Chapter 3.5, Problem 54HP

To determine

To find: Slope of function $f\left(x\right)$

Answer to Problem 54HP

The slope of function $f\left(x\right)$ must be $1$ .

Explanation of Solution

Given information:

  $f\left(x\right)$

  $g\left(x\right)$ is obtained by vertical translation of $f\left(x\right)$ by $5$ units downwards along Y-axis.

  $g\left(x\right)$ is obtained by horizontal translation of $f\left(x\right)$ by $5$ units towards right along X-axis.

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:

We know that,

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.

Since the function $f\left(x\right)$ is moved $5$ units downwards to obtain $g\left(x\right)$ ,

  $\Rightarrow g\left(x\right)=f\left(x\right)-5$

Similarly, we know that

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.

Since the function $f\left(x\right)$ is moved $5$ units towards right to obtain $g\left(x\right)$ ,

  $\Rightarrow g\left(x\right)=f\left(x-5\right)$

Considering $f\left(x\right)=mx+b$ ,

  $g\left(x\right)=f\left(x\right)-5$

  $\Rightarrow g\left(x\right)=mx+b-5$

And $g\left(x\right)=f\left(x-5\right)$

  $\Rightarrow g\left(x\right)=m\left(x-5\right)+b$

  $\begin{array}{l}\Rightarrow mx+b-5=m\left(x-5\right)+b\\ \Rightarrow mx+b-5=mx-5m+b\\ \Rightarrow \left(mx+b\right)-5=\left(mx+b\right)-5m\\ \Rightarrow -5=-5m\\ \Rightarrow m=1\end{array}$

Therefore, the slope of function $f\left(x\right)=mx+b$ is $1$ .

We conclude that, if the function $f\left(x\right)$ is shifted by $5$ units downwards along Y-axis and also if the function $f\left(x\right)$ is shifted by $5$ units towards right along X-axis then the slope of the function $f\left(x\right)$ must be $1$ .