Q. 102 - Precalculus Enhanced with Graphing Utilities

Question

Horizontal Stretches Open the horizontal stretch applet.

(a) Use your mouse to grab the slider and change the value of $a$. Note the role $a$ plays in the graph of $g\left(x\right)=f\left(ax\right)=\sqrt{ax},$ where $f\left(x\right)=\sqrt{x}$ What happens to the points on the graph of $g$when $0<a<1$?

What happens to the points on the graph when $a>1$

(b) To further understand the concept of horizontal compressions, fill in the spreadsheet to the right of the graph as follows:

(i) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{2x}$, if the $y$-coordinate is to be $1$?

(ii) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{2x}$, if the $y$-coordinate is to be $2$ ?

(iii) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{2x}$, if the $y$-coordinate is to be $3$ ?

(iv) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{\frac{1}{2}x}$, if the $y$-coordinate is to be $1$ ?

(v) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{\frac{1}{2}}x$, if the $y$-coordinate is to be $2$?

(vi) What $x$-coordinate is required on the graph of $g\left(x\right)=\sqrt{\frac{1}{2}}x$, if the $y$-coordinate is to be $3$ ?

Step-by-Step Solution

Answer

The values of $a$ decide the translation along the axes and the values are  $\frac{1}{2},2\frac{9}{2},2,8,18$ respectively

Part (a) Step 1: Given information

Given the function $f\left(x\right)=\sqrt{x}$

Part (a) Step 2: Checking the value on various values

Here,

When $0<a<1$, stretch the graph $f$ horizontally. When $a>1$, compress the graph $f$ horizontally.

The graph is 

Part (b) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{2x}$

Part (b) Step 2: Substitute g ( x ) = 1 and calculate the value

Substituting, we get

$$\begin{gathered} \sqrt{2x}=1 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

Part (b)(ii) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{2x}$

Part (b)(ii) Step 2: Substitute g ( x ) = 2 and calculate the value

Substituting, we get

$$\begin{gathered} \sqrt{2x}=2 \\ 2x=4 \\ x=2 \end{gathered}$$

Part (b)(iii) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{2x}$

Part (b)(iii) Step 2: Substitute g ( x ) = 3 and calculate the value

Substituting the value, we get

$$\begin{gathered} g\left(x\right)=\sqrt{2x} \\ 3=\sqrt{2x} \\ 2x=9 \\ x=\frac{9}{2} \\ x=4.5 \end{gathered}$$

Part (b) (iv) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{\frac{1}{2}}x$

Part (b)(iv) Step 2: Substitute g ( x ) = 1 and calculate the value

Substituting, we get

$$\begin{gathered} g\left(x\right)=\sqrt{\frac{1}{2}}x \\ \sqrt{\frac{1}{2}x}=1 \\ \frac{1}{2}x=1 \\ x=2 \end{gathered}$$

Part (b)(v) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{\frac{1}{2}}x$

Part (b)(v) Step 2: Substitute g ( x ) = 2 and calculate the value

Substituting, we get

$$\begin{gathered} g\left(x\right)=\sqrt{\frac{1}{2}}x \\ 2=\sqrt{\frac{1}{2}}x \\ \frac{1}{2}x=4 \\ x=8 \end{gathered}$$

Part (b)(vi) Step 1: Given information

Given the equation $g\left(x\right)=\sqrt{\frac{1}{2}}x$

Part (b)(vi) Step 2: Substituting g ( x ) = 3 and calculating the value

Substituting, we get

$$\begin{gathered} g\left(x\right)=\sqrt{\frac{1}{2}}x \\ \sqrt{\frac{1}{2}x}=3 \\ \frac{1}{2}x=9 \\ x=18 \end{gathered}$$