Texas Grab-and-Go Substitute Packet · Teacher Answer Key
Answer Key: Linear Functions in Four Forms
Course: Algebra I (Grades 9–12)
Subject: Mathematics
For teacher use only
Math
How to use this key
Answers are grouped by section and question number, with full worked steps. Accept any correct method that reaches the right value — alternate strategies (using different table rows, counting up on the graph, or reasoning from the context) are noted where they apply. Watch-for notes flag common misconceptions. A basic or graphing calculator is allowed, so grade the reasoning and steps, not just the arithmetic. Estimated grading time: ~6–8 min per packet.
Start Warm-Up: Retrieval ~1 min
1 Evaluate f(x) = 2x + 3 at x = 4, i.e. find f(4).
f(4) = 2 × 4 + 3 = 8 + 3 = 11. Full credit for showing the
multiply-then-add order. Watch-for: a student who reads f(4) as "f times 4" or who
writes 2 × (4 + 3) = 14, adding before multiplying. Remind them f(4) names the output at
x = 4, and mx is done before adding b.
2 Slope through (1, 5) and (4, 11).
m = (11 − 5) ÷ (4 − 1) = 6 ÷ 3 = 2. Accept the points taken in
either order as long as x-differences and y-differences use the same order:
(5 − 11) ÷ (1 − 4) = (−6) ÷ (−3) = 2. Watch-for: reversing to run over rise,
(4 − 1) ÷ (11 − 5) = 3 ÷ 6 = 0.5, which is wrong.
Build Study the Math ~1 min
3 Find f(7) for f(x) = 2x + 3.
f(7) = 2 × 7 + 3 = 14 + 3 = 17 (so 7 hours costs $17).
Alternate: extend the table by adding 2 each hour (11 → 13 → 15 → 17). Both are
correct.
Apply Use What You Know ~3–4 min
4 Match the four forms (repair cost, Table B).
4a. Using rows (2, 55) and (4, 85): m = (85 − 55) ÷ (4 − 2) = 30 ÷ 2 =
15 dollars per part. (Any two rows work: e.g., (0, 25) and (6, 115) →
90 ÷ 6 = 15.)
4b. y-intercept b = 25 (the cost at x = 0 parts, read straight from the table). Equation: g(x) = 15x + 25.
4c. The slope means the cost goes up $15 per part (each part adds $15). The y-intercept means the $25 diagnostic fee charged before any parts (at 0 parts the cost is already $25).
4d. Solve g(x) = 145: 15x + 25 = 145 → subtract 25: 15x = 120 → divide by 15: x = 8 parts. Check: 15(8) + 25 = 120 + 25 = 145. ✔ Alternate: x = (145 − 25) ÷ 15 = 120 ÷ 15 = 8.
4e. It is NOT proportional, because the y-intercept is 25, not 0 — the line does not pass through the origin (0, 0). (A proportional function must have b = 0.)
4b. y-intercept b = 25 (the cost at x = 0 parts, read straight from the table). Equation: g(x) = 15x + 25.
4c. The slope means the cost goes up $15 per part (each part adds $15). The y-intercept means the $25 diagnostic fee charged before any parts (at 0 parts the cost is already $25).
4d. Solve g(x) = 145: 15x + 25 = 145 → subtract 25: 15x = 120 → divide by 15: x = 8 parts. Check: 15(8) + 25 = 120 + 25 = 145. ✔ Alternate: x = (145 − 25) ÷ 15 = 120 ÷ 15 = 8.
4e. It is NOT proportional, because the y-intercept is 25, not 0 — the line does not pass through the origin (0, 0). (A proportional function must have b = 0.)
5 Write the equation from a context (gym membership).
5a. Slope m = 12 (dollars per month) and y-intercept b = 40 (the
$40 sign-up fee, the cost at 0 months). Equation: h(x) = 12x + 40.
5b. h(9) = 12(9) + 40 = 108 + 40 = $148 after 9 months. Watch-for: a student who writes h(9) = 12 + 40(9) by swapping which number multiplies x — anchor that the per-month rate multiplies the months and the sign-up fee is the constant b.
5b. h(9) = 12(9) + 40 = 108 + 40 = $148 after 9 months. Watch-for: a student who writes h(9) = 12 + 40(9) by swapping which number multiplies x — anchor that the per-month rate multiplies the months and the sign-up fee is the constant b.
6 Error analysis: a mis-found slope for Table B.
The mistake: The student reversed rise and run — they divided
run ÷ rise instead of rise ÷ run. Slope is the change in the output (rise)
over the change in x (run), so the differences are flipped.
Correct work: m = rise ÷ run = (85 − 55) ÷ (4 − 2) = 30 ÷ 2 = 15 dollars per part (not 0.07).
Full credit requires (1) naming that they flipped rise and run (used run/rise), and (2) the correct slope of 15. Sense-check to offer: 0.07 dollars per part would mean about 7¢ a part, which does not match the table jumping $30 for every 2 parts.
Correct work: m = rise ÷ run = (85 − 55) ÷ (4 − 2) = 30 ÷ 2 = 15 dollars per part (not 0.07).
Full credit requires (1) naming that they flipped rise and run (used run/rise), and (2) the correct slope of 15. Sense-check to offer: 0.07 dollars per part would mean about 7¢ a part, which does not match the table jumping $30 for every 2 parts.
Block Extra Applied Task block only · ~1–2 min
7 Real-world modeling: draining tank (block schedule).
7a. Starting amount b = 60 gallons; the tank loses 4 gallons per minute, so
the rate of change is −4. Function: V(x) = −4x + 60. Watch-for: a
student who writes +4 and forgets the water is decreasing.
7b. V(6) = −4(6) + 60 = −24 + 60 = 36 gallons left after 6 minutes.
7c. Solve V(x) = 0: −4x + 60 = 0 → −4x = −60 → x = 15 minutes. Check: V(15) = −4(15) + 60 = −60 + 60 = 0. ✔ Interpretation: the x-intercept (15, 0) is the moment the tank is empty — the input that makes the output 0. Alternate: x = (0 − 60) ÷ (−4) = (−60) ÷ (−4) = 15.
7b. V(6) = −4(6) + 60 = −24 + 60 = 36 gallons left after 6 minutes.
7c. Solve V(x) = 0: −4x + 60 = 0 → −4x = −60 → x = 15 minutes. Check: V(15) = −4(15) + 60 = −60 + 60 = 0. ✔ Interpretation: the x-intercept (15, 0) is the moment the tank is empty — the input that makes the output 0. Alternate: x = (0 − 60) ÷ (−4) = (−60) ÷ (−4) = 15.
Explain Justify Your Strategy (Q8) ~1–2 min
8 3 premium channels: Plan A f(x) = 3x + 8 vs. Plan B g(x) = 5x.
Model answer. Claim: For 3 premium channels, Plan B costs less; and
Plan B is the proportional one.
Evidence: Plan A: f(3) = 3(3) + 8 = 9 + 8 = $17. Plan B: g(3) = 5(3) =
$15. Since $15 < $17, Plan B is cheaper for 3 channels.
Reasoning: Plan B has y-intercept b = 0, so it passes through the origin and is
proportional (g(x) = 5x). Plan A has y-intercept b = 8 (an $8 base charge), so it
is not proportional. Even though Plan A's slope ($3/channel) is smaller than Plan
B's ($5/channel), Plan A's $8 head start makes it cost more at 3 channels.
Note: Full credit requires both computed costs ($17 and $15), naming Plan B as cheaper, and identifying Plan B as proportional (b = 0). Extension worth noting: the two plans are equal when 3x + 8 = 5x → 2x = 8 → x = 4 channels; beyond 4 channels Plan A becomes cheaper. Students are not required to find this, but it is a strong discussion point.
Note: Full credit requires both computed costs ($17 and $15), naming Plan B as cheaper, and identifying Plan B as proportional (b = 0). Extension worth noting: the two plans are equal when 3x + 8 = 5x → 2x = 8 → x = 4 channels; beyond 4 channels Plan A becomes cheaper. Students are not required to find this, but it is a strong discussion point.
Justification scoring rubric (3 points)
| Score | Claim | Evidence | Reasoning |
|---|---|---|---|
| 3 | Names Plan B as cheaper for 3 channels AND names Plan B as proportional. | Both costs computed correctly ($17 and $15) with the numbers shown. | Explains proportional means b = 0 and correctly links slope and y-intercept to the comparison. |
| 2 | Names Plan B as cheaper (proportional part missing or partly right). | One cost computed, or both with a minor arithmetic slip. | Reasoning present but incomplete or missing the y-intercept link. |
| 1 | A plan named with weak or no support. | Evidence largely missing or incorrect. | Little or flawed reasoning. |
| 0 | No/incorrect claim. | No evidence. | No reasoning. |
Close ACE ~1 min
ACE Articulate / Connect / Extend.
Articulate: Accept any correct explanation: "Evaluating f(x) means you are
given the input x and you compute the output (e.g., f(4) = 11).
Solving f(x) = k means you are given the output k and you find the
input x that produces it (e.g., 2x + 3 = 15 → x = 6). Evaluating gives an output;
solving gives an input."
Connect: Valid items where function notation was used to find an output include Q1 (f(4)), Q3 (f(7)), Q5b (h(9)), Q7b (V(6)), or Q8 (f(3), g(3)). Any of these earns credit if correctly named.
Extend: Any genuine non-proportional linear function with a starting amount and a steady rate — e.g., a taxi with a base fare plus a per-mile rate, a phone plan with a monthly fee plus a per-gigabyte charge, a savings account starting at some balance and growing by a fixed weekly deposit. Must have a nonzero starting amount (b ≠ 0) to count as non-proportional.
Connect: Valid items where function notation was used to find an output include Q1 (f(4)), Q3 (f(7)), Q5b (h(9)), Q7b (V(6)), or Q8 (f(3), g(3)). Any of these earns credit if correctly named.
Extend: Any genuine non-proportional linear function with a starting amount and a steady rate — e.g., a taxi with a base fare plus a per-mile rate, a phone plan with a monthly fee plus a per-gigabyte charge, a savings account starting at some balance and growing by a fixed weekly deposit. Must have a nonzero starting amount (b ≠ 0) to count as non-proportional.
Challenge Early Finisher (optional)
EF Write a context for f(x) = 7x + 20 (must be non-proportional).
(1) A sample situation: "A caterer charges a $20 setup fee plus $7 per
guest. Let x = number of guests and f(x) = total charge in dollars." Many correct
stories exist (a $20 flat delivery fee plus $7 per item; a gift-card starting balance
plus deposits, etc.).
(2) Meaning: The slope 7 is the rate of $7 per guest (each guest adds $7). The y-intercept 20 is the $20 starting amount / setup fee charged before any guests (the cost at x = 0).
(3) Evaluate f(5): f(5) = 7(5) + 20 = 35 + 20 = $55.
(4) Solve f(x) = 90: 7x + 20 = 90 → 7x = 70 → x = 10 guests. Check: 7(10) + 20 = 90. ✔
Full credit requires a clear x and f(x), a story with a real starting amount (so it is non-proportional), a correct interpretation of 7 and 20, and the correct values 55 and 10.
(2) Meaning: The slope 7 is the rate of $7 per guest (each guest adds $7). The y-intercept 20 is the $20 starting amount / setup fee charged before any guests (the cost at x = 0).
(3) Evaluate f(5): f(5) = 7(5) + 20 = 35 + 20 = $55.
(4) Solve f(x) = 90: 7x + 20 = 90 → 7x = 70 → x = 10 guests. Check: 7(10) + 20 = 90. ✔
Full credit requires a clear x and f(x), a story with a real starting amount (so it is non-proportional), a correct interpretation of 7 and 20, and the correct values 55 and 10.
Watch Common misconceptions
- Slope vs. y-intercept. Students may swap the two — reporting the starting amount as the rate, or the rate as the starting amount. In g(x) = 15x + 25, m = 15 is the per-part rate and b = 25 is the diagnostic fee. Anchor: the number multiplied by x is the slope; the number added on is the y-intercept (the output at x = 0).
- Reversing rise and run. The Q6 error: computing run ÷ rise (giving ≈0.07) instead of rise ÷ run (giving 15). Slope is change in output over change in x — always the y-difference on top. Subtract the x-values and y-values in the same order.
- Confusing f(x) with multiplication. f(4) does not mean "f times 4"; it names the output when the input is 4. Reinforce this on Q1, Q3, Q5b, Q7b, Q8. A student who "distributes" f across (4) has misread the notation.
- Assuming all lines are proportional. Students often treat any straight line as proportional. A linear function is proportional only if it passes through (0, 0), i.e., b = 0. Figure 1 and Table B both have b ≠ 0, so they are linear but not proportional (Q4e and Q8 check this).
- Evaluate vs. solve. Some students "solve" when asked to evaluate, or plug in the output where the input goes. Anchor: given x → evaluate (find output); given output → solve (find x). Q4d, Q7c, and the ACE Articulate item test this.
- Dropping the negative rate. In Q7, water is draining, so m = −4. A student who writes V(x) = 4x + 60 has missed that the output decreases. Encourage a check: after 15 minutes the tank should be empty, not overflowing.
- Stopping at the wrong step when solving. In Q4d/Q7c, after isolating the mx term a student may forget to divide by m. Encourage a check by substituting the answer back into the equation.
Teacher follow-up based on likely errors
If many students miss Q6, do a quick 5-minute practice labeling rise and run on two or three graphs, always keeping the output-difference on top. If Q4e or Q8 show the "all lines are proportional" idea, re-anchor that only a line through (0, 0) is proportional (b = 0), and contrast it with a line that has a fee. The Q4c interpretations reveal who can attach meaning to slope and y-intercept in context. The Q4d and Q7c items show who can move from evaluating to solving f(x) = k, and Q8 shows who can compare two functions using both slope and y-intercept — all good warm-up discussions next class.
HS_ALG1_LinearFunctions_01 — Linear Functions in Four Forms
Teacher Answer Key