Power functions

Exercise 1

The graph of the function $f (x) = 2 {(x + 8)}^{2} - 8$ can be derived by transforming the graph of $y = x^{2}$ .

a

Which transformations do you have to apply?

b

Swap the order of the last two transformations around and then plot the graph of the function $g$ that has now been created. Why is the order of transformations so important?

Exercise 2

Look at the graph of $f (x) = -2 {(x + 4)}^{2} + 5$

a

Determine the maximum, or minimum, of $f$ and the value of $x$ at this vertex.

b

Solve $-2 {(x + 4)}^{2} + 5 = -5$ .

c

Solve: $f (x) = 5$

d

Solve: $f (x) = 10$

e

Solve: $f (x) > -3$

f

Solve: $f (x) < 0$

g

Solve: $f (x) < 20$

Exercise 3

Given the function $f (x) = -3 {(x + 2)}^{2} + 10$ .

a

Over which interval is this function decreasing?

b

Is the function therefore increasing over the remainder of the domain?

c

Determine the zeros (x-intersects) of $f$ .

Exercise 4

Look at the graph of the function $f (x) = 2 {(x - 1)}^{2} - 3$ .

a

Solve algebraically: $2 {(x - 1)}^{2} - 3 = 0$

b

Draw the graph of $f$ by plotting every transformation starting at $y = x^{2}$ .

c

Compare your answers in a and b. Find every transformation in your solution to a.

Exercise 5

Algebraically solve the following inequalities:

a

$5 {(x - 1)}^{2} - 9 > 4$

b

$5 - x^{2} > -21$

c

$3 {(x - 1)}^{2} < 40$

d

$-4 {(x + 80)}^{2} - 40 < -100$

Exercise 6

A basketball player scores a three-pointer without touching the backboard (so he throws the ball straight through the hoop). The trajectory of the ball is (roughly) parabolic (see figure). The highest point of the curve is shown. The player releases the ball at $2,5$ m above the ground.

a

Write down a formula for the function $h (x)$ that describes the trajectory of the ball.

b

The hoop of the basket is $3,05$ m above the ground. How far away was the player from the (middle of the) hoop?

Exercise 7

Given the function $f (x) = - \frac{1}{2} {(x - 3)}^{2} + c$ . In this formula, $c$ is an unknown constant.

a

What is the value of the extremum of this function $f$ ?

b

For which values of $c$ does this function have $f$ two x-intersects? Explain your answer.

c

For what values of $c$ does the function $f$ not have any intersects with the line $y = 4$ ?

d

For what value of $c$ does the vertex of the function $f$ lie on the line $y = 4 x - 5$ ?

Exercises