From an equation
y = 5(0.8)x
The variable is in the exponent, so the function is exponential. Each step of 1 in x multiplies y by 0.8. It keeps 80% of its previous value: a 20% decrease per step.
Free Algebra 1 practice
Comparing Linear, Quadratic, and Exponential Functions Worksheet
Identify functions from equations, tables, and graphs, then use them to compare real situations. There are 18 core questions and 6 extension questions, with an answer key that explains the reasoning.
Download worksheet + answers · Free PDFs · No sign-up
By Berke Sahbazoglu · Revised October 14, 2026
24 questions18 core + 6 extension
Algebra 1Also useful for Algebra 2 review
Print at 100%US letter · Black-and-white friendly
Three families, three patterns
Start with what changes when the input increases by the same amount. A graph gives a useful clue; an equation or table helps you check it.
| Family | Basic equation | Table pattern | Graph |
|---|---|---|---|
| Linear | y = mx + b | Equal first differences | A straight line |
| Quadratic | y = ax² + bx + c a ≠ 0 | Equal, nonzero second differences | A parabola with a vertex and a vertical line of symmetry |
| Exponential | y = a·bx a ≠ 0; b > 0; b ≠ 1 | Equal ratios between consecutive outputs | A curve approaching a horizontal asymptote |
Check the x-values first. Differences and ratios should be compared over equal input steps. For the basic exponential form above, the asymptote is y = 0. With a positive starting value, a base above 1 gives growth; a base between 0 and 1 gives decay.
In these questions, choose among the three stated families. A short table suggests the intended model; by itself, it does not prove what happens at every other input.
Try one of each
Look for a specific piece of evidence before choosing a family.
From a table
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 3 | 5 | 11 | 21 | 35 |
First differences: 2, 6, 10, 14
Second differences: 4, 4, 4
The inputs step evenly. The first differences change, but the second differences stay at 4. This is a quadratic pattern; y = 2x² + 3 fits the values.
From a graph
The curve turns at (1, −2). Points equally far from x = 1 have the same height. That turning point and symmetry make it quadratic: y = (x − 1)² − 2.
Practice questions
Try each question before opening its answer. Name the family and explain what gave it away. For the extension questions, show your calculation and include units when needed.
The questions below match the PDFs. Print the student worksheet for writing space.
Equations
Questions 1–6 · Name the family and give a reason.01y = 4x - 7
02y = 12 - 3x
03y = 2x² - 5x + 1
04y = -(x - 3)² + 4
05y = 5(1.4)x
06y = 16(1/2)x
Tables
Questions 7–12 · Check the input steps first.07Identify the pattern and name the family.
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | -1 | 2 | 5 | 8 | 11 |
08Identify the pattern and name the family.
| x | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| y | 12 | 8 | 4 | 0 | -4 |
09Identify the pattern and name the family.
| x | -2 | -1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | 9 | 3 | 1 | 3 | 9 |
10Identify the pattern and name the family.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 5 | 4 | 1 | -4 | -11 |
11Identify the pattern and name the family.
| x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| y | 3 | 6 | 12 | 24 | 48 |
12Identify the pattern and name the family.
| x | 0 | 2 | 4 | 6 | 8 |
|---|---|---|---|---|---|
| y | 256 | 64 | 16 | 4 | 1 |
Graphs
Questions 13–18 · Use the turning point, symmetry, or change in height.13Graph G1
Name the function family. Describe the graph feature you used.
14Graph G2
Name the function family. Describe the graph feature you used.
15Graph G3
Name the function family. Describe the graph feature you used.
16Graph G4
Name the function family. Describe the graph feature you used.
17Graph G5
Name the function family. Describe the graph feature you used.
18Graph G6
Name the function family. Describe the graph feature you used.
Extension: models and comparisons
Questions 19–24 · A calculator is useful here.19A culture starts with 200 bacteria and doubles every 6 hours. Assume this doubling pattern continues.
a. Write B(t), the number of bacteria after t hours, and name the function family.
b. How many bacteria does the model predict after 18 hours?
20A phone plan charges $30 per month plus $0.10 for each gigabyte of data used. For this problem, ignore taxes and other fees.
a. Write C(g), the monthly cost for g gigabytes, and name the function family.
b. What is the cost for 45 gigabytes?
21A car is worth $24,000 when purchased. A model predicts that it loses 15% of its value each year.
a. Write V(t), the predicted value after t years, and name the function family.
b. Find the predicted value after 3 years.
22Two hypothetical account models start at $500. In model A, $25 is added each month: A(n) = 500 + 25n. In model B, the balance grows by 4% each month, with no deposits: B(n) = 500(1.04)n. Here n is the number of months.
a. Name the family for each model.
b. Compare the balances after 12 months and after 24 months. Round only the final amounts to the nearest cent.
23Compare Q(x) = x² and E(x) = 2x.
a. Find both values at x = 2, 3, 4 and 5. State which is larger at each input, or say they are equal.
b. Is an exponential function always larger than a quadratic at the same input? Use your results to explain.
24Compare L(x) = 50x, Q(x) = x² and E(x) = 1.2x for positive whole-number inputs. Use a calculator or a spreadsheet for this problem.
a. Find all three values at x = 10 and at x = 100. Round exponential values to two decimal places. Order the values from least to greatest at each input.
b. Starting at x = 1, check consecutive whole-number inputs. What is the first input where E(x) is greater than both L(x) and Q(x)? Show the values at that input and the preceding input.
Answers and reasoning
Open the answer when you are ready to check your work. A correct family name is only part of the answer; your evidence matters too.
Download the complete answer key for a printable copy.
01Show answer to question 1
Linear
The equation has the form y = mx + b. Each increase of 1 in x adds 4 to y.
02Show answer to question 2
Linear
Rewrite it as y = -3x + 12. The slope is -3, so the outputs decrease at a constant rate.
03Show answer to question 3
Quadratic
The highest power of x is 2, and its coefficient is 2. This is y = ax² + bx + c with a ≠ 0.
04Show answer to question 4
Quadratic
The squared expression makes it quadratic. Expanding gives y = -x² + 6x - 5; the graph opens downward.
05Show answer to question 5
Exponential growth
The variable x is in the exponent. The base 1.4 is greater than 1, so each increase of 1 in x multiplies y by 1.4.
06Show answer to question 6
Exponential decay
The variable x is in the exponent. The base 1/2 lies between 0 and 1, so each increase of 1 in x halves y.
07Show answer to question 7
Linear; first difference +3
The x-values increase by 1. The y-values change by +3, +3, +3, +3. A matching model is y = 3x + 5.
08Show answer to question 8
Linear; first difference -4 for each x-step of 2
The x-values increase by 2. The y-values change by -4 each time, so the slope is -4/2 = -2. A matching model is y = -2x + 12.
09Show answer to question 9
Quadratic; second difference +4
For x-steps of 1, the first differences are -6, -2, +2, +6. Their differences are +4, +4, +4. A matching model is y = 2x² + 1.
10Show answer to question 10
Quadratic; second difference -2
For x-steps of 1, the first differences are -1, -3, -5, -7. Their differences are -2, -2, -2. A matching model is y = -x² + 5.
11Show answer to question 11
Exponential growth; ratio 2
The x-values increase by 1, and each output is twice the preceding output. A matching model is y = 3(2)x.
12Show answer to question 12
Exponential decay; ratio 1/4 for each x-step of 2
The x-values increase by 2. The output ratios are all 1/4. The factor for an x-step of 1 is 1/2, so a matching model is y = 256(1/2)x.
13Show answer to question 13
Linear
G1 is a straight line. It rises at a constant rate as x increases.
14Show answer to question 14
Linear
G2 is a straight line with a negative slope. A decreasing function can still be linear.
15Show answer to question 15
Quadratic
G3 is an upward-opening parabola with one turning point. It is symmetric about x = -1.
16Show answer to question 16
Quadratic
G4 is a downward-opening parabola with one turning point. It is symmetric about x = 1.
17Show answer to question 17
Exponential growth
G5 rises more steeply as x increases and approaches the x-axis on the left. Its outputs double for each increase of 1 in x.
18Show answer to question 18
Exponential decay
G6 decreases and flattens toward the x-axis. Its outputs halve for each increase of 1 in x; it has no turning point.
19Show answer to question 19
a) B(t) = 200(2)t/6, exponential growth. b) 1,600 bacteria.
The exponent t/6 counts the number of 6-hour doubling periods. At 18 hours, B(18) = 200 × 2³ = 1,600.
20Show answer to question 20
a) C(g) = 30 + 0.10g, linear. b) $34.50.
Each extra gigabyte adds the same $0.10. C(45) = 30 + 0.10 × 45 = 34.50.
21Show answer to question 21
a) V(t) = 24,000(0.85)t, exponential decay. b) $14,739.
Losing 15% leaves 85% of the preceding value each year. V(3) = 24,000 × 0.85³ = 14,739.
22Show answer to question 22
a) A is linear; B is exponential growth. b) At 12 months: A = $800.00; B ≈ $800.52. At 24 months: A = $1,100.00; B ≈ $1,281.65. B is larger at both times.
A adds a fixed amount each month; B multiplies the preceding balance by 1.04. Substitute n = 12 and n = 24 into both expressions. Retain full calculator precision until the final rounding.
23Show answer to question 23
a) x = 2: Q = 4, E = 4, equal. x = 3: Q = 9, E = 8, Q is larger. x = 4: Q = 16, E = 16, equal. x = 5: Q = 25, E = 32, E is larger. b) No.
At x = 3, the quadratic value 9 exceeds the exponential value 8. This counterexample rules out the claim that an exponential is always larger. For these two models, the exponential is larger at x = 5.
24Show answer to question 24
a) x = 10: L = 500, Q = 100, E = (6/5)10 ≈ 6.19; E < Q < L. x = 100: L = 5,000, Q = 10,000, E = (6/5)100 ≈ 82,817,974.52; L < Q < E. b) x = 42.
At x = 41, E ≈ 1,763.73, Q = 1,681 and L = 2,050; E has passed Q but is still below L. At x = 42, E ≈ 2,116.47, Q = 1,764 and L = 2,100, so E is above both. Checking the consecutive inputs 1 through 41 confirms that none meets both conditions.
Check these before you finish
Check where the variable sits
In x², the variable is the base. In 2x, it is the exponent. A negative coefficient can flip a parabola without changing its family.
A curve is only a clue
A narrow graph window can make a parabola and an exponential look similar. Look for a turning point, symmetry, or the output pattern.
Compare at the given input
A positive exponential with a base above 1 eventually exceeds linear and quadratic growth. It does not have to be largest at the input in the question.
For teachers and tutors: ask students to mark the input intervals before calculating differences. For graphs, have them point to the feature they used. Start with questions 1–18 and save the modeling questions for a second session if needed.
About this resource
Prepared for Burke Tutoring
Berke Sahbazoglu is the founder of Burke Tutoring in Fremont. He teaches Algebra 1 and Algebra 2 and has more than 10 years of tutoring experience.
Found an error or a confusing question? Email Burke Tutoring with the question number and the part you would like clarified.
Originally published July 28, 2026. This revised worksheet has 24 questions; the online answers and PDFs use the same problem set.
For a longer explanation of the exponential family, see Paul’s Online Math Notes.
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