A comprehensive Grade 8 linear functions review unit aligned to Indiana Academic Standards (8.AF), featuring two paired 16-question study guide forms and fully worked teacher solutions.
Function A rate of change:
Function B rate of change:
Function has the greater rate of change.
Show Work:
4. Select all statements describing Interval 2. 5 pts
Examine the graph shown below across its three distinct intervals.
x y (1) (2) (3)
The function is increasing. The slope is negative. The function is decreasing. The function is constant. The slope is positive.
Topic 3 Assessment Review • Form B Page 1 of 4
Topic 3 Study Guide: Linear Functions
Form B • Page 2 of 4
Name:
5. Write linear equation from two points. 2 pts
The graph of a linear function passes through the coordinates:
(3, 14) and (9, 38)
Write an equation in the form y = mx + b for this function.
Show Work:
Equation: y =
6. Sketch a qualitative graph. 1 pt
Devon rides his scooter along a trail. He glides quickly downhill for 3 minutes. Then, he pushes at a slow, steady uphill climb for 12 minutes. Finally, he pauses and rests at a scenic overlook for 5 minutes.
Sketch a graph of Devon's elevation over time.
Time (minutes) Elevation (ft) 0
7. Compare initial values. 2 pts
Maya claims that Function A and Function B have the exact same initial value (y-intercept). Is Maya correct? Justify your response.
Function A x y 0 5 10 2 4
Function B
Initial value of B is the value of y when x = 0.
Justification:
8. Linear function from a graph. 3 pts
A party rental service charges an equipment sanitizing fee plus an hourly rental fee. The graph displays total cost.
hrs Cost ($) 0 40 90 140 190 2 4 6
Sanitizing fee: $40
Hourly rate: $25/hr
Points on line:
(0, 40), (2, 90), (4, 140)
Equation: y = x +
Show Work:
Topic 3 Assessment Review • Form B Page 2 of 4
Topic 3 Study Guide: Linear Functions
Form B • Page 3 of 4
Name:
9. Is the relation shown below a function? 3 pts
Plot the points on the grid to justify your response and explain your reasoning.
(1, 3), (3, 6), (5, 6), (7, 2), (9, 4)
x y 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
Justification & Explanation:
10. Consider the two functions below. 4 pts
Which function is linear? Write its equation in y = mx + b form.
Function A
Function B
Function is linear.
Equation: y = x +
Show Work:
11. Which function has a greater rate of change? 2 pts
Compare the rates of change of Function A and Function B.
Function A
Function B: y = 5.5x + 9
Function A rate of change:
Function B rate of change:
Function has the greater rate of change.
Show Work:
12. Select all statements describing Interval 3. 5 pts
Examine the graph shown below across its three intervals.
x y (1) (2) (3)
The function is increasing. The slope is negative. The function is decreasing. The function is constant. The slope is equal to zero.
Topic 3 Assessment Review • Form B Page 3 of 4
Topic 3 Study Guide: Linear Functions
Form B • Page 4 of 4
Name:
13. Write linear equation from two points. 2 pts
The graph of a linear function passes through the coordinates:
(2, 9) and (8, 39)
Write an equation in the form y = mx + b for this function.
Show Work:
Equation: y =
14. Sketch a qualitative graph. 1 pt
A hot air balloon pilot takes off over flat terrain. The balloon ascends rapidly for 6 minutes. Next, the pilot maintains a constant cruising altitude for 15 minutes. Finally, the balloon slowly descends toward a landing field for 8 minutes.
Sketch a graph of the balloon's altitude over time.
Time (minutes) Altitude (ft) 0
15. Compare initial values. 2 pts
Lucas claims that Function A has a smaller initial value than Function B. Is Lucas correct? Justify your response.
Function A x y 0 4 8 2 4
Function B
Notice the change in y per increase in x.
Justification:
16. Linear function from a graph. 3 pts
An appliance repair technician charges a travel service fee plus an hourly labor rate. The graph displays total cost.
hrs Cost ($) 0 50 95 140 185 1 2 3
Service fee: $50
Hourly labor: $45/hr
Points on line:
(0, 50), (1, 95), (2, 140)
Equation: y = x +
Show Work:
Topic 3 Assessment Review • Form B Page 4 of 4
Scoring: 1 pt for calculating both rates; 1 pt for justified comparison.
Q4: Interval Description (5 pts) IAS 8.AF.7
Interval 2 Evaluation: Segment decreases from left to right.
❌ Increasing: False ✔ Slope is negative: True ✔ Decreasing: True ❌ Constant: False ❌ Slope is positive: False
Scoring: 1 pt per correct statement checked/unchecked (5 pts total).
Function A: The graph crosses the \(y\)-axis at \((0, 4)\), so initial value \(b = 4\).
Function B: Slope \(m = \frac{12-8}{4-2} = 2\). At \(x = 0\), \(y = 8 - 2(2) = 4\). Both functions have an initial value of 4.
Scoring: 1 pt for "Yes"; 1 pt for finding initial value of 4 for both.
Q8: Real-World Linear Modeling (3 pts) IAS 8.AF.6
Context: Kayak rental fee plus hourly rate.
Initial Value (\(b\)): Point \((0, 20) \Rightarrow b = 20\) (reservation fee).
Rate of Change (\(m\)): \(m = \frac{50 - 20}{2 - 0} = \frac{30}{2} = 15\) ($15 per hour).
Equation: y = 15x + 20
Scoring: 1 pt for \(m=15\); 1 pt for \(b=20\); 1 pt for correct linear equation.
Function Focus Teacher Resource • Topic 3 Answer Key Page 1 of 4
Teacher Solutions & Scoring Rubric
Topic 3 Study Guide Solutions: Form A (Q9–Q16)
Form A Key Page 2 of 2
Q9: Relation / Function Check (3 pts) IAS 8.AF.1
Answer: YES, it IS a function.
Reasoning: Every input \(x \in \{0, 2, 4, 6, 8\}\) corresponds to exactly one output value. Even though \(y = 5\) occurs twice (for \(x=2\) and \(x=8\)), no \(x\)-value repeats with different \(y\)-values. Passes the Vertical Line Test.
Q10: Linear vs Non-Linear & Equation (4 pts) IAS 8.AF.3, 8.AF.5
Answer: Function B is linear.
Equation: y = 3x + 2
Work: In Function B, \(\frac{\Delta y}{\Delta x} = \frac{20-11}{6-3} = \frac{9}{3} = 3\). Intercept: \(11 = 3(3) + b \Rightarrow b = 2\). Function A rate is not constant (\(\frac{7-3}{2-1}=4\) then \(\frac{13-7}{3-2}=6\)).
Function A: Crosses \(y\)-axis at \((0, 6) \Rightarrow b = 6\).
Function B: Slope \(m = \frac{19 - 13}{6 - 3} = 2\). At \(x = 0\), \(y = 13 - 2(3) = 7\). Since \(6 < 7\), Function A has a smaller initial value than Function B.
Scoring: 1 pt for "No"; 1 pt for finding \(b=6\) and \(b=7\).
Q16: Real-World Linear Modeling (3 pts) IAS 8.AF.6
Context: Mobile pet grooming dispatch fee plus per-pet fee.
Initial Value (\(b\)): Point \((0, 30) \Rightarrow b = 30\) (dispatch fee).
Rate of Change (\(m\)): \(m = \frac{70 - 30}{2 - 0} = \frac{40}{2} = 20\) ($20 per pet).
Equation: y = 20x + 30
Scoring: 1 pt for \(m=20\); 1 pt for \(b=30\); 1 pt for correct linear equation.
Function Focus Teacher Resource • Topic 3 Answer Key Page 2 of 4
Teacher Solutions & Scoring Rubric
Topic 3 Study Guide Solutions: Form B (Q1–Q8)
Form B Key Total: 44 Points
Q1: Relation / Function Check (3 pts) IAS 8.AF.1
Answer: NOT a function.
Reasoning: The input value x = 2 is paired with two distinct output values: y = 4 and y = 7. Fails the Vertical Line Test at x = 2.
Q2: Linear vs Non-Linear & Equation (4 pts) IAS 8.AF.3, 8.AF.5
Answer: Function B is linear.
Equation: y = 2x + 1 (or y = 2.00x + 1.00)
Work: In Function B, \(\frac{\Delta y}{\Delta x} = \frac{11-5}{5-2} = \frac{6}{3} = 2\). Intercept: \(5 = 2(2) + b \Rightarrow b = 1\). Function A has changing rate (\(\frac{7-3}{3-1}=2\) then \(\frac{13-7}{5-3}=3\)).
Function A Rate: \(\frac{10 - 18}{4 - 2} = \frac{-8}{2} = -4\)
Function B Rate: \(-5.2\) (from \(y = -5.2x + 20\))
Conclusion: Function B has a greater magnitude / steeper rate of change (\(|-5.2| = 5.2 > |-4| = 4\)). Note: Algebraically, \(-4 > -5.2\); accept either Function B (steeper magnitude) or Function A (algebraic value) when justified.
Scoring: 1 pt for calculating both rates; 1 pt for justified comparison.
Q4: Interval Description (5 pts) IAS 8.AF.7
Interval 2 Evaluation: Segment rises from left to right.
✔ Increasing: True ❌ Slope is negative: False ❌ Decreasing: False ❌ Constant: False ✔ Slope is positive: True
Scoring: 1 pt per correct statement checked/unchecked (5 pts total).
Function A: Crosses \(y\)-axis at \((0, 5) \Rightarrow b = 5\).
Function B: Slope \(m = \frac{17 - 11}{6 - 3} = 2\). At \(x = 0\), \(y = 11 - 2(3) = 5\). Both functions have an initial value of 5.
Scoring: 1 pt for "Yes"; 1 pt for demonstrating both have initial value 5.
Q8: Real-World Linear Modeling (3 pts) IAS 8.AF.6
Context: Bounce house sanitizing fee plus hourly rental.
Initial Value (\(b\)): Point \((0, 40) \Rightarrow b = 40\) (sanitizing fee).
Rate of Change (\(m\)): \(m = \frac{90 - 40}{2 - 0} = \frac{50}{2} = 25\) ($25 per hour).
Equation: y = 25x + 40
Scoring: 1 pt for \(m=25\); 1 pt for \(b=40\); 1 pt for correct linear equation.
Function Focus Teacher Resource • Topic 3 Answer Key Page 3 of 4
Teacher Solutions & Scoring Rubric
Topic 3 Study Guide Solutions: Form B (Q9–Q16)
Form B Key Page 2 of 2
Q9: Relation / Function Check (3 pts) IAS 8.AF.1
Answer: YES, it IS a function.
Reasoning: Every input \(x \in \{1, 3, 5, 7, 9\}\) corresponds to exactly one output value. Even though \(y = 6\) is shared by \(x=3\) and \(x=5\), no input has more than one output. Passes the Vertical Line Test.
Q10: Linear vs Non-Linear & Equation (4 pts) IAS 8.AF.3, 8.AF.5
Answer: Function A is linear.
Equation: y = 2x + 1
Work: In Function A, \(\frac{\Delta y}{\Delta x} = \frac{17-9}{8-4} = \frac{8}{4} = 2\). Intercept: \(9 = 2(4) + b \Rightarrow b = 1\). Function B is quadratic (\(y = x^2\)), so its rate is non-constant.
Function A: Crosses \(y\)-axis at \((0, 2) \Rightarrow b = 2\).
Function B: Slope \(m = \frac{11 - 7}{4 - 2} = 2\). At \(x = 0\), \(y = 7 - 2(2) = 3\). Since \(2 < 3\), Function A does have a smaller initial value than Function B.
Scoring: 1 pt for "Yes"; 1 pt for finding \(b=2\) and \(b=3\).
Q16: Real-World Linear Modeling (3 pts) IAS 8.AF.6
Context: Appliance repair travel fee plus hourly labor.
Initial Value (\(b\)): Point \((0, 50) \Rightarrow b = 50\) (service call fee).
Rate of Change (\(m\)): \(m = \frac{95 - 50}{1 - 0} = 45\) ($45 per hour).
Equation: y = 45x + 50
Scoring: 1 pt for \(m=45\); 1 pt for \(b=50\); 1 pt for correct linear equation.
Function Focus Teacher Resource • Topic 3 Answer Key Page 4 of 4
Function B: y = -3x + 10
Function A rate of change:
Function B rate of change:
Function has the greater rate of change.
Show Work:
Skill 4 • Qualitative Intervals 8.AF.7
Select all statements that correctly describe Interval 2.
x y (1) (2) (3)
The function is increasing. The slope is negative. The function is decreasing. The function is constant. The slope is equal to zero.
Topic 3 Linear Functions Mastery • Quick Check Page 1 of 2
Indiana Academic Standards • 8.AF Skills 5–8
Topic 3 Quick Check: Linear Functions
Name:
Page 2 Score: / 4
Skill 5 • Line Through Two Points 8.AF.5
A linear function passes through the coordinates:
(2, 9) and (5, 24)
Write an equation in the form y = mx + b. Show all steps below.
Show Work:
Equation: y =
Skill 6 • Sketch Qualitative Graph 8.AF.7
Liam is filling a container. For the first 3 minutes, water pours in rapidly at a constant rate. For the next 4 minutes, he lowers the hose to a slow, steady trickle. For the last 3 minutes, he turns the water completely off.
Sketch a graph of water volume over time.
Time (minutes) Volume (gal) 0
Skill 7 • Compare Initial Values 8.AF.4
Compare the initial values (y-intercepts) of Function A and Function B. Show work to justify.
Function A x y 0 5 10 2 4
Function B
Find the value of y when x = 0.
Initial value of A: | Initial value of B:
Justification:
Skill 8 • Linear Modeling 8.AF.6
A food truck rental company charges an equipment reservation fee plus an hourly rental fee. The graph displays total cost.
hrs Cost ($) 0 30 70 110 2 4 6
Initial fee: $30
Hourly rate: $20/hr
Points on line:
(0, 30), (2, 70), (4, 110)
Equation: y = x +
Show Work:
Topic 3 Linear Functions Mastery • Quick Check Page 2 of 2
Step 2: Solve for \(y\)-intercept (\(b\)):
\(y = mx + b \Rightarrow 9 = 5(2) + b\)
\(9 = 10 + b \Rightarrow \mathbf{b = -1}\)
Final Equation: y = 5x - 1
Skill 6 • Qualitative Sketch (1 pt) IAS 8.AF.7
Liam filling container: fast pour (0–3 min), trickle (3–7 min), off (7–10 min).
Time (min) Volume Fast Trickle Off (Flat)
Rubric: Continuous graph from origin showing steep increase, less steep increase, and horizontal plateau.
