Absolute Value, Piecewise & Step Functions — Drills
Absolute Value, Piecewise & Step Functions has 8 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Absolute value — |x| — distance from zero, always positive or zero.
- Isolate — Get the variable alone on one side.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Parabola — The U-shaped graph of a quadratic function.
Absvalue
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Evaluate |1|.
Worked answer
1
|1| = 1
Problem 2
Evaluate |11 - 4|.
Worked answer
7
11 - 4 = 7
|7| = 7
Solving
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve |2x - 8| = 20.
Worked answer
14 and -6
2x - 8 = 20 gives x = 14
2x - 8 = -20 gives x = -6
Problem 2
Solve |x - 5| = 8.
Worked answer
13 and -3
x - 5 = 8 or = -8
x = 13 or x = -3
Vertex
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the vertex of y = |x + 6| + 3.
Worked answer
(-6, 3)
x + 6 = 0 gives x = -6
Then y = 3
Vertex (-6, 3), opening upward.
Problem 2
Find the vertex of y = -3|x + 1| + 0.
Worked answer
(-1, 0)
x + 1 = 0 gives x = -1
Then y = 0
Vertex (-1, 0), opening downward.
Piecewise
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For f(x) = x + 1 if x < -2, and f(x) = 3x if x ≥ -2, find f(7).
Worked answer
21
7 ≥ -2, so use the second rule.
f(7) = 3(7) = 21
Problem 2
For f(x) = x - 5 if x < 0, and f(x) = 2x if x ≥ 0, find f(4).
Worked answer
8
4 ≥ 0, so use the second rule.
f(4) = 2(4) = 8
Isolate
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 5|x - 2| - 6 = 54.
Worked answer
14 and -10
5|x - 2| = 60
|x - 2| = 12
x = 14 or x = -10
Problem 2
Solve 2|x + 1| + 7 = 27.
Worked answer
9 and -11
2|x + 1| = 20
|x + 1| = 10
x = 9 or x = -11
No solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve |x - 6| = -6. If there is no solution, answer "none".
Worked answer
none
Absolute value measures distance, so it is never negative.
So there is no solution.
Problem 2
Solve |x + 8| = -5. If there is no solution, answer "none".
Worked answer
none
Absolute value measures distance, so it is never negative.
So there is no solution.
From a graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
The graph of y = -|x| + 5 has a vertex at (0, 5) and
- opens upward
- opens downward
- is a parabola
- is a straight line
Worked answer
opens downward
-|x| turns the V upside down.
So (0, 5) is a maximum.
Problem 2
The graph of y = -|x| + 7 has a vertex at (0, 7) and
- opens downward
- is a parabola
- opens upward
- is a straight line
Worked answer
opens downward
-|x| turns the V upside down.
So (0, 7) is a maximum.
Step
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A garage charges $9 for the first hour and $5 for each hour after. What do 4 hours cost?
Worked answer
24
9 + 5(3)
= 9 + 15
= $24
Problem 2
A garage charges $3 for the first hour and $4 for each hour after. What do 8 hours cost?
Worked answer
31
3 + 4(7)
= 3 + 28
= $31