Chapter 3.5, Problem 55HP

To determine

To find: Reflection of given line about Y-axis and equation of the reflection.

Answer to Problem 55HP

The equation of the reflection of line $y=2x$ is $y=-2x$

Explanation of Solution

Given information:

A line with slope of $2$ .

Concept:

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

If $g\left(x\right)=f\left(x+l\right)$ it is known as horizontal shift.

If $l<0$ moves it right.

If $l>0$ moves it left.

If $g\left(x\right)=-f\left(x\right)$ it is known as Vertical reflection it means reflects about X-axis

If $g\left(x\right)=f\left(-x\right)$ it is known as Horizontal reflection it means reflects about Y-axis

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

If $g\left(x\right)=f\left(bx\right)$

If $b>1$ it is known as Horizontal compression.

If $0<b<1$ it is known as Horizontal Stretch.

Calculation:

Consider a line $y=2x$ with slope of $2$ .

We know that,

If $g\left(x\right)=f\left(-x\right)$ it is known as Horizontal reflection it means reflects about Y-axis

Let the line $y=2x$ be the function $f\left(x\right)=2x$

The Horizontal reflection of function $f\left(x\right)$ is function $g\left(x\right)$ .

  $\begin{array}{l}\Rightarrow g\left(x\right)=f\left(-x\right)\\ \Rightarrow g\left(x\right)=2\left(-x\right)\\ \Rightarrow g\left(x\right)=-2x\end{array}$

Therefore, the equation of the reflection is $y=-2x$ .

The graph of lines $y=2x$ and its reflection about Y-axis $y=-2x$ is

Where, the Red line represents the line $y=2x$

And the Blue line represents the reflection of line $y=2x$ about Y-axis which is $y=-2x$ .