Linear functions: what slope and intercept mean in context
The exam hands you the model and asks what one number means. The answer is never a number — it is a sentence about the situation.
Which of the following is the best interpretation of 180 in this context?
The function f is defined by f(x) = 45x + 180, where f(x) is the total cost, in dollars, of renting a floor sander for x days.
180 is the value of f(0): the cost before a single day has been counted, so it is a one-time charge on top of the daily rate. A gives 180 the slope's job — 45 is the per-day cost, not 180. D is the cost of one day, which is f(1) = 45 + 180 = 225 dollars, not 180. C reads 180 as a number of days, but 180 is measured in dollars; the only quantity on the x-axis is days, and nothing in the model caps it.
1
Every linear model says “a starting amount, plus a rate times how many”.
In f(x) = mx + b the b is where the story starts and the m is how fast it moves. Nothing else is in there.
f(x) = 12x + 250
f(x) is the number of trees in a nursery x weeks after planting began.
250 — the intercept
the nursery held 250 trees when planting began
the value when x = 0
12 — the slope
12 trees are added each week
the change for every 1 added to x
b answers “how much at the start?”m answers “how much per one?”
Say the sentence out loud before you look at the choices. “There were 250 trees at the start and 12 more go in every week” is the whole model, and the correct choice will be one half of that sentence written formally.
2
The intercept is the output when the input is zero — named in the situation's words.
Whatever x counts, set it to 0 and ask what moment that is. The intercept is the value at that moment.
Model
x = 0 means
The intercept means
f(x) = 45x + 180
no days rented yet
180 dollars charged before any day
f(x) = 12x + 250
the week planting began
250 trees already in the nursery
P(t) = 900 − 75t
the moment the machine started
900 parts were in the bin to begin with
The units of the intercept are always the units of the output — dollars, trees, parts, degrees. That alone kills a choice such as “the sander can be rented for 180 days”.
“At the start” is not always “at zero cost”
An intercept of 0 is a real answer, and so is a negative one. In P(x) = 9x − 400, the −400 says the venture is 400 dollars down before it sells anything; the break-even point is where P(x) = 0, at x = 4009 ≈ 44.4, which is a different question entirely.
3
The slope is the change per one unit — and its units tell you the sentence.
Slope = (change in output) ÷ (change in input), so its units are output-units per input-unit: dollars per day, gallons per minute, parts per hour.
P(t) = 900 − 75t, parts in a bin after t hours
the slope is −75
units: parts ÷ hours = parts per hour
that is the phrase the choice must contain
sign: negative
“removes”, “decreases by”, “loses”
in words: 75 parts leave the bin each hour
rate first, then the direction
If the model arrives as a table or a graph instead, compute the slope from two points before you interpret it. From (0, 900) to (4, 600): slope = 600 − 9004 − 0 = −3004 = −75 parts per hour. The 300 is the drop over four hours, not the rate.
The two choices that swap slope and intercept
Almost every interpretation item includes one choice that gives the intercept the slope's job (“the cost is 180 dollars per day”) and one that reports a change over the whole interval rather than per unit (“300 parts per hour”). Read the words “each”, “per” and “every” as carefully as you read the number.
4
Read the same two sentences straight off the graph.
The graph shows the amount in a savings account w weeks after it was opened. The axis titles are the situation, so the interpretation is already half written.
intercept (0, 150)
the account held $150 when it was opened
slope = 1004 = 25
$100 gained over 4 weeks
in words: $25 is added each week
dollars per week, the units of the axes
A(w) = 25w + 150
and the model falls out of the reading
$150 at the start, $25 more each week
5
Going the other way: turn the sentence into the function.
“Which equation represents this situation?” is the same skill run backwards. Find the fixed amount, find the rate, and decide the rate's sign.
A printer is loaded with 1,200 sheets of paper and prints 18 sheets each minute. Write the number of sheets left, S, in terms of the number of minutes, m.
fixed amount: 1,200 sheets at m = 0
that is the constant term
rate: 18 sheets per minute, leaving
so the coefficient is −18
S(m) = 1,200 − 18m
start, then the running change
test m = 10: 1,200 − 180 = 1,020 ✓
ten minutes of printing is 180 sheets
“starts with”, “initial”, “flat fee” → the constant“each”, “per”, “every” → the coefficient“uses”, “loses”, “removes” → negative
Where the calculator earns its keep
When the choices are four similar equations, graph the one you built in Desmos and check two points from the story — here, (0, 1200) and (10, 1020). A choice that misses either point is out, and this is faster than re-reading the sentence a third time.
6
Sometimes the question is about a whole term, not a single number.
A term is a quantity in its own right, so it gets a full sentence too — usually one containing “total” or “so far”.
In C(h) = 65h + 95
Means
95
the fixed call-out fee, in dollars
65
the charge, in dollars, for each hour of work
65h
the total charge, in dollars, for h hours of work
C(h)
the total bill, fee and labor together
And when the model is written as a product, the number inside the bracket is where the output is zero. In R(x) = 14(x − 40), R(40) = 0, so 40 is the number of items at which the revenue balances out — read it by asking what makes the bracket vanish, not by expanding.
R(x) = 14(x − 40)
a factored linear model
R(40) = 14(0) = 0
40 is the break-even count
R(x) = 14x − 560
the same model; now 14 is clearly the rate
Do not name a number that is not there
In R(x) = 14(x − 40) there is no choice that reads “the starting revenue is −40 dollars”, and “−40” is not the intercept either — expanding shows the intercept is −560. If a choice describes a quantity you have not actually computed, compute it before you trust the choice.
✓
Check: does your sentence survive a substitution?
f(x) = 45x + 180f(0) = 180the one-time charge ✓
A(w) = 25w + 150A(1) − A(0) = 25$25 gained in one week ✓
S(m) = 1,200 − 18mS(0) = 1,200a full tray before printing ✓
Remember the flow
Set the input to 0 · Divide the change by the change · Say it in the situation's units
The answer to an interpretation item is a sentence, and the units decide which one.
Now you try
Practice problem
Which of the following is the best interpretation of 32t in this context?
The function E is defined by E(t) = 1,450 − 32t, where E(t) is the number of tickets remaining for a concert t hours after tickets went on sale.
32 tickets leave the pile every hour, so after t hours 32t is the running total sold — the amount subtracted from the original 1,450. Check at t = 5: 32(5) = 160 tickets sold, and E(5) = 1,450 − 160 = 1,290 left. B describes E(t) itself, the whole expression rather than the term. C describes 32 alone, without the t, which is the per-hour rate. D is a different quantity: solving 1,450 − 32t = 0 gives t ≈ 45.3 hours, a number the term 32t never reports.
Hint: decide first whether the question is about a number, a term or the whole function. A number carries the units of one side of the story; a term carries a total; the function carries the quantity named in the stem.
Every lesson in the SAT Math Review has one of these. 10 questions, marked as you answer, with the working shown for each.
Question 1 of 10
The function f is defined by f(x) = 9x − 14. What is the value of f(6)?
f(6) = 9(6) − 14 = 54 − 14 = 40.
Question 2 of 10
The graph of the linear function f is shown, where y = f(x). What is the x-intercept of the graph of f?
The x-intercept is where the line crosses the x-axis (y = 0); the graph crosses at x = 3, so the x-intercept is (3, 0). (0, 6) is the y-intercept.
Question 3 of 10
A gym charges an enrollment fee of 50 dollars plus 28 dollars per month. Which function gives the total cost c(m), in dollars, of a membership for m months?
The monthly charge multiplies the months and the enrollment fee is paid once: c(m) = 28m + 50.
Question 4 of 10
The linear function g is defined by g(x) = kx + 9, where k is a constant. If g(4) = 1, what is the value of k?
g(4) = 4k + 9 = 1, so 4k = −8 and k = −2.
Question 5 of 10
In the xy-plane, the graph of the linear function f contains the points (0, −5) and (6, 13). Which equation defines f, where y = f(x)?
Slope = (13 − (−5))/(6 − 0) = 3 and the y-intercept is −5, so f(x) = 3x − 5.
Question 6 of 10
The function h is defined by h(x) = 12,400 − 780x, where h(x) is the altitude, in feet, of an airplane x minutes after it began descending. Which of the following is the best interpretation of the y-intercept of the graph of y = h(x) in the xy-plane?
The y-intercept is h(0) = 12,400, the altitude at the moment the descent began. The slope −780 is the descent per minute, and the x-intercept (about 15.9) is when it reaches the ground.
Question 7 of 10
For the linear function f, the table gives three values of x and their corresponding values of f(x). If h(x) = f(x) + 8, which equation defines h?
The functions f and g are defined by f(x) = (2/3)x + 7 and g(x) = (1/3)x − 12. If the function h is defined by h(x) = f(x) + g(x), what is the x-coordinate of the x-intercept of the graph of y = h(x) in the xy-plane?
A car rental costs 65 dollars for the first day and 42 dollars for each additional day. Which function f gives the total cost, in dollars, of renting the car for x days, where x ≥ 1?
There are x − 1 additional days: f(x) = 65 + 42(x − 1) = 42x + 23. Check: f(1) = 65 and f(2) = 107.
Question 10 of 10
The graph of y = f(x) − 4 is shown, where f is a linear function. Which equation defines f?
The line shown passes through (0, 2) and (4, 12), so f(x) − 4 = (5/2)x + 2. Adding 4 gives f(x) = (5/2)x + 6.
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