\) term is negative, the parabola opens downward.

Definition: PARABOLA ORIENTATION

For the quadratic equation \(y=ax^2+bx+c\), if:

The image shows two statements. The first statement reads “a greater than 0, the parabola opens upwards”. This statement is followed by the image of an upward opening parabola. The second statement reads “a less than 0, the parabola opens downward”. This statement is followed by the image of a downward opening parabola.

Example\(\PageIndex{4}\)

Determine whether each parabola opens upward or downward:

  • \(y=−3x^2+2x−4\)
  • \( y=6x^2+7x−9\)

Example\(\PageIndex{5}\)

  • \(y=2x^2+5x−2\)
  • \(y=−3x^2−4x+7\)

Example \(\PageIndex{6}\)

  • \(y=−2x^2−2x−3\)
  • \(y=5x^2−2x−1\)

Find the Axis of Symmetry and Vertex of a Parabola

Look again at Figure . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.

We show the same two graphs again with the axis of symmetry in red. See Figure .

This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (-2, -1). Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line that goes through the center of the parabola at the point (2, 7). Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3.

The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of \(y=ax^2+bx+c\) is x=\(−\frac{b}{2a}\).

So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula x=\(−\frac{b}{2a}\).

The figure shows the steps to find the axis of symmetry for two parabolas. On the left side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times 1, which simplifies to x equals negative 2. On the right side the standard form of a quadratic equation which is y equals a x squared plus b x plus c is written above the given equation y equals negative x squared plus 4 x plus 3. The axis of symmetry is the equation x equals negative b divided by the quantity two times a. Plugging in the values of a and b from the quadratic equation the formula becomes x equals negative 4 divided by the quantity 2 times -1, which simplifies to x equals 2.

The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the vertex of the parabola.

We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its x -coordinate is \(−\frac{b}{2a}\). To find the y -coordinate of the vertex, we substitute the value of the x -coordinate into the quadratic equation.

The figure shows the steps to find the vertex for two parabolas. On the left side is the given equation y equals x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals -2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of -2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with -2 plugged in for the x value which is y equals -2 squared plus 4 times -2 plus 3. This simplifies to y equals -1. Below this is the statement “vertex is (-2, -1)”. On the right side is the given equation y equals negative x squared plus 4 x plus 3. Below the equation is the statement “axis of symmetry is x equals 2”. Below that is the statement “vertex is” next to the statement is an ordered pair with x-value of 2, the same as the axis of symmetry, and the y-value is blank. Below that the original equation is rewritten. Below the equation is the equation with 2 plugged in for the x value which is y equals negative the quantity 2 squared, plus 4 times 2 plus 3. This simplifies to y equals 7. Below this is the statement “vertex is (2, 7)”.

Definition: AXIS OF SYMMETRY AND VERTEX OF A PARABOLA

For a parabola with equation \(y=ax^2+bx+c\):

  • The axis of symmetry of a parabola is the line x=\(−\frac{b}{2a}\).
  • The vertex is on the axis of symmetry, so its x -coordinate is \(−\frac{b}{2a}\).

To find the y -coordinate of the vertex, we substitute x=\(−\frac{b}{2a}\) into the quadratic equation.

Example\(\PageIndex{7}\)

For the parabola \(y=3x^2−6x+2\) find:

  • the axis of symmetry and
  • the vertex.

Example \(\PageIndex{8}\)

For the parabola \(y=2x^2−8x+1\) find:

  • (2,−7)

Example \(\PageIndex{9}\)

For the parabola \(y=2x^2−4x−3\) find:

  • (1,−5)

Find the Intercepts of a Parabola

When we graphed linear equations, we often used the x - and y -intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.

Remember, at the y -intercept the value of x is zero. So, to find the y -intercept, we substitute x=0 into the equation.

Let’s find the y -intercepts of the two parabolas shown in the figure below.

This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals 0 squared plus4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “x equals 0”. Next to that is the equation of the graph with 0 plugged in for x which gives y equals negative quantity 0 squared plus 4 times 0 plus 3. This simplifies to y equals 3. Below the equation is the statement “y-intercept (0, 3)”.

At an x -intercept , the value of y is zero. To find an x -intercept, we substitute \(y=0\) into the equation. In other words, we will need to solve the equation \(0=ax^2+bx+c\) for x.

\[\begin{array} {ll} {y=ax^2+bx+c} \\ {0=ax^2+bx+c} \\ \nonumber \end{array}\]

But solving quadratic equations like this is exactly what we have done earlier in this chapter.

We can now find the x -intercepts of the two parabolas shown in Figure .

First, we will find the x -intercepts of a parabola with equation \(y=x^2+4x+3\).

Now, we will find the x -intercepts of the parabola with equation \(y=−x^2+4x+3\).

We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph.

\[\begin{array} {l} {(2+\sqrt{7},0) \approx (4.6,0)} & {(2−\sqrt{7},0) \approx (-0.6,0)}\\ \nonumber \end{array}\]

Do these results agree with our graphs? See Figure .

This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Three points are plotted on the curve at (-3, 0), (-1, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (-1, 0) and (-3, 0)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Three points are plotted on the curve at (-0.6, 0), (4.6, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (2 plus square root of 7, 0) is approximately equal to (4.6, 0) and (2 minus square root of 7, 0) is approximately equal to (-0.6, 0).”

Definition: FIND THE INTERCEPTS OF A PARABOLA

To find the intercepts of a parabola with equation \(y=ax^2+bx+c\):

\[\begin{array}{ll} {\textbf{y-intercept}}& {\textbf{x-intercept}}\\ {\text{Let} x=0 \text{and solve the y}}& {\text{Let} y=0 \text{and solve the x}}\\ \nonumber \end{array}\]

Example \(\PageIndex{10}\)

Find the intercepts of the parabola \(y=x^2−2x−8\).

Example \(\PageIndex{11}\)

Find the intercepts of the parabola \(y=x^2+2x−8\).

y:(0,−8); x:(−4,0), (2,0)

Example \(\PageIndex{12}\)

Find the intercepts of the parabola \(y=x^2−4x−12\).

y:(0,−12); x:(6,0), (−2,0)

In this chapter, we have been solving quadratic equations of the form \(ax^2+bx+c=0\). We solved for xx and the results were the solutions to the equation.

We are now looking at quadratic equations in two variables of the form \(y=ax^2+bx+c\). The graphs of these equations are parabolas. The x -intercepts of the parabolas occur where y=0.

For example:

\[\begin{array}{cc} {\textbf{Quadratic equation}}&{\textbf{Quadratic equation in two variable}}\\ {}&{y=x^2−2x−15}\\ {x^2−2x−15}&{\text{Let} y=0, 0=x^2−2x−15}\\ {(x−5)(x+3)=0}&{0=(x−5)(x+3)}\\ {x−5=0, x+3=0}&{x−5=0, x+3=0}\\ {x=5, x=−3}&{x=5, x=−3}\\ {}&{(5,0) \text{and} (−3,0)}\\ {}&{\text{x-intercepts}}\\ \end{array}\]

The solutions of the quadratic equation are the x values of the x -intercepts.

Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the x -intercepts of the graphs, the number of x -intercepts is the same as the number of solutions.

Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form \(ax^2+bx+c=0\). Now, we can use the discriminant to tell us how many x -intercepts there are on the graph.

This figure shows three graphs side by side. The leftmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. Below the graph is the inequality b squared minus 4 a c greater than 0. Below that is the statement “Two solutions”. Below that is the statement “ Two x-intercepts”. The middle graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is on the x-axis. Below the graph is the equation b squared minus 4 a c equals 0. Below that is the statement “One solution”. Below that is the statement “ One x-intercept”. The rightmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper left quadrant. Below the graph is the inequality b squared minus 4 a c less than 0. Below that is the statement “No real solutions”. Below that is the statement “ No x-intercept”.

Before you start solving the quadratic equation to find the values of the x -intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.

Example \(\PageIndex{13}\)

Find the intercepts of the parabola \(y=5x^2+x+4\).

Example \(\PageIndex{14}\)

Find the intercepts of the parabola \(y=3x^2+4x+4\).

y:(0,4); x:none

Example \(\PageIndex{15}\)

Find the intercepts of the parabola \(y=x^2−4x−5\).

y:(0,−5); x:(5,0)(−1,0)

Example \(\PageIndex{16}\)

Find the intercepts of the parabola \(y=4x^2−12x+9\).

Example \(\PageIndex{17}\)

Find the intercepts of the parabola \(y=−x^2−12x−36.\).

y:(0,−36); x:(−6,0)

Example \(\PageIndex{18}\)

Find the intercepts of the parabola \(y=9x^2+12x+4\).

y:(0,4); x:\((−\frac{2}{3},0)\)

Graph Quadratic Equations in Two Variables

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.

How To Graph a Quadratic Equation in Two Variables

Example \(\PageIndex{19}\)

Graph \(y=x2−6x+8\).

The image shows the steps to graph the quadratic equation y equals x squared minus 6 x plus 8. Step 1 is to write the quadratic equation with y on one side. This equation has y on one side already. The value of a is one, the value of b is -6 and the value of c is 8.

Example \(\PageIndex{20}\)

Graph the parabola \(y=x^2+2x−8\).

y:(0,−8); x:(2,0),(−4,0); axis: x=−1; vertex: (−1,−9);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, -9). Three points are plotted on the curve at (0, -8), (2, 0) and (-4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

Example \(\PageIndex{21}\)

Graph the parabola \(y=x^2−8x+12\).

y:(0,12); x:(2,0),(6,0); axis: x=4; vertex:(4,−4);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (4, -4). Three points are plotted on the curve at (0, 12), (2, 0) and (6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 4.

Definition: GRAPH A QUADRATIC EQUATION IN TWO VARIABLES.

  • Write the quadratic equation with yy on one side.
  • Determine whether the parabola opens upward or downward.
  • Find the axis of symmetry.
  • Find the vertex.
  • Find the y -intercept. Find the point symmetric to the y -intercept across the axis of symmetry.
  • Find the x -intercepts.
  • Graph the parabola.

We were able to find the x -intercepts in the last example by factoring. We find the x -intercepts in the next example by factoring, too.

Example \(\PageIndex{22}\)

Graph \(y=−x^2+6x−9\).

Example \(\PageIndex{23}\)

Graph the parabola \(y=−3x^2+12x−12\).

y:(0,−12); x:(2,0); axis: x=2; vertex:(2,0);

The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -1 to 10. The vertex is at the point (2, 0). One other point is plotted on the curve at (0, -12). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2.

Example \(\PageIndex{24}\)

Graph the parabola \(y=25x^2+10x+1\).

y:(0,1); x:(−15,0); axis: x=−15; vertex:(−15,0);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (-1 fifth, 0). One other point is plotted on the curve at (0, 1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1 fifth.

For the graph of \(y=−x^2+6x−9\) the vertex and the x -intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation \(0=−x^2+6x−9\) is 0, so there is only one solution. That means there is only one x -intercept, and it is the vertex of the parabola.

How many x -intercepts would you expect to see on the graph of \(y=x^2+4x+5\)?

Example \(\PageIndex{25}\)

Graph \(y=x^2+4x+5\).

Example \(\PageIndex{26}\)

Graph the parabola \(y=2x^2−6x+5\).

y:(0,5); x:none; axis: \(x=\frac{3}{2}\); vertex:\((\frac{3}{2},\frac{1}{2})\);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 10. The vertex is at the point (3 halves, 1 half). One other point is plotted on the curve at (0, 5). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 3 halves.

Example \(\PageIndex{27}\)

Graph the parabola \(y=−2x^2−1\).

y:(0,−1); x:none; axis: x=0; vertex:(0,−1);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (0, -1). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 0.

Finding the y -intercept by substituting x=0 into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the x -intercepts in Example . We will use the Quadratic Formula again in the next example.

Example \(\PageIndex{28}\)

Graph \(y=2x^2−4x−3\).

.

Example \(\PageIndex{29}\)

Graph the parabola \(y=5x^2+10x+3\).

y:(0,3); x:(−1.6,0),(−0.4,0); axis: x=−1; vertex:(−1,−2);

The graph shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -5 to 5. The y-axis of the plane runs from -5 to 5. The vertex is at the point (-1,-2). Three other points are plotted on the curve at (0, 3), (-1.6, 0), (-0.4, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

Example \(\PageIndex{30}\)

Graph the parabola \(y=−3x^2−6x+5\).

y:(0,5); x:(0.6,0),(−2.6,0); axis: x=−1; vertex:(−1,8);

The graph shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from -10 to 10. The y-axis of the plane runs from -10 to 10. The vertex is at the point (-1, 8). Three other points are plotted on the curve at (0, 5), (0.6, 0) and (-2.6, 0). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -1.

Solve Maximum and Minimum Applications

Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The y -coordinate of the vertex is the minimum y -value of a parabola that opens upward. It is the maximum y -value of a parabola that opens downward. See Figure .

This figure shows two graphs side by side. The left graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper right quadrant. The vertex is labeled “maximum”. The right graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. The vertex is labeled “minimum”.

Definition: MINIMUM OR MAXIMUM VALUES OF A QUADRATIC EQUATION

The y -coordinate of the vertex of the graph of a quadratic equation is the

  • minimum value of the quadratic equation if the parabola opens upward.
  • maximum value of the quadratic equation if the parabola opens downward.

Example \(\PageIndex{31}\)

Find the minimum value of the quadratic equation \(y=x^2+2x−8\).

Example \(\PageIndex{32}\)

Find the maximum or minimum value of the quadratic equation \(y=x^2−8x+12\).

The minimum value is −4 when x=4.

Example \(\PageIndex{33}\)

Find the maximum or minimum value of the quadratic equation \(y=−4x^2+16x−11\).

The maximum value is 5 when x=2.

We have used the formula

\[\begin{array} {l} {h=−16t^2+v_{0}t+h_{0}}\\ \nonumber \end{array}\]

to calculate the height in feet, h, of an object shot upwards into the air with initial velocity, \(v_{0}\), after t seconds.

This formula is a quadratic equation in the variable tt, so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height.

Example \(\PageIndex{34}\)

The quadratic equation \(h=−16t^2+v_{0}t+h_{0}\) models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.

  • How many seconds will it take the volleyball to reach its maximum height?
  • Find the maximum height of the volleyball.

\(h=−16t^2+176t+4\)

Since a is negative, the parabola opens downward.

The quadratic equation has a maximum.

1. \[\begin{array} {ll} {}&{t=−\frac{b}{2a}}\\ {\text{Find the axis of symmetry.}}& {t=−\frac{176}{2(−16)}}\\ {}&{t=5.5}\\ {}&{\text{The axis of symmetry is} t = 5.5}\\ {\text{The vertex is on the line} t=5.5}& {\text{The maximum occurs when} t =5.5 \text{seconds.}}\\ \nonumber \end{array}\]

Example \(\PageIndex{35}\)

The quadratic equation \(h=−16t^2+128t+32\) is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height? Round answers to the nearest tenth.

It will take 4 seconds to reach the maximum height of 288 feet.

Example \(\PageIndex{36}\)

A toy rocket shot upward from the ground at a rate of 208 ft/sec has the quadratic equation of \(h=−16t^2+208t\). When will the rocket reach its maximum height? What will be the maximum height? Round answers to the nearest tenth.

It will take 6.5 seconds to reach the maximum height of 676 feet.

  • Graphing Quadratic Functions
  • How do you graph a quadratic function?
  • Graphing Quadratic Equations

Key Concepts

  • The graph of every quadratic equation is a parabola.
  • a>0, the parabola opens upward.
  • a<0, the parabola opens downward.
  • The axis of symmetry of a parabola is the line \(x=−\frac{b}{2a}\).
  • To find the y -coordinate of the vertex we substitute \(x=−\frac{b}{2a}\) into the quadratic equation.
  • Find the Intercepts of a Parabola To find the intercepts of a parabola with equation \(y=ax^2+bx+c\): \[\begin{array} {ll} {\textbf{y-intercept}}&{\textbf{x-intercepts}}\\ {\text{Let} x=0 \text{and solve for y}}&{\text{Let} y=0 \text{and solve for x}}\\ \nonumber \end{array}\]
  • The y - coordinate of the vertex of the graph of a quadratic equation is the

Graphing Quadratics in Vertex Form Algebra 1 Practice Worksheet

Show preview image 1

Also included in

algebra 1 10 1 worksheet graphing quadratics answers

Description

This 8-question algebra 1 worksheet provides students with organized practice graphing quadratics in Vertex Form. Students will first identify the vertex and then complete a table of values in order to graph the parabola. I have also provided an option that does not include the function table.

Works great as a class work or homework assignment after using this ★FOLDABLE★ to introduce the concept.

This resource is also included in the following bundles:

★ Algebra 1 Practice Worksheets Bundle

★ Algebra 1 Activities Bundle

You may be interested in some of my other algebra 1 resources:

★ Algebra 1 (+ Pre-Algebra) Foldable Bundle

★ Algebra 1 Semester 1 Google Form Bundle

★ Algebra 1 Semester 2 Google Form Bundle

★ Algebra 1 Puzzle Bundle

★ Algebra 1 Task Cards Bundle

★ Algebra 1 Scavenger Hunt Bundle

★ Algebra 1 BOOM Cards Bundle

Questions, concerns, or requests? Feel free to email me at

[email protected]

Questions & Answers

Lisa davenport.

  • We're hiring
  • Help & FAQ
  • Privacy policy
  • Student privacy
  • Terms of service
  • Tell us what you think

Kidsmart Education

  • How Does Online Tutoring Work?
  • Testimonials
  • Packages & Pricing
  • Contact Confirmation
  • Hannah Mangham
  • Start Your Session
  • Schedule Tutoring
  • English Worksheet Examples
  • Spanish Worksheet Examples
  • Schedule English Tutoring
  • Mission Statement
  • Privacy Policy
  • Tutoring Agreement
  • Free Worksheets
  • Number Basics
  • Integer Math
  • Rational Numbers
  • Variable Expressions
  • Proportions & Percents
  • Solving Single Variable Equations
  • Inequalities
  • The Coordinate Plane & Linear Functions
  • Plane Figures
  • Volumes of Solids
  • Solving Equations
  • Linear Equations and Inequalities
  • Solving Linear Systems
  • Polynomial and Quadratic Factoring
  • Solving Quadratics

Graphing Quadratics

  • Radicals & Rational Exponents
  • Rational Expressions
  • Geometry Essentials
  • Reasoning & Proofs
  • Parallel & Perpendicular Lines
  • Congruent Triangles
  • Triangle Relationships
  • Right Triangle Relationships & Trigonometry
  • Quadrilaterals
  • Transformations
  • Perimeter, Circumference & Area
  • Surface Area & Volume
  • Equations & Inequalities
  • Linear Equations & Functions
  • Linear Systems & Matrices
  • Quadratic Functions & Factoring
  • Polynomials & Polynomial Functions
  • Rational Exponents & Radical Functions
  • Exponential & Logarithmic Functions
  • Rational Functions
  • Trigonometric Ratios & Functions
  • Trigonometric Graphs, Identities & Equations
  • Probabilities & Statistics
  • Sequences & Series
  • Functions & Graphs
  • Polynomials, Power & Rational Functions
  • Exponential, Logistic & Logarithmic Functions
  • Trigonometric Functions
  • Analytic Trigonometry
  • Trigonometric Applications
  • Systems & Matrices
  • Analytic Geometry
  • Discrete Mathematics
  • Introduction to Calculus
  • Functions, Continuity & Limits
  • Derivatives
  • Applications of Differentiation
  • Definite Integral
  • Applications of Integration
  • Logarithmic & Exponential Functions
  • Techniques of Integration
  • Seperable Differential Equations
  • Taylor’s Theorem, L’Hopital’s Rule & Improper Integrals
  • Infinite Sequences & Theories
  • Conic Sections & Polar Coordinates
  • Curves & Vectors in the Plane
  • Vectors in Space & Solid Analytic Geometry
  • Partial Derivatives
  • Maxima & Minima with Two or Three Variables
  • Multiple Integrals
  • Vector Analysis

Kid Smart Education

We are dedicated to helping students reach their full potential in Math, English, College Application Essays, & SAT Prep. Get Started Today!

Testimonial

"Every now and then you come across someone who is truly amazing at what they do. Nancy Gleason is one of those people. Whether you have a child who loves math or struggles with math, she is a resource to consider." - Judy Lazaro

Download Free Algebra I > Graphing Quadratics Worksheets Below:

All worksheets are free to download and use for practice or in your classroom.  All we ask is that you don’t remove the KidSmart logo.

Click on for Answers

  • Graphing Quadratics in Vertex Form –
  • Graphing Quadratics in Intercept Form –
  • Graphing Quadratics in Standard Form –
  • Graphing Quadratics Mixed –

Need a little extra help?

Schedule an online tutoring session with Nancy or Hannah. Professional homework help is just a few clicks away!

Mathwarehouse Logo

Algebra Worksheets

Free worksheets with answer keys.

Enjoy these free printable sheets . Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

  • Distance Formula
  • Equation of Circle
  • Factor Trinomials Worksheet
  • Domain and Range
  • Mixed Problems on Writing Equations of Lines
  • Slope Intercept Form Worksheet
  • Standard Form Worksheet
  • Point Slope Worksheet
  • Write Equation of Line from the Slope and 1 Point
  • Write Equation of Line from Two Points
  • Equation of Line Parallel to Another Line and Through a Point
  • Equation of Line Perpendicular to Another Line and Through a Point
  • Slope of a Line
  • Perpendicular Bisector of Segment
  • Write Equation of Line Mixed Review
  • Word Problems
  • Multiplying Monomials Worksheet
  • Multiplying and Dividing Monomials Sheet
  • Adding and Subtracting Polynomials Worksheet
  • Multiplying Monomials with Polynomials Worksheet
  • Multiplying Binomials Worksheet
  • Multiplying Polynomials
  • Simplifying Polynomials
  • Factoring Trinomials
  • Operations with Polynomials Worksheet
  • Dividing Radicals
  • Simplify Radicals Worksheet
  • Adding Radicals
  • Mulitplying Radicals Worksheet
  • Radicals Review (Mixed review worksheet on radicals and square roots)
  • Solve Systems of Equations Graphically
  • Solve Systems of Equations by Elimination
  • Solve by Substitution
  • Solve Systems of Equations (Mixed review)
  • Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
  • Real World Connections (Compare cell phone plans)
  • Scientific Notation
  • Operations with Scientific Notation

Ultimate Math Solver (Free) Free Algebra Solver ... type anything in there!

Popular pages @ mathwarehouse.com.

Surface area of a Cylinder

IMAGES

  1. Graphing Quadratics Notes and Worksheets

    algebra 1 10 1 worksheet graphing quadratics answers

  2. Graphing Quadratics Review Worksheet Answers

    algebra 1 10 1 worksheet graphing quadratics answers

  3. Graphing Quadratics Worksheet Answers

    algebra 1 10 1 worksheet graphing quadratics answers

  4. Graphing Quadratic Functions In Standard Form Worksheet

    algebra 1 10 1 worksheet graphing quadratics answers

  5. Graphing Quadratic Functions Worksheet Answers

    algebra 1 10 1 worksheet graphing quadratics answers

  6. Graphing Quadratic Functions Worksheet Answer Key Algebra 1 Algebra

    algebra 1 10 1 worksheet graphing quadratics answers

VIDEO

  1. Graphing Basic Quadratic

  2. Quadratic Equations

  3. A19.19 Solving Quadratic Equations by Graphing

  4. Quadratic Equation

  5. Math 20-1 Quadratics Review #1

  6. Forming and Solving Quadratic Equations

COMMENTS

  1. PDF Graphing Quadratic Functions.ks-ia1

    ©W 42 Y01Z20 2K Guht XaP uS Ho efJtSwbaFrmeI 4L dL 8Cb. w U RApl Olm sr miTgeh KtIs O yrhe 7swelr YvRejdC. 3 0 bMuaXdIei dwIi kt5hX yIon kfPiLn vi3t Ae7 5A ylng 9eBb VrjaC i1 D.K Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Graphing Quadratic Functions Date_____ Period____

  2. PDF Answers (Lesson 10-1)

    Glencoe Algebra 1 Translating Quadratic Graphs When a figure is moved to a new position without undergoing any rotation, then the figure is said to have been translated to that position.

  3. PDF Infinite Algebra 1

    Worksheet by Kuta Software LLC Algebra 1 Practice: Graphing Quadratic Functions ... Algebra 1 Practice: Graphing Quadratic Functions Name_____ ID: 1 ©g l2t0z1D5a DK[uxtqaA GSaoffVtmwgaxrseQ TL^LsCu.B f AAqlwlJ jrOizgvh^tEsu UrleXsweYrnvheGdQ.-1- Sketch the graph of each function. ...

  4. PDF Name: Period: 10.1 Notes-Graphing Quadratics

    For question 1 - 6, identify the maximum or minimum point, the axis of symmetry, and the roots (zeros) of the graph of the quadratic function shown, as indicated. Section 1: 1. Maximum point; ( ___, ___ ) Axis of Symmetry: Roots: 3. Minimum point; ( ___, ___ ) Axis of Symmetry: Roots: y 5 4 3 2 1 -5 4 -3 2 -1 -1 2 3 4 5 x -5

  5. Free Printable Math Worksheets for Algebra 1

    Finding angles of triangles. Finding side lengths of triangles. Statistics. Visualizing data. Center and spread of data. Scatter plots. Using statistical models. Free Algebra 1 worksheets created with Infinite Algebra 1. Printable in convenient PDF format.

  6. Graphing quadratics review (article)

    Check out this video. Example: Non-vertex form Graph the function. g ( x) = x 2 − x − 6 First, let's find the zeros of the function—that is, let's figure out where this graph y = g ( x) intersects the x -axis. g ( x) = x 2 − x − 6 0 = x 2 − x − 6 0 = ( x − 3) ( x + 2)

  7. Quadratic functions & equations

    Familiar Attempted Not started Quiz Unit test About this unit We've seen linear and exponential functions, and now we're ready for quadratic functions. We'll explore how these functions and the parabolas they produce can be used to solve real-world problems. Intro to parabolas Learn Parabolas intro Interpreting a parabola in context

  8. Quadratic equations & functions

    Algebra (all content) 20 units · 412 skills. Unit 1 Introduction to algebra. Unit 2 Solving basic equations & inequalities (one variable, linear) Unit 3 Linear equations, functions, & graphs. Unit 4 Sequences. Unit 5 System of equations. Unit 6 Two-variable inequalities. Unit 7 Functions. Unit 8 Absolute value equations, functions, & inequalities.

  9. PDF 0,-$#*1%/ +%234#)%5)/631

    Find the vertex. Is the vertex a maximum or a minimum? Can you tell (without graphing) if your vertex is going to be a maximum or a minimum? a) y = x2 b) y =1 2 x2c) y = ! 2 x2 1 Example 3: Without graphing, order the quadratic functions from widest to narrowest: y = ! 4 x2,y= 1 4 x2,y= x2 Example 4: Graph the following functions.

  10. Infinite Algebra 1

    Infinite Algebra 1 covers all typical algebra material, over 90 topics in all, from adding and subtracting positives and negatives to solving rational equations. Suitable for any class with algebra content. Designed for all levels of learners from remedial to advanced. Test and worksheet generator for Algebra 1.

  11. 10.5: Graphing Quadratic Equations

    Answer. We will graph the equation by plotting points. Choose integers values for x, substitute them into the equation and solve for y. Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation \ (y=x^2−1\) Example \ (\PageIndex {2}\) Graph \ (y=−x^2\). Answer.

  12. Algebra 1 Worksheets

    This Algebra 1 - Quadratic Functions Worksheets will produce problems for practicing graphing quadratic function from their equations. These Quadratic Functions Worksheets are a good resource for students in the 5th Grade through the 8th Grade. Graphing Quadratic Inequalities Worksheets

  13. PDF Algebra 1

    L 3 GMca sd de4 Ew 0i vt ihe WIDnlf Lijn uiJtIe R yA8l Ggke 8brsa V f13. b Worksheet by Kuta Software LLC Algebra 1 ID: 1 ... Graphing Quadratics Extra Practice List the concavity, y-intercept, axis of symmetry, and vertex for each function. 1) y = x2 + 8x + 15 2) y = −3x2 − 24 x − 49

  14. PDF Unit 2-2: Writing and Graphing Quadratics Worksheet Practice PACKET

    Find a quadratic model for each set of values. 1.1. (-1, 1), (1, 1), (3, 9) 2. (-4, 8), (-1, 5), (1, 13) 3. (-1, 10), (2, 4), (3,-6) 4.

  15. PDF Graphing Quadratics Review Worksheet Name

    8. Solutions to quadratic equations are called _______________________. Determine whether the quadratic functions have two real roots, one real root, or no real roots. If possible, list the zeros of the function. 9. Number of roots: _____ 10. Number of roots: _____ 11. Number of roots: ____ Zero(s): ____________ Zero(s): ____________

  16. Graph quadratics: standard form

    Lesson 10: Quadratic standard form. Finding the vertex of a parabola in standard form. Graphing quadratics: standard form. Math >. Algebra 1 >. Quadratic functions & equations >.

  17. 10.1 Intro to Quadratics

    Common Core Standard: A-SSE.A.2, A-SSE.B.3, A-APR.B.3, F-IF.B.4, F-IF.C.7, F-IF.C.8

  18. Graphing Quadratics in Vertex Form Algebra 1 Practice Worksheet

    This 8-question algebra 1 worksheet provides students with organized practice graphing quadratics in Vertex Form. Students will first identify the vertex and then complete a table of values in order to graph the parabola. I have also provided an option that does not include the function table. Works great as a class work or homework assignment ...

  19. Free Math Worksheets

    Download Free Algebra I > Graphing Quadratics Worksheets Below: All worksheets are free to download and use for practice or in your classroom. All we ask is that you don't remove the KidSmart logo. Click on for Answers Graphing Quadratics Graphing Quadratics in Vertex Form - Graphing Quadratics in Intercept Form -

  20. Algebra 1

    Completing the Square Worksheet #1. Completing the Square Worksheet #2. Solve by Graphing Worksheet and Review - To solve by graphing, the answers come from where the curved line crosses the x-axis. All Graphs are provided.

  21. Algebra 1 Worksheets

    CHAPTER 2 WORKSHEETS. F ractions Review WS # 1 (Solns on back of WS) 2-1 Solving One-Step Equation s. 2-2 Solving Two-Step Equations. 2-3 Solving Multi-Step Equations . 2- 4 Solving Equations with Variables on Both Sides ( SOLUTIONS) 2-5 Literal Equations and Formulas. 2-6 Ratios, Rates, and Conversions ( SOLUTIONS)

  22. Algebra Workshets -- free sheets(pdf) with answer keys

    Enjoy these free printable sheets. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key. Algebra. Distance Formula. Equation of Circle. Factoring. Factor Trinomials Worksheet. Functions and Relations.

  23. Fillable Online Algebra 1 10.1 worksheet graphing quadratics answers

    Do whatever you want with a Algebra 1 10.1 worksheet graphing quadratics answers: fill, sign, print and send online instantly. Securely download your document with other editable templates, any time, with PDFfiller. No paper. No software installation. On any device & OS. Complete a blank sample electronically to save yourself time and money. Try