Question

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you will need grid paper.

1. each graph below is a graph of a linear relation. describe the relationship between the variables in each graph.

a) $y = 4x - 1$ b) $y = -3x + 9$
graph of $y=4x-1$ graph of $y=-3x+9$
two graphs are shown, one for $y = 4x - 1$ with a grid and plotted points, and one for $y = -3x + 9$ with a grid and plotted points

Explanation:

Part (a) - \( y = 4x - 1 \)

Step 1: Identify the slope and intercept

The equation is in slope - intercept form \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y = 4x-1 \), the slope \( m = 4 \) (which is positive) and the y - intercept \( b=- 1 \).

Step 2: Analyze the relationship

A positive slope means that as the value of \( x \) (the independent variable) increases, the value of \( y \) (the dependent variable) also increases. The rate of increase of \( y \) with respect to \( x \) is 4. That is, for every 1 - unit increase in \( x \), \( y \) increases by 4 units. Also, when \( x = 0 \), \( y=-1 \), which is the starting point (y - intercept) of the linear relationship.

Part (b) - \( y=-3x + 9 \)

Step 1: Identify the slope and intercept

The equation is in slope - intercept form \( y = mx + b \). For \( y=-3x + 9 \), the slope \( m=-3 \) (which is negative) and the y - intercept \( b = 9 \).

Step 2: Analyze the relationship

A negative slope means that as the value of \( x \) (the independent variable) increases, the value of \( y \) (the dependent variable) decreases. The rate of decrease of \( y \) with respect to \( x \) is 3. That is, for every 1 - unit increase in \( x \), \( y \) decreases by 3 units. Also, when \( x = 0 \), \( y = 9 \), which is the starting point (y - intercept) of the linear relationship.

Answer:

s:
a) In the linear relationship \( y = 4x-1 \), \( y \) is a dependent variable and \( x \) is an independent variable. As \( x \) increases, \( y \) increases at a rate of 4 units per 1 - unit increase in \( x \), with a y - intercept of - 1.
b) In the linear relationship \( y=-3x + 9 \), \( y \) is a dependent variable and \( x \) is an independent variable. As \( x \) increases, \( y \) decreases at a rate of 3 units per 1 - unit increase in \( x \), with a y - intercept of 9.