A constant rate of change is a straight line in disguise. The work is finding the rate, then finding where the story started.
Which equation defines g?
The table gives four values of x and the corresponding values of g(x) for the linear function g. Which equation defines g?
x
3
5
7
9
g(x)
26
38
50
62
g(x) climbs 12 every time x climbs 2, so the rate is 122 = 6 per unit of x, not 12. Then 26 = 6(3) + b gives b = 8, and g(x) = 6x + 8. Check the far end: 6(9) + 8 = 62 ✓. C uses 12 as the slope — the difference between two rows, never divided by the two-unit step across — and it fits the first row, 12(3) − 10 = 26, which is exactly why it is tempting. B has the right rate but treats 26 as the starting value, when 26 is g at x = 3, not at x = 0. D makes both slips at once.
1
A table is linear when equal steps across give equal steps down.
Write the differences in the margin. Two columns of differences — one for x, one for y — settle the question before any algebra starts.
x
g(x)
step across
step down
3
26
—
—
5
38
+2
+12
7
50
+2
+12
9
62
+2
+12
38 − 26 = 12, 50 − 38 = 12, 62 − 50 = 12
the same step down every row
5 − 3 = 2, 7 − 5 = 2, 9 − 7 = 2
and the same step across every row
Both columns constant means linear. If the steps down grow while the steps across stay the same, the table is not linear at all — that is a different lesson.
2
The rate of change is the step down divided by the step across.
m = Δy/Δx. The difference between two rows is only the slope when the rows are one x-unit apart.
m = 38 − 265 − 3
any two rows will do — the rate is constant
m = 122 = 6
6 per unit of x, not 12
check with the outer rows: 62 − 269 − 3 = 366 = 6
same answer, which is what “linear” promises
Δy/Δxany two rowsdivide, never just subtract
The step across is hardly ever 1
In the opening table the x-values were 3, 5, 7, 9. Reading the slope as 12 — the step down, undivided — produced choice C, g(x) = 12x − 10, which even matches the first row. It fails at the second: 12(5) − 10 = 50, but the table says 38. One extra row is all the check you need, and it costs five seconds.
3
Run the table back to x = 0 to find the starting value.
SAT tables almost never include x = 0, because the y-intercept is the part they want you to work for. There are two ways back, and they agree.
substitute a row: 26 = 6(3) + b
put any (x, y) pair into y = mx + b
26 = 18 + b, so b = 8
one line of arithmetic
or step backwards: 26 − 6 − 6 − 6 = 8
x = 3 → 2 → 1 → 0, subtracting the rate each time
g(x) = 6x + 8
and 6(7) + 8 = 50 ✓ against the table
Where the calculator earns its keep
Desmos takes a table directly: click the plus, choose Table, type the x-values in one column and the g-values in the other, and the four dots appear. Type y ~ mx + b underneath and Desmos reports m = 6 and b = 8 for you. It is worth knowing for a table you cannot see the pattern in — but on a two-row difference like this one, the pencil is faster.
4
A sentence becomes a model the same way: one number multiplies, one stands alone.
“A repair service charges a fee of 45 dollars plus 22 dollars for each hour of work.”
“for each hour” → 22 multiplies h
per, each, every, an hour — all mean the rate
“a fee of 45 dollars” → charged once
it does not depend on h, so it is the constant
C(h) = 22h + 45
and C(0) = 45, the fee for showing up
C(3) = 22(3) + 45 = 66 + 45 = 111
a three-hour job costs 111 dollars
Units settle every argument. 22 is dollars per hour, so it must meet something measured in hours; 45 is dollars already, so it must stand by itself. C(h) = 45h + 22 fails that test on sight.
Watch the direction of the story
“A tank starts with 60 gallons and loses 4 gallons a minute” is V(t) = 60 − 4t, not 60 + 4t. A rate that shrinks the quantity is negative, and the SAT counts on a reader who spots the number 4 and stops reading. Decreases by, loses, drops, is reduced by — all mean a minus sign.
5
Once you have the model, the prediction is one substitution.
A tank is being filled. The table gives the volume V, in gallons, after t minutes.
t (minutes)
4
6
8
10
V (gallons)
58
73
88
103
m = 73 − 586 − 4 = 152 = 7.5
7.5 gallons per minute — decimals are fine here
58 = 7.5(4) + b = 30 + b, b = 28
28 gallons were already in the tank
V(t) = 7.5t + 28
check: 7.5(10) + 28 = 103 ✓
V(25) = 7.5(25) + 28 = 187.5 + 28 = 215.5
t = 25 is well past the last row — that is the point
215.5 gallons after 25 minutes
The SAT asks for a value outside the table on purpose. Continuing the table row by row would take another eight rows; the equation does it in one line, and it is the only method that works when the question asks for t = 400.
running it backwards: 7.5t + 28 = 163
“after how many minutes does the tank hold 163 gallons?”
7.5t = 135, t = 18
same model, solved for the other letter
6
Say what each number means in the situation's own words.
Interpretation items hand you the finished model and ask what one piece of it represents.
V(t) = 7.5t + 28
Piece
What it is
In this context
28
value when t = 0
the tank held 28 gallons before filling began
7.5
rate of change
the volume increases by 7.5 gallons each minute
V(t)
the output
the volume, in gallons, after t minutes
constant → the value at zerocoefficient → per one unitsay the units out loud
A model is only honest inside its story
V(t) = 7.5t + 28 says the tank holds 7,528 gallons after 1,000 minutes, which no tank does. Interpretation choices sometimes offer “the total volume of the tank” for the constant 28 — it is not; it is the volume at t = 0. Read the constant as a starting value, and read a far-out prediction as what the model says rather than what the world guarantees.
✓
Check: put a table row back into your equation.
the opener6(7) + 8 = 50matches the x = 7 row ✓
the repair fee22(0) + 45 = 45no hours worked, fee still charged ✓
the tank7.5(8) + 28 = 88matches the t = 8 row ✓
Remember the flow
Differences both ways · Divide for the rate · Back to x = 0 for b · Substitute to predict
Then test your equation against a row you did not use.
Now you try
Practice problem
What is the value of y?
The table gives three values of x and the corresponding values of y for a linear relationship. What is the value of y when x = 20?
x
5
9
13
y
41
29
17
y drops 12 while x rises 4, so the rate is −124 = −3. Then 41 = −3(5) + b gives b = 56 and y = −3x + 56, so at x = 20, y = −60 + 56 = −4. Check a row you did not use: −3(13) + 56 = 17 ✓. A takes −12 as the slope without dividing by the four-unit step, giving y = −12x + 101 and −139. C treats x = 20 as the next row and subtracts 12 once more from 17, but the next row would be x = 17, not x = 20. D is 56, the y-intercept — a real number in this problem, just not the one asked for.
Hint: the rows step by 4, not by 1. Find the rate per single unit of x first, then either build y = mx + b or step up from x = 13 seven units at −3 each — both land in the same place.
