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:

For Question 3:

Step 1: Find x - intercepts

To find the x - intercepts, we set \( p(x)=(x - 9)(x - 1)=0 \). Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So, \( x-9 = 0 \) gives \( x = 9 \) and \( x - 1=0 \) gives \( x = 1 \). So the x - intercepts are \( x = 1 \) and \( x = 9 \), and the points are \( (1,0) \) 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 \( (0,9) \).

Step 3: Analyze the graph

The function \( p(x)=(x - 9)(x - 1)=x^{2}-10x + 9 \) is a quadratic function with a positive leading coefficient (\( a = 1>0 \)), so the parabola opens upwards. The vertex of a quadratic function \( ax^{2}+bx + c \) has its x - coordinate at \( x=-\frac{b}{2a} \). For \( p(x)=x^{2}-10x + 9 \), \( a = 1 \), \( b=-10 \), so \( x=-\frac{-10}{2\times1}=5 \). Substitute \( x = 5 \) into the function: \( p(5)=(5 - 9)(5 - 1)=(-4)\times(4)=-16 \). So the vertex is at \( (5,-16) \). The graph is a parabola opening upwards with x - intercepts at \( (1,0) \) and \( (9,0) \) and y - intercept at \( (0,9) \) and vertex at \( (5,-16) \).

Answer:

• x - intercepts: \( (1,0) \) and \( (9,0) \)
• y - intercept: \( (0,9) \)
• The graph is a parabola opening upwards with vertex at \( (5, - 16) \), crossing the x - axis at \( (1,0) \) and \( (9,0) \), and crossing the y - axis at \( (0,9) \)