Exercise 5 Page 11

a Let's examine the given function.

y = x - 4

Note that it is a square root function. With this information, we can identify its parent function as follows.

\underline{Parent Function} y = x \underline{Transformation} y = x - 4

As we can see,

4

has been subtracted from the output of the parent function. Therefore, the graph of

y = x - 4

is a vertical translation of the graph of the parent square root function

4

units down. Now we will verify our answer by using a graphing calculator in a square viewing window.

Notice that a square viewing window has a height-to-width ratio of $2$ to $3 .$ To draw the graphs on a calculator, we first press the $Y =$ button and type the functions in any of the rows. Having written the functions, we can push $GRAPH$ to draw it.

Looking at the graphs, we can see that the graph of $y = x - 4$ is the graph of $y = x$ translated $4$ units down. Therefore, our answer is correct.

b The second function is also a square root function, so we can immediately identify its parent function.

\underline{Parent Function} y = x \underline{Transformation} y = x + 4

4

is added to the input, so the graph of

y = x + 4

is a horizontal translation of the graph of its parent function

4

units to the left. Let's verify our answer by using a graphing calculator.

Thus, our answer is correct. The graph of the parent function has been translated $4$ units to the left to obtain the graph of $y = x + 4 .$

c Next, we will compare

y = - x

to its parent function.

\underline{Parent Function} y = x \underline{Transformation} y = - x

The output has been multiplied by

- 1,

so its a reflection in the

x -

axis. By drawing these two functions on a graphing calculator, we will verify our answer.

This graph verifies that the graph of $y = - x$ is the graph of the parent function flipped over the $x -$ axis.

d Now, we will continue with a different function family.

y = x^{2} + 1

The given function is a quadratic function. Therefore, its parent function is

y = x^{2} .

\underline{Parent Function} y = x^{2} \underline{Transformation} y = x^{2} + 1

Because

1

is added to the output, the graph of

y = x^{2} + 1

is a vertical translation of the graph of the parent function

1

unit up. Now, we will check whether our answer is correct.

Looking at the vertex of the parabola, we can see that the graph of the parent function has been translated $1$ unit up. Therefore, our answer is correct.

e The next function is also a quadratic function.