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Algebra 1

8.9.3 Writing an Equation to Represent a Quadratic Function with Only One Solution

Algebra 18.9.3 Writing an Equation to Represent a Quadratic Function with Only One Solution

8.9.3 Writing an Equation to Represent a Quadratic Function with Only One Solution

8.9.3 • Writing an Equation to Represent a Quadratic Function with Only One Solution

Activity

In previous lessons, we saw that a quadratic equation can have zero, one, or two solutions.

1.

Sketch graphs that represent three quadratic functions: one that has no zeros, one with one zero, and one with two zeros.

2.

Graph the function defined by f ( x ) = x 2 − 2 x + 1 f ( x ) = x 2 − 2 x + 1 . Examine the x x -intercepts of the graph.

Use the Desmos graphing tool or technology outside the course.

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Explain what the x x -intercepts reveal about the function.

3.

Solve x 2 − 2 x + 1 = 0 x 2 − 2 x + 1 = 0 by using the factored form and zero product property. What solutions do you get?

4.

Try to find another quadratic function that you think will only have one zero.

Use the Desmos graphing tool or technology outside the course.

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Graph it to check your prediction, then enter your final answer.

Video: Quadratic Equations with Only One Solution

Watch the following video to learn more about writing an equation to represent a quadratic function with only one solution.

8.9.3

Self Check

Without graphing, identify the function whose graph has only one zero.
  1. f ( x ) = x 2 + 2 x − 15
  2. f ( x ) = x 2 − 14 x + 49
  3. f ( x ) = x 2 − 1
  4. f ( x ) = x 2 − 7 x − 18

Additional Resources

Writing an Equation to Represent a Quadratic Function with Only One Solution

Identifying the Number of Zeros

How many zeros does the graph of the function f ( x ) = x 2 + 5 x − 14 f ( x ) = x 2 + 5 x − 14 have? How can you find out without graphing the function?

Let's find the factored form of the function and solve it.

Step 1 - Set the function equal to 0.

x 2 + 5 x − 14 = 0 x 2 + 5 x − 14 = 0

Step 2 - Find the factored form.

( x + 7 ) ( x − 2 ) = 0 ( x + 7 ) ( x − 2 ) = 0

Step 3 - Set each factor equal to 0 and solve.

x = − 7 x = − 7 and x = 2 x = 2

Since there are two solutions, we know that the function f ( x ) = x 2 + 5 x − 14 f ( x ) = x 2 + 5 x − 14 will have zeros at x = − 7 x = − 7 and x = 2 x = 2 .

In other words, the x x -intercepts of the function are ( − 7 , 0 ) ( − 7 , 0 ) and ( 2 , 0 ) ( 2 , 0 ) .

For any function, the number of solutions is equal to the number of zeros when graphed.

Writing an Equation to Represent a Quadratic Function with One Solution

We can also find a quadratic function if we know the zeros. For example, if we want to write a quadratic function that has one zero or one solution, we can start with a factored form of the function representing one zero or solution.

Any factored form of a function with a factor that repeats itself, such as ( x − 3 ) ( x − 3 ) ( x − 3 ) ( x − 3 ) , has only one solution or one zero.

We can graph the function f ( x ) = ( x − 3 ) ( x − 3 ) f ( x ) = ( x − 3 ) ( x − 3 ) to check our prediction.

We know the solution is x = 3 x = 3 . Let's rewrite the function in standard form.

Factored form:

f ( x ) = ( x − 3 ) ( x − 3 ) f ( x ) = ( x − 3 ) ( x − 3 )

Multiply to find standard form:

f ( x ) = x 2 − 6 x + 9 f ( x ) = x 2 − 6 x + 9

Now, we graph the function f ( x ) = x 2 − 6 x + 9 f ( x ) = x 2 − 6 x + 9 .

The graph of the function has one zero at x = 3 x = 3 . The x x -intercept of the function is ( 3 , 0 ) ( 3 , 0 ) . The equation is said to have a double root at this point.

So, our prediction was correct.

Try it

Writing an Equation to Represent a Quadratic Function with Only One Solution

1.

Write a quadratic function in standard form whose graph has two zeros.

2.

Write a quadratic function in standard form whose graph has one zero.

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