Writing Equations that Represent a Graph

7.13.3 Writing Equations that Represent a Graph

7.13.3 • Writing Equations that Represent a Graph

Activity

Use the graphing tool or technology outside the course. Identify the function that matches each graph. Make sure your graph goes through all three points shown!

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Identify the function of Graph A.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 1, 1), (0, 0), and (1, 1). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 6 to 6.

Compare your answer:

$y = x^{2}$

Identify the function of Graph B.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 3, 0), (0, negative 9), and (3, 0). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 10 to 2.

Compare your answer:

$y = x^{2} - 9$ or $y = (x - 3) (x + 3)$

Identify the function of Graph C.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 2, 0), (0.5, negative 6.25), and (3, 0). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 10 to 2.

Compare your answer:

$y = (x + 2) (x - 3)$

Identify the function of Graph D.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 6, 0), (negative 3, negative 9), and (0, 0). The x-axis scale is 1 and extends from negative 9 to 4. The y-axis scale is 1 and extends from negative 10 to 2.

Compare your answer:

$y = x (x + 6)$

Identify the function of Graph E.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 2, 0), (0, 4), and (2, 0). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 6 to 6.

Compare your answer:

$y = - x^{2} + 4$ or $y = (- x + 2) (x + 2)$

Identify the function of Graph F.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (0, 4), (2, 0), and (4, 4). The x-axis scale is 1 and extends from negative 4 to 8. The y-axis scale is 1 and extends from negative 6 to 6.

Compare your answer:

$y = (x - 2) (x - 2)$ or $y = x^{2} - 4 x + 4$

Identify the function of Graph G.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 1, 0), (0, 3), and (3, 0). The x-axis scale is 1 and extends from negative 4 to 8. The y-axis scale is 1 and extends from negative 6 to 6.

Compare your answer:

$y = (- x - 1) (x - 3)$

Identify the function of Graph H.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (0, 5), (1, 0), and (5, 0). The x-axis scale is 1 and extends from negative 4 to 8. The y-axis scale is 1 and extends from negative 5 to 7.

Compare your answer:

$y = (x - 1) (x - 5)$

Identify the function of Graph I.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 1, 2), (0, 0), and (1, 2). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 4 to 8.

Compare your answer:

$y = 2 x^{2}$

10.

Identify the function of Graph J.

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 4, negative 4), (0, 0), and (4, negative 4). The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 8 to 4.

Compare your answer:

$y = - 0.25 x^{2}$

Video: Writing Equations that Represent a Quadratic Graph

Watch the following video to learn more about writing equations that represent a quadratic graph.

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Self Check

Which equation describes the graph below?

$GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A $y$-intercepts OF NEGATIVE 12 AND $x$-intercepts OF NEGATIVE 4 AND 3.$

$f(x) = x(x + 3)$
$f(x) = x + 12$
$f(x) = (x - 3)(x + 4)$
$f(x) = (x + 3)(x - 4)$

Additional Resources

Writing Quadratic Equations from Graphs

Write an equation for the graph below:

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 2, 0), (1, negative 9), and (4, 0). The x-axis scale is 1 and extends from negative 10 to 10. The y-axis scale is 1 and extends from negative 15 to 10.

First, locate the $x$ -intercepts at $(- 2, 0)$ and $(4, 0)$ .

Next, write the factor for each $x$ -intercept.

The factors that result in the given zeros are $(x + 2)$ and $(x - 4)$ .

The equation for this quadratic function is $f (x) = (x + 2) (x - 4)$ .

Use graphing technology to check your answer.

Try it

Writing Quadratic Equations from Graphs

Write an equation for the graph below:

A parabola on a coordinate grid. The parabola passes through 3 points that are labeled: (negative 6, 0), (negative 2, negative 16), and (2, 0). The x-axis scale is 1 and extends from negative 15 to 15. The y-axis scale is 1 and extends from negative 20 to 7.5.

Here is how to write an equation for the function graphed:

First, identify the $x$ -intercepts: $x = - 6$ , $x = 2 .$
Next, write a factor for each zero:
$(x + 6)$ and $(x - 2)$

The function in factored form is $f (x) = (x + 6) (x - 2)$ .