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Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.F.A.1, 8.F.A.2, 8.F.A.3 · Level: Grade 8
Problem 1 Ordered pairs
(a) The inputs 1, 2, 3, 4 are all different, so each has one output. (b) The input 2 is paired with both 5 and 8.
(a) function (b) not a function
Problem 2 Mapping diagram
Every input has exactly one arrow. The outputs go up by 3 as the inputs go up by 1, and input 1 gives 5, so output = 3 · input + 2. Check: 3(4) + 2 = 14.
Yes, a function; output = 3 · input + 2
Problem 3 Table rule
Each step of 1 in x adds 3 to y, and y = 6 when x = 0.
y = 3x + 6
Problem 4 Function notation
f(5) = 4(5) − 7 = 13. f(−2) = 4(−2) − 7 = −15. f(0) = −7.
13, −15 and −7
Problem 5 Vertical line test
The dashed vertical line crosses the curve at two points, which means one input has two outputs.
Not a function
Problem 6 Input and output
2(3) + 9 = 15, 2(6) + 9 = 21 and 2(10) + 9 = 29.
15, 21 and 29
Problem 7 Quadratic table
Compute: 4 − 3 = 1, 1 − 3 = −2, 0 − 3 = −3, 4 − 3 = 1, 16 − 3 = 13. The output 1 appears for both −2 and 2, which is allowed.
1, −2, −3, 1, 13; yes, 1 repeats
Problem 8 Linear or nonlinear tables
Table 1: the outputs rise 4, 4, 4, so the rate is constant. Table 2: the outputs rise 3, 5, 7, so the rate changes.
Table 1 is linear; Table 2 is nonlinear.
Problem 9 Side and area
From (1, 1) to (3, 9): 8 ÷ 2 = 4. From (2, 4) to (3, 9): 5 ÷ 1 = 5. Different pairs of points give different rates, so the graph is not a line.
Rates 4 and 5; nonlinear
Problem 10 Compare two functions
B: the rate is (9 − 3) ÷ 2 = 3 and the initial value is 3. A: rate 2.5 and initial value 4. Since 3 > 2.5 and 4 > 3:
B has the greater rate; A has the greater initial value.
Problem 11 Graph versus equation
C: rate (10 − 2) ÷ 4 = 2 and y-intercept 2. D: rate 3 and y-intercept −1.
D is steeper (3 > 2); C starts higher (2 > −1).
Problem 12 Spot the linear equations
(a) The form mx + b with m = 7: linear. (b) x is squared, so the graph is a curve: nonlinear. (c) This equals −0.5x + 4: linear.
(a) and (c) are linear; (b) is nonlinear.
Problem 13 Find the rule
The outputs rise 6 when x rises 2, so the rate is 3. Then 7 = 3(2) + b gives b = 1. Check: 3(8) + 1 = 25.
y = 3x + 1
Problem 14 Repeated outputs
Each name has one month, so the first is a function even though June repeats. In the reverse, the input June is paired with both Ava and Ben.
First: function. Reverse: not a function.
Problem 15 Pool water level
d(t) = 4t + 12. d(6.5) = 26 + 12 = 38. For 60: 4t + 12 = 60, so 4t = 48 and t = 12.
d(t) = 4t + 12; 38 inches; after 12 hours
Problem 16 Two phone plans
B rises $10 per 100 minutes, which is $0.10 per minute, so the rate matches A. B: 28 = 0.10(100) + b, so b = 18. A has a $15 fee and B has an $18 fee, so B costs $3 more at every number of minutes.
Same rate; Plan A costs $3 less.
Problem 17 Perimeter and area
P: 4, 8, 12, 16, 20 (steps of 4: linear). A: 1, 4, 9, 16, 25 (steps of 3, 5, 7, 9: nonlinear). At s = 4 both are 16.
Perimeter is linear; they are equal when s = 4.
Problem 18 Constant versus vertical
The graph of y = 5 is a horizontal line: every input has the single output 5. The graph of x = 5 is a vertical line: the input 5 is paired with every y, so it fails the vertical line test.
y = 5 is a function; x = 5 is not.
Problem 19 Points on a line
(14 − 6) ÷ (3 − 1) = 4 and (26 − 14) ÷ (6 − 3) = 4. The rate is constant. From 6 = 4(1) + b, b = 2. So f(x) = 4x + 2 and f(10) = 42.
f(x) = 4x + 2; f(10) = 42
Problem 20 Fence and seed
At s = 6: C = 72 and S = 0.5(36) = 18. They match when 12s = 0.5s², so s = 24 (cost $288 for each). At s = 30: C = 360 and S = 450.
$72 and $18; match at s = 24; seed costs more at s = 30