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Linear functions practice
SAT, PSAT and ACT Algebra · 43 questions in the bank
Linear functions are the single most tested topic in SAT Algebra, and they appear throughout ACT Math too. Almost every question reduces to one of three tasks: find the slope, find the intercept, or interpret one of them in context.
The interpretation questions are where points are lost, because they require you to translate y = mx + b back into the situation the question describes.
How to answer these
- Slope is a rate. In context it is always 'per': dollars per hour, seedlings per day, degrees per minute. Read the units off the axes and the sentence writes itself.
- The intercept is the starting value. It is what you have when the other variable is zero, which is usually a fee, a head start, or an initial amount.
- From a table, take two rows. Slope is the change in output over the change in input. Then work backwards to the value at zero.
- Check a point. Once you have an equation, plug in one row from the table. It takes five seconds and catches most sign errors.
What the wrong answers look like
- Sign errors on the intercept. Especially when the table starts away from zero and you have to work backwards.
- Swapping slope and intercept in interpretation questions. The question asks what one number means, and the other number's meaning is offered as a distractor.
- Rate per wrong unit. The table steps in twos and the rate is asked per one.
- Reading the graph rather than the question. Some questions describe a situation with no graph at all, and the answer depends on the sentence, not a picture.
Practice questions
Four real questions from the CruxStudy bank, with the keyed answer marked and the explanation behind a click so you can try them first.
Example 1Easier
A plumber charges customers according to the function C(h) = 60h + 45, where C(h) is the total charge in dollars for a job that takes h hours.
What is the best interpretation of the number 45 in this context?
- The cost, in dollars, of each hour of labor
- The number of hours a typical job takes
- The total charge, in dollars, for a one-hour job
- The fixed fee, in dollars, charged before any labor correct
Show the explanation
When h = 0, C(0) = 45, so 45 is the fixed fee charged regardless of labor time; a one-hour job costs 105 dollars.
Example 2Medium
A grower counted the number of sprouted seedlings n in a germination tray on several days d after planting. The counts increase at a constant rate.
| Day (d) | Seedlings sprouted (n) |
|---|
| 2 | 34 |
| 4 | 61 |
| 6 | 88 |
| 8 | 115 |
Which equation gives n in terms of d for the values in the table?
- n = 13.5d + 7 correct
- n = 13.5d + 34
- n = 27d - 20
- n = 27d + 7
Show the explanation
The count rises 27 seedlings every 2 days, a rate of 13.5 per day, and 34 - 13.5(2) = 7, so n = 13.5d + 7.
Example 3Harder
The table shows the water depth d, in feet, at a harbor dock h hours after midnight during a falling tide. Over this interval the depth is a linear function of h.
| Hours after midnight, h | Water depth, d (feet) |
|---|
| 1 | 14.2 |
| 3 | 11.8 |
| 5 | 9.4 |
According to the model, what was the water depth at midnight?
- 15.4 feet correct
- 16.6 feet
- 17.8 feet
- 19.0 feet
Show the explanation
The depth falls 2.4 feet every 2 hours, a rate of -1.2 feet per hour, so going back one hour from h = 1 gives 14.2 + 1.2 = 15.4 feet.
Example 4Medium
The value of a delivery van, in dollars, t years after purchase is modeled by v(t) = 24,000 - 1,800t.
What is the best interpretation of the number 1,800 in this model?
- The van's value decreases by $1,800 each year correct
- The van's value decreases by $24,000 each year
- The van's value increases by $1,800 each year
- The van is worth $1,800 one year after purchase
Show the explanation
The slope -1,800 means the value drops by 1,800 dollars for each one-year increase in t.
Drill all 43 of them
Open the drill inside CruxStudy and it serves linear functions questions one at a time, with an explanation after every answer and your accuracy tracked. Free, no account.
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Common questions
What does slope mean in a word problem?
It is the rate of change: how much the output changes for each one-unit increase in the input. In context it is always expressible as something 'per' something.
How do I find a linear equation from a table?
Take two rows, divide the change in output by the change in input to get the slope, then work backwards to the value where the input is zero to get the intercept.
Are linear functions on both the SAT and the ACT?
Yes, heavily on both. It is the highest-frequency algebra topic on either test.