Chapter 1.3B, Problem 3GCE

To determine

To explain: the effect of different values of m and b on graph with the help of sketch illustration.

Answer to Problem 3GCE

The table representing the effect of the values of m and b on the equation $y=mx+b$ is shown below.

$\begin{array}{c}\text{Case 1: }m<0\text{ and }b<0\\ \text{Example: }y=-x-2\end{array}$	$\begin{array}{c}\text{Case 2: }m<0\text{ and }b>0\\ \text{Example: }y=-x+2\end{array}$
$\begin{array}{c}\text{Case 3: }m=0\text{ and }b>0\\ \text{Example: }y=2\end{array}$	$\begin{array}{c}\text{Case 4: }m=0\text{ and }b<0\\ \text{Example: }y=-2\end{array}$
$\begin{array}{c}\text{Case 5: }m>0\text{ and }b<0\\ \text{Example: }y=x-2\end{array}$	$\begin{array}{c}\text{Case 6: }m>0\text{ and }b>0\\ \text{Example: }y=x+2\end{array}$
$\begin{array}{c}\text{Case 7: }m=\text{undefined and }b>0\\ \text{Example: }x=2\end{array}$	$\begin{array}{c}\text{Case 8: }m=\text{undefined and }b<0\\ \text{Example: }x=-2\end{array}$

Explanation of Solution

Given information:

The given graph equation is as follows:

  $y=mx+b$

Calculation:

The effect of different values of m and b on the given equation will be as follows:

CASE 1: When $m<0\text{ and }b<0$ , the slope would be negative and the line will intersect the negative value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=-x-2$ is $-1$ and $-2$ respectively. The graph for $m<0\text{ and }b<0$ is show below.

  

CASE 2: When $m<0\text{ and }b>0$ , the slope would be negative and the line will intersect the positive value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=-x+2$ is $-1$ and $2$ respectively. The graph for $m<0\text{ and }b>0$ is show below.

  

CASE 3: When $m=0\text{ and }b>0$ , the slope would be zero and the line will intersect the positive value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=2$ is $0$ and $2$ respectively. The graph for $m=0\text{ and }b>0$ is show below.

  

The line will be horizontal i.e. parallel to x- axis. Since the y- intercept is positive so, the line will be on the upper side of the x -axis.

CASE 4: When $m=0\text{ and }b<0$ , the slope would be zero and the line will intersect the positive value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=-2$ is $0$ and $-2$ respectively. The graph for $m=0\text{ and }b<0$ is show below.

  

The line will be horizontal i.e. parallel to x- axis. Since the y- intercept is negative so, the line will be on the lower side of the x -axis.

CASE 5: When $m>0\text{ and }b<0$ , the slope would be negative and the line will intersect the negative value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=x-2$ is $1$ and $-2$ respectively. The graph for $m>0\text{ and }b<0$ is show below.

  

CASE 6: When $m>0\text{ and }b>0$ , the slope would be negative and the line will intersect the positive value of y- axis at $x=0$ . Example: The slope value and y- intercept for equation $y=x+2$ is $1$ and $2$ respectively. The graph for $m>0\text{ and }b>0$ is show below.

  

CASE 7: When $m=\text{undefined and }b>0$ , the line will intersect the positive value of x- axis at $y=0$ . Example: The slope value and y- intercept for equation $x=2$ is undefined and $0$ respectively. The graph for $m=\text{undefined and }b>0$ is show below.

  

The line will be vertical i.e. parallel to y- axis. Since the b is positive, so the line will be on the right side of the y -axis.

CASE 8: When $m=\text{undefined and }b<0$ , the line will intersect the negative value of x- axis at $y=0$ . Example: The value of m and b for equation $x=-2$ is undefined and $0$ respectively. The line will be vertical i.e. parallel to y- axis. Since the b is negative, so the line will be on the left side of the y -axis. The graph for $m=\text{undefined and }b<0$ is show below.