Power functions

Exercise 1

The research department of a wholesale company investigates to what extent tomato sales depend on price. Somebody claims that the following formula is true: $a = \frac{500}{p}$ . In this formula, $a$ is the amount sold per day in kg and $p$ the price per kg in euro.

a

Rewrite the formula in such a way that you can see that sales are directly proportional to a power of the price.

b

Use your graphing calculator to plot the graph of this function for prices between € 1 and € 5 per kg. If you double the price, are sales reduced by less or more than half? How can you immediately see this in the graph?

c

The company keeps a stock of $300$ kg of tomatoes. Calculate at which price this stock would be sold within a day. Write down the formula that allows you to calculate this directly.

d

How many kg of tomatoes are sold at a price of € 0.01? And at a price of € 100.00? What does this mean for the usefulness of the above formula?

Exercise 2

Given the function $f (x) = \frac{3}{\sqrt{x - 1}} + 5$ .

a

Explain how you can arrive at this function by transformation of the graph $y = x^{- \frac{1}{2}}$ .

b

What transformation do you have to apply to get the graph of $f$ ?

c

Write down the domain and range of $f$ .

d

Solve: $f (x) \leq 10$ .

Exercise 3

Look at the graphs of functions $f (x) = -5 + 2 \sqrt{(x - 3)}$ and $g (x) = \sqrt{x}$ .

a

Write $f$ and $g$ as power functions, and explain how you can get the graph of $f (x)$ through transforming the graph of $g (x)$ .

b

Write down the domain and range of both $f$ and $g$ .

c

Solve: $f (x) \geq 100$ .

Exercise 4

Given the function $f (x) = \frac{100}{{(x - 10)}^{2}} + 25$ .

a

Demonstrate that graph of this function can be derived from a power function. Write down the required transformations.

b

Which asymptotes does the graph of $f$ have?

c

Write down the domain and range of $f$ .

d

Solve: $f (x) \leq 50$ .

Exercise 5

One function that can be derived from a power function by transformation is: $h (x) = a {(x - b)}^{c} + d$ .

a

For which values of $c$ does this function have a maximum or a minimum?

b

What determines whether there is a maximum or a minimum?

c

Where in the formula can you find the position of the vertex? Give the coordinates of this vertex.

Exercises