Question

1. (continued) identify the x - intercepts and the y - intercept of each graph.

function j
$5x - x^2$
$x(5 - x)$
function k
$x^2 + 4x + 4$
$(x + 2)(x + 2)$

1. 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.
2. here is an expression that models function p, another quadratic function: $(x - 9)(x - 1)$. predict the x - intercepts and the y - intercept of the graph that represent this function.

Explanation:

Problem 3: Predict x - intercepts and y - intercept of \( p(x)=(x - 9)(x - 1) \)

Step 1: Find x - intercepts

To find the x - intercepts, we set \( p(x)=0 \). So, we solve the equation \((x - 9)(x - 1)=0\). By the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \).
If \( x-9=0 \), then \( x = 9 \). If \( x - 1=0 \), then \( x = 1 \). So the x - intercepts are the points where \( x = 1 \) and \( x = 9 \), or in coordinate form \((1,0)\) and \((9,0)\).

Step 2: Find y - intercept

To find the y - intercept, we set \( x = 0 \) in the function \( p(x)=(x - 9)(x - 1) \).
Substitute \( x = 0 \) into the function: \( p(0)=(0 - 9)(0 - 1)=(-9)\times(-1)=9 \). So the y - intercept is the point \((0,9)\).

Answer:

x - intercepts: \( x = 1 \) (or \((1,0)\)) and \( x = 9 \) (or \((9,0)\)); y - intercept: \( y = 9 \) (or \((0,9)\))