Writing & Transforming Linear Functions — Practice
Once a linear model is written, every question about it — rate, starting value, prediction, shift — is a short calculation.
What these questions cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Arithmetic sequence — A list that grows by adding the same amount each time.
- Function — A rule pairing each input with exactly one output.
- Slope — How much y changes for each 1 that x increases.
- Linear function — One whose graph is a straight line: y = mx + b.
- Rate of change — The slope, in the units of the situation.
- Average rate of change — f(b) - f(a)b - a over an interval.
- Transformation — A change to the function that moves or stretches its graph.
- Sequence — An ordered list of numbers. The nth one is an.
- Explicit formula — A rule giving an directly from n.
- Common difference — The constant added each time, in an arithmetic sequence.
Problem 1
A plumber charges $60 to visit plus $45 an hour. Write C(h).
Worked answer
C(h) = 45h + 60
b = 60, m = 45
C(h) = 45h + 60
Problem 2
A pool has 900 litres and loses 25 litres an hour. Write V(t).
Worked answer
V(t) = -25t + 900
b = 900, m = -25
V(t) = -25t + 900
Problem 3
For f(x) = 7x - 3, what is the rate of change?
Worked answer
7
m = 7
Problem 4
Find the 10th term of 5, 9, 13, 17, ….
Worked answer
41
5 + 9(4)
= 5 + 36 = 41
Problem 5
Find the common difference of 20, 17, 14, 11, ….
Worked answer
-3
17 - 20 = -3
The sequence is decreasing
Problem 6
For f(x) = x2, find the average rate of change from x = 2 to x = 5.
Worked answer
7
f(5) = 25, f(2) = 4
213 = 7
Problem 7
For f(x) = 2x2 - 1, find the average rate of change from x = 1 to x = 4.
Worked answer
10
f(1) = 1, f(4) = 31
303 = 10
Problem 8
The graph of f(x) is shifted 4 units up. Write the new function.
Worked answer
f(x) + 4
f(x) + 4
Problem 9
The graph of f(x) is shifted 3 units right. Write the new function.
Worked answer
f(x - 3)
f(x - 3)
This is the one that feels backwards
Problem 10
The graph of g(x) = f(x + 5) is the graph of f moved
- 5 units left
- 5 units right
- 5 units up
- 5 units down
Worked answer
5 units left
f(x + 5) reaches its old output 5 units earlier
So the graph shifts left
Problem 11
Write an explicit formula for 8, 14, 20, 26, ….
Worked answer
a_n = 8 + 6(n - 1)
an = 8 + (n-1)6
Or equivalently 6n + 2
Problem 12
A taxi charges $3.50 plus $2.25 a mile. Find the cost of a 12-mile trip.
Worked answer
30.5
2.25(12) = 27
27 + 3.5 = $30.50
Problem 13
Two gyms: A charges $20 plus $35 a month; B charges $80 plus $25 a month. After how many months do they cost the same?
Worked answer
6
20 + 35m = 80 + 25m
10m = 60
m = 6 months
Problem 14
A function has f(3) = 12 and f(11) = -4. Find the average rate of change.
Worked answer
-2
-4 - 1211 - 3
= -168 = -2
Problem 15
An arithmetic sequence has a4 = 19 and a9 = 44. Find a1.
Worked answer
4
44 - 19 = 25 over 5 steps, so d = 5
a1 = 19 - 3(5)
= 4
Problem 16
Describe the transformation from f(x) = x2 to g(x) = (x - 2)2 + 5.
Worked answer
right 2 and up 5
(x - 2) shifts right 2
+5 shifts up 5