Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

8.11.2 Finding a Quadratic Function from Its Zeros

8.11.2 • Finding a Quadratic Function from Its Zeros

Activity

In previous lessons, you learned about the relationship between a quadratic function's zeros and the binomial factors in the factored form of the equation. Use this information to answer the following questions. If needed, review activities 8.9.3 and 8.10.3.

Questions 1 – 3 refer to a quadratic equation that has the zeros –3 and 4.

1.

Write the factored form of this equation set equal to zero.

Compare your answer:

$(x + 3) (x - 4) = 0$

2.

Now, write the standard form of the same quadratic equation.

Compare your answer:

$x^{2} - x - 12 = 0$

3.

Write the quadratic equation as the function $f (x)$ .

Compare your answer:

$f (x) = x^{2} - x - 12$

So a quadratic equation that has the zeros –3 and 4 can be expressed as a function $f (x) = x^{2} - x - 12$ .

For questions 4 – 8, identify a quadratic function in standard form, $f (x)$ , with the given zeros.

4.

8 and 4

Compare your answer:

$f (x) = x^{2} - 12 x + 32$

The factored form derived from the zeros is $f (x) = (x - 8) (x - 4)$ .

5.

−6 and 3

Compare your answer:

$f (x) = x^{2} + 3 x - 18$

The factored form derived from the zeros is $f (x) = (x + 6) (x - 3)$ .

6.

$\sqrt{3}$ and $- \sqrt{3}$

Compare your answer:

$f (x) = x^{2} - 3$

The factored form derived from the zeros is $f (x) = (x + \sqrt{3}) (x - \sqrt{3})$ .

7.

5

Compare your answer:

$f (x) = x^{2} - 10 x + 25$

The factored form derived from the zeros is $f (x) = (x - 5) (x - 5)$ .

8.

−4

Compare your answer:

$f (x) = x^{2} + 8 x + 16$

The factored form derived from the zeros is $f (x) = (x + 4) (x + 4)$ .

9.

Examine the graph below, then identify a quadratic function in standard form, $f (x)$ , with the given zeros.

Compare your answer:

$f (x) = x^{2} - 2 x - 24$

The zeros are –4 and 6.

The factored form derived from the zeros is $f (x) = (x + 4) (x - 6)$ .

10.

Examine the graph below, then identify a quadratic function in standard form, $f (x)$ , with the given zeros.

Compare your answer:

$f (x) = x^{2} - 6 x + 9$

The zero is 3.

The factored form derived from the zero is $f (x) = (x - 3) (x - 3)$ .

11.

Examine the graph below, then identify a quadratic function in standard form, $f (x)$ , with the given zeros.

Compare your answer:

$f (x) = x^{2} - 3 x - 10$

The zeros are –2 and 5.

The factored form derived from the zeros is $f (x) = (x + 2) (x - 5)$ .

Self Check

Identify a quadratic function,

g(x)

, in standard form, that has the zeros 5 and –7.

$g(x) = x^2 - 12x + 35$
$g(x) = x^2 + 12x + 35$
$g(x) = x^2 - 2x + 35$
$g(x) = x^2 + 2x - 35$

Additional Resources

Finding a Quadratic Function from Its Zeros

Given the zeros of a quadratic function, we can write it in standard form.

Let's look at an example.

Example 1

Write a function, $h (x)$ , that has the zeros 5 and –3.

We know that if the zeros are 5 and –3, then the factor equations that were solved are:

$x - 5 = 0$ and $x + 3 = 0$ .

When these equations are solved, they provide the zeros 5 and –3.

We can use these equations to write the function in factored form.

$h (x) = (x - 5) (x + 3)$

By using the distributive property, or FOIL, we can find the standard form of the function.

$h (x) = x^{2} - 2 x - 15$

Recall that sometimes a function might have only one zero.

Example 2

Write a function, $d (x)$ , that has one zero at 8.

From previous lessons, we know that when the factor equations were identical, there was only one zero.

$x - 8 = 0$ and $x - 8 = 0$

The factored form of this equation is $d (x) = (x - 8) (x - 8)$ .

After using the distributive property, or FOIL, the standard form of the function is $d (x) = x^{2} - 16 x + 64$ .

Finding the zeros from a graph follows the same process.

Example 3

Write a function, $z (x)$ , that corresponds to the graph shown.

The zeros of the graph are –4 and –1. The resulting factor equations are:

$x + 4 = 0$ and $x + 1 = 0$ .

The factored form of this equation is $z (x) = (x + 4) (x + 1)$ .

After using FOIL, the standard form of the function is $z (x) = x^{2} + 5 x + 4$ .

Try it

Finding a Quadratic Function from Its Zeros

1.

Identify a quadratic function in standard form, $f (x)$ , with zeros –6 and 1.

Here is how to find the standard form of the quadratic functions:

$x = - 6$ and $x = 1$
$f (x) = (x + 6) (x - 1)$
$f (x) = x^{2} + 5 x - 6$

2.

Identify a quadratic function in standard form, $f (x)$ , with the zero –9.

Here is how to find the standard form of the quadratic functions:

$x = - 9$
$f (x) = (x + 9) (x + 9)$
$f (x) = x^{2} + 18 x + 81$

3.

Identify a quadratic function in standard form, $f (x)$ , with the given zeros.

Here is how to find the standard form of the quadratic functions:

The zeros of the graph are $x = 3$ and $x = 7$ .
$f (x) = (x - 3) (x - 7)$
$f (x) = x^{2} - 10 x + 21$