Exercise 4 Page 441

a Let's review the effects of the parameters

a

and

h

on a function of the form

y = a (x - h)^{2} .

A quadratic function of the form $y = a (x - h)^{2}$ is a transformation of the quadratic parent function $f (x) = x^{2} .$

If $a < 0$ the graph is reflected in the $x - axis .$
If $0 < ∣ a ∣ < 1$ the graph is vertically shrunk.
If $1 < ∣ a ∣$ the graph is verticalyl stretched.
If $h > 0$ the graphs is horizontally translated $h$ units to the right.
If $h < 0$ the graphs is horizontally translated $∣ h ∣$ units to the left.

We can now identify the value of the parameters by comparing our given function

y = (x - 3)^{2}

with the general form

y = a (x - h)^{2} .

y = a (x - h)^{2} y = (1) (x - 3)^{2}

For this case

a = 1

and

h = 3 .

Therefore, the graph will be the same as the parent function

f (x) = x^{2},

but translated

3

units to the right.

We can use a graphing calculator to verify our prediction.

As we can see in the graph above, the function $y = (x - 3)^{2}$ is the same as that of $f (x) = x^{2}$ but shifted $3$ units to the right, as predicted.

b We will proceed just as we did in Part A. First let's identify the parameters

a

and

h

for the given function

y = (x + 3)^{2}

by comparing it to the general form

y = a (x - h)^{2} .

y = a (x - h)^{2} y = (1) (x - (- 3))^{2}

Hence, for this case

a = 1

and

h = - 3 .

Therefore, the graph will be just as that of the parent function

f (x) = x^{2}

but translated

3

units to the left.

We can use a graphing calculator to verify our prediction.

As we can see in the graph above, the function $y = (x + 3)^{2}$ is the same as that of $f (x) = x^{2}$ but shifted $3$ units to the left, as predicted.

c We will proceed just as we did in previous in the parts. First let's identify the parameters

a

and

h

for the given function

y = - (x - 3)^{2}

by comparing it to the general form

y = a (x - h)^{2} .

y = a (x - h)^{2} y = (- 1) (x - 3)^{2}

Then, for this case

a = - 1

and

h = 3 .

Therefore, the graph will be similar to that of the parent function

f (x) = x^{2}

but translated

3

units to the right, and reflected in the

x - axis .

We can use a graphing calculator to verify our prediction.

As we can see in the graph above, the function $y = - (x - 3)^{2}$ is the same as that of $f (x) = x^{2},$ but shifted $3$ units to the right and reflected in the $x - axis .$ This confirms our prediction.

Subchapter links

Communicate Your Answer p.441

Monitoring Progress p.442-445

Exercises p.446-448