Question

Swimming Laps Miyuki practices swimming laps with her team. The function$y=f\left(t\right)$ graphed below gives her distance (in meters) from the starting edge of the pool tseconds after she starts her lap.

(a) Describe in words Miyuki’s swim practice. What is her average speed for the first 30s?

(b) Graph the function$y=1.2f\left(t\right)$. How is the graph of the new function related to the graph of the original function?

(c) What is Miyuki’s new average speed for the first 30s?

Short answer

(a) From the graph, notice that the distance cover by the Miyuki is same in every 30s of interval is 50 m.

Miyuki average speed for the first 30 seconds is$\frac{5}{3}\text{m/sec.}$

(b) The graph of function $y=1.2f\left(t\right)$is shown below:

Here, the new graph is vertical stretching with the graph of original graph.

(c) Miyuki’s new average speed for the first 30 sec is$2\text{m/}sec.$

Step-by-step solution

Part a. Step 1. Describe in words Miyuki’s swim practice.

From the graph notice the following points:

• Miyuki’s starts swimming at time $t=0$, and travels 50 m distance in$t=30$seconds.
• Again he came back to his starting point after 30 seconds, that is at time $t=60s$.
• Next starts swimming at time $t=60$, and travels 50 m distance in 60 seconds that is at $t=120$seconds.

Part a. Step 2. Find the average speed.

Take any points that lie on the given graph, say $\left(0,0\right)\text{and}\left(30,50\right)$.

To find the average speed, use the slope formula between two points $\left(x_1,y_1\right)=\left(0,0\right),\left(x_2,y_2\right)=\left(30,50\right)$.

So,

$\begin{array}{c}\text{averagespeed}=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{50-0}{30-0}\\ =\frac{5}{3}\end{array}$

So, Miyuki average speed for the first 30 seconds is$\frac{5}{3}\text{m/sec.}$

Part b. Step 1. Transformation property.

To graph a function $y=cf\left(t\right)$, stretch the graph of$y=f\left(t\right)$ vertically by a factor of c if$c>1.$

To graph a function $y=cf\left(t\right)$, shrink the graph of$y=f\left(t\right)$ vertically by a factor of c if$0<c<1.$

Part b. Step 2. Graph the given function.

Given the function is $y=1.2f\left(t\right)$.

Comparing$y=1.2f\left(t\right)$ with$y=cf\left(t\right)$ one can get $c=1.2$.

Since $c=1.2>1$, so, vertically stretches the graph of$y=f\left(t\right)$ by a factor of 1.2.

Then the new graph is vertical stretching with the graph of original graph.

Part c. Step 1. Formula to find the slope.

The slope between two points$\left(x_1,y_1\right),\left(x_2,y_2\right)$is defined as follows:

width="116">slope=y2−y1x2−x1.

Part c. Step 2. Find the average speed.

Take any points that lie on the given graph – 1, say $\left(0,0\right)\text{and}\left(30,60\right)$.

To find the average speed, use the slope formula between two points $\left(x_1,y_1\right)=\left(0,0\right),\left(x_2,y_2\right)=\left(30,60\right)$.

$\begin{array}{c}\text{averagespeed}=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{60-0}{30-0}\\ =2\end{array}$

So, Miyuki new average speed for the first 30 seconds is$2\text{m/sec.}$