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.
Write the factored form of this equation set equal to zero.
Enter the factored form of the equation.
Compare your answer:
Now, write the standard form of the same quadratic equation.
Enter the standard form of the equation.
Compare your answer:
Write the quadratic equation as the function .
Enter the quadratic equation as the function.
Compare your answer:
So a quadratic equation that has the zeros –3 and 4 can be expressed as a function .
For questions 4 – 8, identify a quadratic function in standard form, , with the given zeros.
8 and 4
Enter a quadratic function in standard form.
Compare your answer:
The factored form derived from the zeros is .
−6 and 3
Enter a quadratic function in standard form.
Compare your answer:
The factored form derived from the zeros is .
and
Enter a quadratic function in standard form.
Compare your answer:
The factored form derived from the zeros is .
5
Enter a quadratic function in standard form.
Compare your answer:
The factored form derived from the zeros is .
−4
Enter a quadratic function in standard form.
Compare your answer:
The factored form derived from the zeros is .
Examine the graph below, then identify a quadratic function in standard form, , with the given zeros.
Enter a quadratic function in standard form.
Compare your answer:
The zeros are –4 and 6.
The factored form derived from the zeros is .
Examine the graph below, then identify a quadratic function in standard form, , with the given zeros.
Enter a quadratic function in standard form.
Compare your answer:
The zero is 3.
The factored form derived from the zero is .
Examine the graph below, then identify a quadratic function in standard form, , with the given zeros.
Enter a quadratic function in standard form.
Compare your answer:
The zeros are –2 and 5.
The factored form derived from the zeros is .
Self Check
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, , 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:
and .
When these equations are solved, they provide the zeros 5 and –3.
We can use these equations to write the function in factored form.
By using the distributive property, or FOIL, we can find the standard form of the function.
Recall that sometimes a function might have only one zero.
Example 2
Write a function, , that has one zero at 8.
From previous lessons, we know that when the factor equations were identical, there was only one zero.
and
The factored form of this equation is .
After using the distributive property, or FOIL, the standard form of the function is .
Finding the zeros from a graph follows the same process.
Example 3
Write a function, , that corresponds to the graph shown.
The zeros of the graph are –4 and –1. The resulting factor equations are:
and .
The factored form of this equation is .
After using FOIL, the standard form of the function is .
Try it
Try It: Finding a Quadratic Function from Its Zeros
Identify a quadratic function in standard form, , with zeros –6 and 1.
Here is how to find the standard form of the quadratic functions:
and
Identify a quadratic function in standard form, , with the given zeros.
Here is how to find the standard form of the quadratic functions:
The zeros of the graph are and .
Identify a quadratic function in standard form, , with the zero –9.
Here is how to find the standard form of the quadratic functions: