Functions and graphs

Exercise 1

Given the function $f (x) = 8 - 4 x + x^{3}$ .

a

Calculate $f (3)$ .

b

Determine the $x$ -values for which $f (x) = 8$ .

c

What window settings do you need to use in order to see all `x` -intersects and all vertices of the graph of $f$ ?

d

Is there exactly one value of $y$ for every value of $x$ ? Or are there examples where this is not the case?

e

Is there exactly one value of $x$ for every value of $y$ ? Or are there examples where this is not the case?

Exercise 2

For the water you use in your house there is an annual fixed charge of € 42.00 and an additional € 0.25 for every m³ used. Your annual costs $K$ thus depend on the number of m³ ( $a$ ) that you use.

a

Why is $K$ a function of $a$ ?

b

Calculate $K (100)$ .

c

Write down the function rule for $K (a)$ .

d

Most families pay less than $500$ euro annualy for their water usage. How many cubic metres of water can these families be using at most?

Exercise 3

Given the functions $f (x) = 100 - x^{2}$ and $g (x) = x^{2}$ .

a

Determine the `x` -intersects and the vertex of the graph of $f$ .

b

Use your calculator to graph $f$ and $g$ . Write down what window settings are best used if you want to see the `x` -intersects and vertices of both functions.

c

Determine all points where the two graphs intersect, rounded to two decimals.

Exercise 4

Given the functions $y_{1} = x^{4} - 2 x^{2}$ and $y_{2} = - x^{2} + 4 x$ .

a

Calculate the `x` -intersects of both functions.

b

Now use your calculator to graph both functions. Write down what window settings are best used if you want to see the `x` -intersects and vertices of both functions.

c

Determine all points where the two graphs intersect, rounded to one decimal.

Exercise 5

In the following you are given four function rules. For each one, determine the `x` -intersects. Then write down the window settings that give the best view of the graph.

a

$f (x) = 100 x - x^{2}$

b

$g (x) = 10 x (x - 50)$

c

$h (x) = {(x - 10)}^{2} - 1600$

d

$k (x) = 200 + 1.6 x$

Exercises