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Algebra 1

7.10.3 The x- and y-intercepts of Quadratic Expressions

Algebra 17.10.3 The x- and y-intercepts of Quadratic Expressions

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Activity

Here are pairs of expressions in standard and factored forms. Each pair of expressions define the same quadratic function, which can be represented with the given graph.

Identify the x x -intercepts and the y y -intercept of each graph.

1. Function f f
x 2 + 4 x + 3 x 2 + 4 x + 3
( x + 3 ) ( x + 1 ) ( x + 3 ) ( x + 1 )

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 5 to 6. The y-axis scale is 1 and extends from negative 5 to 5.

a. Identify the x x -intercepts of the graph.

b. Identify the y y -value of the y y -intercept of the graph.

2. Function g g
x 2 5 x + 4 x 2 5 x + 4
( x 4 ) ( x 1 ) ( x 4 ) ( x 1 )

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 5 to 6. The y-axis scale is 1 and extends from negative 5 to 5.

a. Identify the x x -intercepts of the graph.

b. Identify the y y -value of the y y -intercept of the graph.

3. Function h h
x 2 9 x 2 9
( x 3 ) ( x + 3 ) ( x 3 ) ( x + 3 )

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 5 to 6. The y-axis scale is 1 and extends from negative 10 to 2.

a. Identify the x x -intercepts of the graph.

b. Identify the y y -value of the y y -intercept of the graph.

4. Function j j
x 2 5 x x 2 5 x
x ( x 5 ) x ( x 5 )

A parabola on a coordinate grid. 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 2.

a. Identify the x x -intercepts of the graph.

b. Identify the y y -value of the y y -intercept of the graph.

5. Function k k
5 x x 2 5 x x 2
x ( 5 x ) x ( 5 x )

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 2 to 8.

a. Identify the x x -intercepts of the graph.

b. Identify the y y -value of the y y -intercept of the graph.

6. Function l l
x 2 + 4 x + 4 x 2 + 4 x + 4
( x + 2 ) ( x + 2 ) ( x + 2 ) ( x + 2 )

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 6 to 6. The y-axis scale is 1 and extends from negative 2 to 8.

a. Identify the x x -intercept of the graph. Answer:  

b. Identify the y y -value of the y y -intercept of the graph.

7. What do you notice about the x x -intercepts, the y y -intercept, and the numbers in the expressions defining each function? Make a couple of observations.

8. Here is an expression that models function p p , another quadratic function: ( x 9 ) ( x 1 ) ( x 9 ) ( x 1 ) .

a. Predict the x x -intercepts of the graph that represents this function.

b. Identify the y y -value of the y y -intercept of the graph that represents this function.

Video: Determining the x- and y-intercepts of Quadratic Expressions

Watch the following video to learn more about the x x - and y y -intercepts of quadratic expressions.

Are you ready for more?

Extending Your Thinking

1.

Find the values of a a , p p , and q q that will make y = a ( x p ) ( x q ) y = a ( x p ) ( x q ) be the equation represented by the graph.

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 2 to 6. The y-axis scale is 1 and extends from negative 4 to 8.

What is the value of a a ?

What is the value of a a?

2.

Find the values of a a , p p , and q q that will make y = a ( x p ) ( x q ) y = a ( x p ) ( x q ) be the equation represented by the graph.

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 2 to 6. The y-axis scale is 1 and extends from negative 4 to 8.

What is the value of p p ?

3.

Find the values of a a , p p , and q y = a ( x p ) ( x q ) q y = a ( x p ) ( x q ) be the equation represented by the graph.

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 2 to 6. The y-axis scale is 1 and extends from negative 4 to 8.

What is the value of q q ?

Self Check

What are the x -values of the x -intercept(s) of 

y = ( x 9 ) ( x + 2 ) ?

  1. x = 9 , x = 2
  2. x = 2 , x = 9
  3. x = 2 , x = 9
  4. x = 2 , x = 9

Additional Resources

Finding x x -intercepts Without Graphs

When a quadratic is in factored form, the x x -intercepts of a quadratic equation will be:

  • the opposite sign of the constant in the factor because if x + a = 0 x + a = 0 , then x = a x = a .
  • x = 0 x = 0 is an option.
  • If a factor is repeated, there is only one x x -intercept.

Example

Find the x x -intercepts of y = ( x 3 ) ( x + 2 ) y = ( x 3 ) ( x + 2 ) .

The x x -intercepts occur when y = 0 y = 0 .

Set each factor to 0 and solve.

x 3 = 0 x + 2 = 0 x = 3 x = 2 x 3 = 0 x + 2 = 0 x = 3 x = 2

Notice that for the factor ( x 3 ) ( x 3 ) or ( x 3 ) ( x + ( 3 ) ( x 3 ) ( x + ( 3 ) ), the constant is – 3, so the x x -intercept is x = + 3 x = + 3 or the point ( 3 , 0 ) ( 3 , 0 ) .

For the factor ( x + 2 ) ( x + 2 ) , the constant is + 2 + 2 , so the x x -intercept is x = 2 x = 2 or the point ( 2 , 0 ) ( 2 , 0 ) .

Try it

Try It: Finding x x -intercepts Without Graphs

Find the x x -intercepts of y = ( x 1 ) ( x + 5 ) y = ( x 1 ) ( x + 5 ) .

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