Question

relating expressions and their g

1. (continued) identify the x-intercepts a

function j
$5x - x^2$
$x(5 - x)$

1. what do you notice about the x

numbers in the expressions defin
observations.

1. here is an expression that

function: $(x - 9)(x - 1)$. predict
graph that represent this f

Explanation:

Problem 1: Identify the x - intercepts of function \( j(x)=5x - x^{2}=x(5 - x) \)

Step 1: Recall the definition of x - intercepts

The x - intercepts of a function \( y = f(x) \) are the values of \( x \) for which \( y=0 \), i.e., we need to solve the equation \( f(x)=0 \).

For the function \( j(x)=x(5 - x) \), we set \( j(x) = 0 \), so we have the equation \( x(5 - x)=0 \).

Step 2: Solve the equation \( x(5 - x)=0 \)

According to the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \) (or both).

In the equation \( x(5 - x)=0 \), let \( a=x \) and \( b = 5 - x \).

• When \( a=x=0 \), we get one solution \( x = 0 \).
• When \( b = 5 - x=0 \), we solve for \( x \):

Subtract 5 from both sides: \( -x=- 5 \)
Multiply both sides by - 1: \( x = 5 \)

Step 1: Recall the zero - product property

To find the x - intercepts of the function \( y=(x - 9)(x - 1) \), we set \( y = 0 \), so we solve the equation \( (x - 9)(x - 1)=0 \).

Step 2: Solve the equation \( (x - 9)(x - 1)=0 \)

Using the zero - product property, if \( (x - 9)(x - 1)=0 \), then either \( x-9 = 0 \) or \( x - 1=0 \).

• If \( x-9=0 \), then \( x=9 \).
• If \( x - 1=0 \), then \( x = 1 \).

Answer:

The x - intercepts of the function \( j(x)=5x - x^{2} \) are \( x = 0 \) and \( x=5 \).

Problem 3: Predict the x - intercepts of the function \( f(x)=(x - 9)(x - 1) \)