Math: Algebra, 3 original questions
Interpret linear rules, tables, starting values and rates of change in three original SAT-style questions, with explained answers and common function traps.
A linear function has a constant rate of change: equal input increases produce equal output increases. In f(x) = mx + b, m describes the output change per input unit and b is the output when x = 0. Translate a table by calculating differences, and interpret coefficients using the context's units. A zero of the function is an input producing output zero. Distinguish the function's rule from one particular value, and verify that a model fits the supplied data.
Method at a glance
Use these steps in order, then apply them to the questions below.
State what x and f(x) represent. A slope is measured in output units per input unit.
From a table, divide the output difference by the input difference. Unequal input steps require this ratio rather than the output difference alone.
Use one known pair to solve for b, or evaluate the function at x = 0 when that input is meaningful.
Evaluate at a requested input or solve f(x) = 0. Explain which quantity the resulting number represents.
These are original SAT-style questions written by PeakSAT, not official College Board questions. Choose an answer and check it to see why each choice is right or wrong, or open the worked answer.
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If f(x) = 7x + 19, what is f(3)?
Correct answer
Choice C: 40
Explanation
Replace x with 3: f(3) = 7 × 3 + 19 = 21 + 19 = 40. The constant 19 is added once; it is not multiplied by 3.
Why each choice is right or wrong
This includes 7 × 3 but omits the constant 19.
This is f(1), not f(3).
Correct: substituting 3 gives 40.
This multiplies both 7 and 19 by 3 instead of substituting only for x.
A linear function has the following values: x: 1, 3, 5 f(x): 14, 22, 30 Which rule gives f(x)?
Correct answer
Choice A: f(x) = 4x + 10
Explanation
Each input increase of 2 gives an output increase of 8, so the slope is 8/2 = 4. Since f(1) = 14, the intercept is 14 − 4 = 10. The rule 4x + 10 reproduces all three values.
Why each choice is right or wrong
Correct: the slope is 4 and the intercept is 10.
This uses an output difference of 8 as slope without dividing by the input difference 2.
14 is the output at x = 1, not the intercept at x = 0.
It fits x = 1 only; at x = 3 it gives 34 rather than 22.
The volume of water in a tank is modeled by V(t) = 180 − 12t for 0 ≤ t ≤ 15, where t is in minutes and V(t) is in liters. What does t = 15 represent?
Correct answer
Choice D: The tank is empty after 15 minutes
Explanation
Set the volume to zero: 180 − 12t = 0 gives t = 15. This is the endpoint of the stated interval, and V(15) = 0. The model therefore says the tank is empty after 15 minutes.
Why each choice is right or wrong
The initial amount is V(0) = 180 liters, not 15.
The loss rate is 12 liters per minute, not 15.
At t = 15 the modeled volume is 0, not 12.
Correct: 180 − 12 × 15 = 0 liters.
Using a table's output difference as slope when the input step is not 1. Divide by the input difference.
Confusing f(3) with multiplying the whole function by 3. Substitute 3 for every input variable.
Treating a negative slope as a negative starting amount. The slope and intercept describe different quantities.
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Compare output change divided by input change across intervals. A consistent ratio supports one linear rule for those data.
No. A context can restrict the domain; extending a tank model past empty, for example, would produce impossible negative volumes.
f(0) is an output at input zero. A zero of f is an input at which the output equals zero.
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