Question

midyear check
algebra 1
unit 3: linear functions and absolute value functions
students will understand the concept of a function and be able to represent and interpret linear functions.

1. compare the two functions ( f(x) ) and ( g(x) ). determine which one has a greater ( y )-intercept, ( x )-intercept, and rate of change.

( f(x) = 4x - 3 )

( x )	( -3 )	( -1 )	( 1 )	( 3 )

part a: determine the intercepts and rate of change for the functions.

	( x )-intercept	( y )-intercept	rate of change
( g(x) )

part b: complete the sentences below.

• function ____ has a greater ( x )-intercept.
• function ____ has a greater ( y )-intercept.
• function ____ has a greater rate of change.

things to remember!

Explanation:

Step1: Analyze \( f(x) = 4x - 3 \)

• x - intercept: Set \( f(x)=0 \), so \( 4x - 3 = 0 \). Solving for \( x \), we get \( 4x=3 \), \( x = \frac{3}{4}=0.75 \).
• y - intercept: Set \( x = 0 \), then \( f(0)=4(0)-3=-3 \).
• Rate of change (slope): For a linear function \( y = mx + b \), the slope \( m = 4 \).

Step2: Analyze \( g(x) \) (using the table)

• Find the equation of \( g(x) \): Let's use two points, e.g., \( (-3,13) \) and \( (-1,8) \). The slope \( m=\frac{8 - 13}{-1-(-3)}=\frac{-5}{2}=-2.5 \). Using point - slope form \( y - y_1=m(x - x_1) \) with \( (x_1,y_1)=(-3,13) \), \( y - 13=-2.5(x + 3) \), \( y - 13=-2.5x-7.5 \), \( y=-2.5x + 5.5 \).
• x - intercept: Set \( g(x)=0 \), \( -2.5x+5.5 = 0 \), \( 2.5x = 5.5 \), \( x=\frac{5.5}{2.5}=2.2 \).
• y - intercept: Set \( x = 0 \), \( g(0)=5.5 \).
• Rate of change: The slope \( m=-2.5 \).

Step3: Fill the table for Part A

Function	x - intercept	y - intercept	rate of change
\( g(x) \)	\( 2.2 \)	\( 5.5 \)	\( - 2.5 \)

Step4: Compare for Part B

• Greater x - intercept: \( 2.2>0.75 \), so \( g(x) \).
• Greater y - intercept: \( 5.5>-3 \), so \( g(x) \).
• Greater rate of change: \( 4>-2.5 \), so \( f(x) \).

Answer:

Part A Table:

Function	x - intercept	y - intercept	rate of change
\( g(x) \)	\( 2.2 \) (or \( \frac{11}{5} \))	\( 5.5 \) (or \( \frac{11}{2} \))	\( - 2.5 \) (or \( -\frac{5}{2} \))

Part B:

• Function \( \boldsymbol{g(x)} \) has a greater x - intercept.
• Function \( \boldsymbol{g(x)} \) has a greater y - intercept.
• Function \( \boldsymbol{f(x)} \) has a greater rate of change.