Chapter 12.5, Problem 26E

a.

To determine

To determine the equation of the line.

Answer to Problem 26E

The equation of the line is $\text{y=2x}-\text{1}$ .

Explanation of Solution

Given information:

Consider the line given below:

Formula used:

The slope is calculated using the formula : $\text{m=}\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}$

The equation of the line is calculated using the formula : $\text{y}-{\text{y}}_1=\text{m}\left(\text{x}-{\text{x}}_1\right)$

Calculation :

The slope is calculated by :

$\begin{array}{l}\text{m=}\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}\\ \text{m=}\frac{-1-1}{-1-0}\left[\text{Substituting the values of any two coordinates}\right]\\ \text{m=}\frac{-2}{-1}\\ \text{m=}2\left[\text{Simplify}\right]\end{array}$

The slope of the line is $\text{m=}2$

The equation of the line can be calculated by :

$\begin{array}{l}\text{y}-{\text{y}}_1=\text{m}\left(\text{x}-{\text{x}}_1\right)\\ \text{y}-\left(-1\right)=2\left(\text{x}-0\right)\left[{\text{Substituting the value of m, x}}_1{\text{ and y}}_1\right]\\ \text{y+1=2x}\\ \text{y=2x}-\text{1 }\left[\text{Simplify}\right]\end{array}$

Hence, the equation of the line is $\text{y=2x}-\text{1}$ .

b.

To determine

To draw the image of the line after reflection in the y-axis and determine the equation of the image.

Answer to Problem 26E

The equation of the line after reflection in y-axis is $\text{y=}-\text{2x}-\text{1}$ .

Explanation of Solution

Given information:

Consider the line below:

Graph:

The image of the above line after reflection in the y-axis is :

Formula used:

The slope is calculated using the formula : $\text{m=}\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}$

The equation of the line is calculated using the formula : $\text{y}-{\text{y}}_1=\text{m}\left(\text{x}-{\text{x}}_1\right)$

Calculation :

The slope is calculated by :

$\begin{array}{l}\text{m=}\frac{{\text{y}}_2-{\text{y}}_1}{{\text{x}}_2-{\text{x}}_1}\\ \text{m=}\frac{-1-1}{1-0}\left[\text{Substituting the values of any two coordinates}\right]\\ \text{m=}\frac{-2}{1}\\ \text{m=}-2\left[\text{Simplify}\right]\end{array}$

The slope of the line is $\text{m=}-2$ .

The equation of the line can be calculated by :

$\begin{array}{l}\text{y}-{\text{y}}_1=\text{m}\left(\text{x}-{\text{x}}_1\right)\\ \text{y}-\left(-1\right)=-2\left(\text{x}-0\right)\left[{\text{Substituting the value of m, x}}_1{\text{ and y}}_1\right]\\ \text{y+1=}-\text{2x}\\ \text{y=}-\text{2x}-\text{1 }\left[\text{Simplify}\right]\end{array}$

Hence, the equation of the line after reflection in y-axisis $\text{y=}-\text{2x}-\text{1}$ .

c.

To determine

To compare the slope and y-intercept of the original line related to the slope and y-intercept of the image.

Answer to Problem 26E

The slope changes it’s sign from positive to negative after reflection in the y-axis.

The y-intercept does not change after reflection in y-axis.

Explanation of Solution

Given information:

The coordinates of the points on the original line are $\left(1,3\right)$ , $\left(0,1\right)$ , $\left(-1,-1\right)$ and $\left(-2,-3\right)$ .

The slope of the original line is $\text{m=}2$ .

The equation of the line is $\text{y=2x}-\text{1}$ .

The coordinates of the line after reflection in the y-axis are $\left(-1,3\right)$ , $\left(0,1\right)$ , $\left(1,-1\right)$ and $\left(2,-3\right)$ .

The slope of the line after reflection in the y-axis is $\text{m=}-2$ .

The equation of the line after reflection in y-axis is $\text{y=}-\text{2x}-\text{1}$ .

The slope changes it’s sign from positive to negative after reflection in the y-axis.

From the equations of the line before and after reflection in the y-axis, the y-intercept, $\text{b=}-1$ .

The y-intercept does not change after reflection in y-axis.