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A graph is used to represent points that make a function true. For example, the graph and table below show points for the quadratic function, y = x² – x – 6.

x	y
-3	6
-2	0
-1	-4
0	-6
1	-6
2	-4
3	0
4	6

table of values and graph for the function y = x2 – x – 6

Both the graph and the table of values can be used to solve the equation, x² – x – 6 = 0, which is related to the function y = x² – x – 6.

In the graph, which horizontal line represents the value y = 0?
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The x-axis
What are the coordinates of the two points on the graph containing a y-value equal to 0?
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(−2, 0) and (3, 0)
Which values of x will make the equation, x² – x – 6 = 0, a true statement?
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x = 3 and x = −2

Interactive exercise. Assistance may be required. In the interactive below, the graphs represent equations related to the function
y = x² – x – 6. The red line represents the graph y = k. Match each graph with the equation by dragging the graph to the appropriate slot. Then, indicate the correct solutions represented in the graph by selecting the correct ordered pair(s) from the dropdown menu. You may not use all of the graphs or all of the ordered pair(s). Your dropdown answer box will remain red until you have chosen the correct answer.

Video segment. Assistance may be required. View the video below to learn more about determining the solutions to a quadratic equation.

Source: Evaluating Functions Using a Graphing Calculator, Gdawy Enterprises, You Tube

Pause and Reflect

What do the x-coordinates of the x-intercepts of a quadratic function, f(x), tell you about the solutions of the related quadratic equation, f(x) = 0?

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The x-coordinates of the x-intercepts, which are also the roots of the quadratic function, f(x), are the same as the solutions of the related quadratic equation, f(x) = 0.

How is using a graph of a quadratic function, f(x), to determine the solutions of the related equation, f(x) = k, like using graphs of systems of equations to determine solutions?

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If you draw the line y = k, it will intersect the graph of f(x) at the points representing the solution to the equation, f(x) = k.

Practice

Determine the solution to the each quadratic equation by graphing. Use the quadratic equation solver below to check your answers.

Quadratic Equation Solver

Directions: Click on the link below. Enter the values of a, b, and c into the equation solver. Click "Submit Query" to view the solution.

This activity might not be viewable on your mobile device. Interactive exercise. Assistance may be required. Quadratic Solver

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Need additional directions?

x² + 5 = 0
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Click to see the graph.

How many real solutions does this equation have?
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Real solutions occur where the graph crosses the x-axis.

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There are 0 real solutions.
What are the real solutions of this equation?
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Where does the graph cross the x-axis?

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There are no real solutions.

x² – 10x − 24 = 0
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Click to see the graph.
- How many real solutions does this equation have?
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  Real solutions occur where the graph crosses the x-axis.
  
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  There are 2 real solutions.
- What are the real solutions of this equation?
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  Where does the graph cross the x-axis?
  
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  x = −2 and x = 12
x² − 4 = 0
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- How many real solutions does this equation have?
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  Real solutions occur where the graph crosses the x-axis.
  
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  Check Your Answer
  There are 2 real solutions.
- What are the real solutions of this equation?
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  Need a hint?
  Where does the graph cross the x-axis?
  
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  x = −2 and x = 2
x² − 6x + 9 = 0
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Click to see the graph.
- How many real solutions does this equation have?
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  Real solutions occur where the graph crosses the x-axis.
  
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  There is 1 real solution.
- What are the real solutions of this equation?
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  Where does the graph cross the x-axis?
  
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  x = 3
x² − 6x + 9 = 4
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How many real solutions does this equation have?
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Real solutions occur where the graph crosses the line y = 4.

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There are 2 real solutions.
What are the real solutions of this equation?
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Where does the graph cross the line y = 4?

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x = 1 and x = 5

1 over 2 1 2 x² − 1 over 2 1 2 x − 6 = -3
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Click to see the graph.

How many real solutions does this equation have?
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Real solutions occur where the graph crosses the line y = -3.

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There are 2 real solutions.
What are the real solutions of this equation?
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Where does the graph cross the line y = -3?

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x = −2 and x = 3