Question

8th math on-level
mr. chavarria
chapter 4 exam

1. write definitions for the following vocabulary words:

a. linear function:
b. evaluate:

1. create a linear function using the given function evaluations

$f(1) = 6$ and $f(2) = 2$

1. create a linear function $g(x)$:

that is parallel to the given graph and passes through the value $g(0) = 8$

Explanation:

Problem 2: Create a linear function using \( f(1) = 6 \) and \( f(2) = 2 \)

Step 1: Recall the linear function form

A linear function is in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept.

Step 2: Calculate the slope (\( m \))

The slope formula between two points \( (x_1,y_1) \) and \( (x_2,y_2) \) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). We have two points from the function evaluations: when \( x_1 = 1,y_1 = 6 \) and \( x_2=2,y_2 = 2 \). So \( m=\frac{2 - 6}{2 - 1}=\frac{- 4}{1}=-4 \).

Step 3: Find the y - intercept (\( b \))

Substitute \( x = 1 \), \( y = 6 \) and \( m=-4 \) into \( f(x)=mx + b \). We get \( 6=-4\times1 + b \). Solving for \( b \): \( 6=-4 + b \), then \( b=6 + 4=10 \).

Step 4: Write the linear function

Using \( m=-4 \) and \( b = 10 \) in the form \( f(x)=mx + b \), we have \( f(x)=-4x + 10 \).

Problem 3: Create a linear function \( g(x) \) parallel to the given graph and passes through \( g(0)=8 \)

First, we need to find the slope of the given graph. From the graph, we can see two points (for example, let's assume we can identify two points on the given line). Let's say the given line passes through \( (0,-2) \) and \( (2,1) \) (from the hand - written notes). The slope \( m \) of the given line is \( m=\frac{1-(-2)}{2 - 0}=\frac{3}{2} \)? Wait, no, looking at the graph, the line passes through \( (0,-2) \) and \( (1,0) \)? Wait, the line in the graph seems to pass through \( (0, - 2) \) and \( (2,1) \)? Wait, actually, from the hand - written part, there is a calculation for the slope. Wait, the line in the graph: let's re - examine. The line passes through \( (0,-2) \) and \( (2,1) \)? No, the line in the graph has a slope. Wait, the key point is that parallel lines have the same slope. From the hand - written work, the original line (the one in the graph) has a slope. Let's assume that the slope of the given graph is \( m=\frac{1-(-2)}{2 - 0}=\frac{3}{2} \)? No, wait, looking at the graph, when \( x = 0 \), \( y=-2 \) and when \( x = 2 \), \( y = 1 \). But the correct way: if we take two points on the given line, say \( (0,-2) \) and \( (1,0) \), then the slope \( m=\frac{0-(-2)}{1 - 0}=2 \). Since the line \( g(x) \) is parallel to the given line, the slope of \( g(x) \) is also \( m = 2 \).

Step 1: Recall the property of parallel lines

Parallel lines have the same slope.

Step 2: Determine the slope of \( g(x) \)

From the given graph, we find the slope of the original line. Let's take two points on the original line, for example, \( (0,-2) \) and \( (1,0) \). The slope \( m=\frac{0-(-2)}{1 - 0}=2 \). So the slope of \( g(x) \) is also \( m = 2 \) (because they are parallel).

Step 3: Find the y - intercept of \( g(x) \)

We know that \( g(0)=8 \). For a linear function \( g(x)=mx + b \), when \( x = 0 \), \( g(0)=b \). So \( b = 8 \).

Step 4: Write the linear function \( g(x) \)

Using \( m = 2 \) and \( b = 8 \) in the form \( g(x)=mx + b \), we get \( g(x)=2x+8 \).

Answer:

(for problem 2): \( f(x)=-4x + 10 \)