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Algebra 1

8.11.2 Finding a Quadratic Function from Its Zeros

Algebra 18.11.2 Finding a Quadratic Function from Its Zeros

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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.

Use the "^" symbol to enter exponents.

1.

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

2.

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

3.

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

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

4.

8 and 4

5.

−6 and 3

6.

33 and 33

7.

5

8.

−4

9.

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

A parabola that opens up with x-intercepts of negative 4 and 6.
10.

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

A parabola that opens up with an x-intercepts of 3.
11.

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

A parabola that opens up with x-intercepts of negative 2 and 5.

Self Check

Identify a quadratic function, g ( x ) , in standard form, that has the zeros 5 and –7.
  1. g ( x ) = x 2 12 x + 35
  2. g ( x ) = x 2 + 12 x + 35
  3. g ( x ) = x 2 2 x + 35
  4. g ( x ) = x 2 + 2 x 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)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:

 x5=0x5=0     and      x+3=0x+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)=(x5)(x+3)h(x)=(x5)(x+3)

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

h(x)=x22x15h(x)=x22x15

Recall that sometimes a function might have only one zero.

Example 2

Write a function, d(x)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.

x8=0x8=0     and      x8=0x8=0

The factored form of this equation is d(x)=(x8)(x8)d(x)=(x8)(x8).

After using the distributive property, or FOIL, the standard form of the function is d(x)=x216x+64d(x)=x216x+64.

Finding the zeros from a graph follows the same process.

Example 3

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

A parabola that opens up with x-intercepts of negative 4 and negative 1.

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

x+4=0x+4=0     and      x+1=0x+1=0.

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

After using FOIL, the standard form of the function is z(x)=x2+5x+4z(x)=x2+5x+4.

Try it

Try It: Finding a Quadratic Function from Its Zeros

1.

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

A parabola that opens up with x-intercepts of 3 and 7.

2.

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

A parabola that opens up with x-intercepts of 3 and 7.

3.

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

A parabola that opens up with x-intercepts of 3 and 7.

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