Question

1. consider the graph of the function ( f(x) = -4x + 12 ).

part a: determine the key features of its graph. select all that apply.
a. the rate of change is 12
b. the y-intercept is ( (0, 12) )
c. the domain can be represented by ( -4 leq x leq 12 )
d. the y-intercept is ( (0, -4) )
e. the range can be represented by ( -infty < y < infty )

Explanation:

Step1: Analyze the slope (rate of change)

The function is in the form \( f(x) = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y - intercept. For \( f(x)=- 4x + 12 \), the slope \( m=-4 \), so the rate of change is - 4, not 12. So option a is incorrect.

Step2: Find the y - intercept

To find the y - intercept, we set \( x = 0 \) in the function \( f(x)=-4x + 12 \). Then \( f(0)=-4(0)+12 = 12 \). So the y - intercept is the point \( (0,12) \), so option b is correct and option d is incorrect.

Step3: Analyze the domain of a linear function

A linear function of the form \( f(x)=mx + b \) (where \( m
eq0 \)) has a domain of all real numbers, \( (-\infty,\infty) \), unless there is a restriction given. Since there is no restriction given for \( f(x)=-4x + 12 \), the domain is not \( - 4\leq x\leq12 \). So option c is incorrect.

Step4: Analyze the range of a linear function with \( m

eq0 \)
For a linear function \( f(x)=mx + b \) where \( m
eq0 \), as \( x \) takes on all real values (since the domain is all real numbers), \( y = f(x) \) will also take on all real values. So the range of \( f(x)=-4x + 12 \) is \( (-\infty,\infty) \) (or \( -\infty

Answer:

b. The y - intercept is \( (0, 12) \)
e. The range can be represented by \( -\infty < y < \infty \)