Linear relations

Exercise 1

Cyclist $1$ rides from A to B with a constant speed of $20$ km/h. Cyclist $2$ rides from B to A with a constant speed of $25$ km/h. The distance between A and B is $150$ km for each cyclist. $a$ is the distance from A and $t$ is the time in hours.

a

Draw the graphs of both rides into the same $a, t$ -coordinate system.

b

For each cyclist, set up the corresponding formula for the relationship between $a$ and $t$ .

c

After how many hours do the two cyclists meet each other? Provide an explanation for your answer.

Exercise 2

For each of the following linear functions, calculate the `x` - and `y` -axis intersects. Then use the `y` -intercept and gradient to draw the corresponding graphs. Use these graphs to check the intersects you calculated earlier.

a

$h (t) = 3 t - 5$

b

$f (x) = x - 4$

c

$g (x) = -0.5 x + 4$

d

$k (x) = -2 (x + 3)$

Exercise 3

Given the function $y_{1} = -4 + 5 x$ .

a

Draw the graph of this function and show the values of the `y` -intercept and the gradient in the graph.

b

The graph of $y_{1}$ is shifted up the $y$ -axis by $10$ units. Determine the function rule of this new graph $y_{2}$ .

c

The graph of $y_{1}$ is reflected about the $y$ -axis. Determine the function rule for this new graph $y_{3}$ .

Exercise 4

You here see a coordinate system with four points $A B C D$ arranged in a square. A linear function in this coordinate system goes through point $(1, 5)$ .

a

Explain why the corresponding formula must be $y = a x + 5 - a$ .

b

Which values of $a$ produce a line that does not have any points in common with the square?

Exercise 5

Given the functions $f (x) = 4 - 0.5 x$ and $g (x) = 2 x - 1$ .

a

Calculate algebraically the intersects of both functions with the axes of the coordinate system.

b

Calculate algebraically the point of intersection of the graphs.

Exercise 6

The so-called 'Elfstedentocht' is an ice-skating tour along eleven Frisian towns. The last part, from Dokkum to Leeuwarden, is an almost straight stretch of $26$ km. A participant arrives in Dokkum after $7$ hours. He manages to keep a constant speed over this last stretch. He arrives in Leeuwarden after three quarters of an hour, and in total he has completed a tour of $200$ km.

a

What was his speed on the last stretch of the tour?

b

Assume that $t$ is the time in hours, with $t = 0$ being the start of the entire tour for this participant. The total distance covered is $a$ . Which function rule $a (t)$ describes the last part of his tour?

c

This participant started the tour at the same time as I did, but I arrived in Dokkum $2$ hours later than him. Like him, I managed to skate with a constant speed over the last stretch, but it took me an hour to do so. Which formula describes my skate from Dokkum to Leeuwarden?

Exercises