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

7.10.3 The x- and y-intercepts of Quadratic Expressions

7.10.3 • The x- and y-intercepts of Quadratic Expressions

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

1. Function $f$
$x^{2} + 4 x + 3$
$(x + 3) (x + 1)$

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

Compare your answer:

Graph of $f$ : $x$ -intercepts: $(- 3, 0)$ and $(- 1, 0)$ .

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

Graph of $f$ : $y$ -intercept $(0, 3)$ .

2. Function $g$
$x^{2} - 5 x + 4$
$(x - 4) (x - 1)$

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

Compare your answer:

Graph of $g$ : $x$ -intercepts: $(1, 0)$ and $(4, 0)$ .

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

Graph of $g$ : $y$ -intercept $(0, 4)$ .

3. Function $h$
$x^{2} - 9$
$(x - 3) (x + 3)$

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

Compare your answer:

Graph of $h$ : $x$ -intercepts: $(3, 0)$ and $(- 3, 0)$ .

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

Graph of $h$ : $y$ -intercept $(0, - 9)$ .

4. Function $j$
$x^{2} - 5 x$
$x (x - 5)$

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

Compare your answer:

Graph of $j$ : $x$ -intercepts: $(0, 0)$ and $(5, 0)$ .

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

Graph of $j$ : $y$ -intercept $(0, 0)$ .

5. Function $k$
$5 x - x^{2}$
$x (5 - x)$

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

Compare your answer:

Graph of $k$ : $x$ -intercepts: $(0, 0)$ and $(5, 0)$ .

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

Graph of $k$ : $y$ -intercept $(0, 0)$ .

6. Function $l$
$x^{2} + 4 x + 4$
$(x + 2) (x + 2)$

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

– 2, Graph of $l$ : $x$ -intercept $(- 2, 0)$ .

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

Graph of $l$ : $y$ -intercept $(0, 4)$ .

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

Compare your answer:

The $y$ -coordinate of the $y$ -intercept is the constant term when the function is expressed in standard form. The $y$ -intercept is $(0, - 9)$ for the function $h$ , $x^{2} - 9$ .
The $x$ -intercepts in each graph seem to match the numbers in the factors of the expressions in factored form, but the signs seem to be the opposite. For example, when the expression is $(x + 3) (x + 1)$ , the $x$ -intercepts are $(- 3, 0)$ and $(- 1, 0)$ .
When one of the factors is just a variable (rather than a sum or a difference), one of the $x$ -intercepts is $(0, 0)$ . For the function $i$ , $x (x - 5)$ , one of the intercepts is $(0, 0)$ .
x(x-5) and x(5-x) have the same $x$ -intercepts, but one graph opens upward and the other opens downward.
When the two factors are the same, there is only one $x$ -intercept. For the function $l$ , $(x + 2) (x + 2)$ , – 2 is the $x$ -intercept.

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

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

Compare your answer:

The $x$ -intercepts will be $(9, 0)$ and $(1, 0)$ .

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

The $y$ -intercept will be $(0, 9)$ because 9 is the constant term when the expression is put in standard form.

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

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

Determining the x- and ð ¦-intercepts of Quadratic Expressions

Self Check

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

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

$x = 9, x=2$
$x = - 2, x = - 9$
$x = - 2, x = 9$
$x = 2, x = - 9$

Additional Resources

Finding $x$ -intercepts Without Graphs

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

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

Example

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

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

Set each factor to 0 and solve.

$x - 3 = 0 x + 2 = 0 x = 3 x = - 2$

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

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

Try it

Finding $x$ -intercepts Without Graphs

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

Here is how to find the $x$ -intercepts:

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

Set each factor to 0 and solve.

$x - 1 = 0 x + 5 = 0 x = 1 x = - 5$

Notice that for the factor $(x - 3)$ or $(x - 1) (x + (- 1)$ ), the constant is – 1, so the $x$ -intercept is $x = + 1$ or the point $(1, 0)$ .

For the factor $(x + 5)$ , the constant is +5, so the $x$ -intercept is $x = - 5$ or the point $(- 5, 0)$ .

Are you ready for more?

Extending Your Thinking

1.

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

What is the value of $a$ ?

Compare your answer:

$a = 2$

2.

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

What is the value of $p$ ?

Compare your answer:

$p = 1$ or $p = 3$

3.

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

What is the value of $q$ ?

Compare your answer:

$q = 1$ or $q = 3$

7.10.3 The x- and y-intercepts of Quadratic Expressions