Digital SAT · Math · Unit 1 · Algebra

Lesson 1.4 · Linear Functions

On the test: the single most frequent topic — expect 4–6 questions every test

Big idea. A linear function changes by the SAME amount every time x goes up by 1. That fixed amount is the slope m; where it starts (at x = 0) is the y-intercept b. So every linear function is y = mx + b: start at b, then add m for every step. Every question here is about reading or finding those two numbers.

🎛 Move it · build a line

Slide rise, run and $b$ . The gold step from the $y$ -intercept shows what the slope MEANS: go right by run, up by rise, and you are on the line again.

y = \frac{2}{3} x + 1

rise2run3b1

Slope = rise ÷ run = 2 ÷ 3. From (0, 1) go 3 right and 2 up — you land on the line again.

🧑‍🏫 Rule 1 · Slope from two pointsstep 1 of 4

1
the idea
$m = \frac{change in y}{change in x} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$
Slope is how much $y$ changes for each 1 that $x$ changes. Measure both changes between two points and divide.

🧑‍🏫 Rule 2 · The intercepts: set the OTHER letter to 0step 1 of 5

1
y-intercept
$x = 0$
Where the line crosses the $y$ -axis, every point has $x = 0$ .

🧑‍🏫 Rule 3 · Slope of Ax + By = C: get y alonestep 1 of 5

1
📋 The question · SAT · December 2023 · Math 800 Book Q34
What is the slope of the graph of $8 x - 4 y = - 12$ ?
1. A) $- 2$
2. B) $- \frac{2}{3}$
3. C) $\frac{2}{3}$
4. D) $2$
the question
$8 x - 4 y = - 12$
December 2023: what is the slope?

Why it works · why “rise over run” is the rate

Between any two points of a straight line the ratio is the same — that is what “straight” means. So one pair of points tells you the rate for the whole line.

Trap · which intercept?

“

y

-coordinate of the

y

-intercept” is just

b

. “

x

-intercept” needs

y = 0

and then solving. Read which one is asked.

Desmos move · read it off

Type the equation, then tap where the line meets each axis — Desmos labels the intercepts. For slope, tap two points and divide.

Fast lane · graph questions

Find two points where the line crosses grid corners exactly (often the intercepts). Never guess between grid lines.

Real question 1SAT · November 2023 · Math 800 Book Q27

$figure from the test$

Figure from the test paper — our labels appear after you answer.

The graph of the linear equation $15 x + B y = 60$ is shown, where $B$ is a constant. What is the value of $B$ ?

🧑‍🏫 Rule 4 · Slope and a point → y = mx + bstep 1 of 5

1
📋 The question · SAT · March 2024 · Math 800 Book Q55
The graph of $y = f (x)$ has slope $\frac{4}{5}$ and passes through $(20, 19)$ . Which equation defines $f$ ?
1. A) $f (x) = \frac{4}{5} x + \frac{19}{20}$
2. B) $f (x) = \frac{4}{5} x + \frac{24}{5}$
3. C) $f (x) = \frac{4}{5} x + 19$
4. D) $f (x) = \frac{4}{5} x + 3$
the question
$m = \frac{4}{5}, through (20, 19)$
March 2024: which equation defines $f$ ?

🧑‍🏫 Rule 5 · Parallel and perpendicularstep 1 of 5

1
parallel
$m_{2} = m_{1}$
Parallel lines rise at the same rate — same slope, different intercept.

🧑‍🏫 Rule 6 · “$408 for the first hour and $204 for each ADDITIONAL hour”step 1 of 5

1
📋 The question · SAT · June 2023 · Math 800 Book Q7
A business consultant charges $408 for the first hour and $204 for each additional hour of work. Which of the following functions gives the charge $C (h)$ , in dollars, for $h$ hours of work, where $h$ is a positive integer?
1. A) $C (h) = 204 h + 204$
2. B) $C (h) = 204 h + 408$
3. C) $C (h) = 408 h + 204$
4. D) $C (h) = 408 h + 612$
the trap
$C (h) \neq = 408 + 204 h$
June 2023. The first hour is already paid for by the 408 — it must not be charged twice.

🧑‍🏫 Rule 7 · “Infinitely many solutions” = the SAME linestep 1 of 4

1
📋 The question · SAT · August 2024 · Math 800 Book Q108
$y = \frac{2}{5} x + 7$ ; second equation $y = m x + b$ ; infinitely many solutions. What is $b$ ?
1. A) $- 7$
2. B) $- \frac{1}{7}$
3. C) $\frac{1}{7}$
4. D) $7$
the question
$y = \frac{2}{5} x + 7, y = m x + b$
August 2024: the system has infinitely many solutions. What is $b$ ?

Fast lane · test the options with one point

For “which equation defines the line”, put a known point (often the

y

-intercept) into each option — the wrong ones usually die at once.

Real question 2SAT · June 2024 · Math 800 Book Q91

$figure from the test$

Figure from the test paper — our labels appear after you answer.

Line $k$ is shown in the $x y$ -plane. Line $j$ (not shown) is perpendicular to line $k$ and passes through the point $(- 12, - 19)$ . Which equation defines line $j$ ?

🧑‍🏫 Rule 8 · f(x) is a machine: put in x, get out ystep 1 of 5

1
read it
$f (x) = 100 x + 2$
“ $f$ of $x$ ” is the OUTPUT when $x$ goes in. It is NOT $f$ times $x$ .

🧑‍🏫 Rule 9 · f(x) + k moves the graph up kstep 1 of 4

1
📋 The question · SAT · May US 2024 · Math 800 Book Q63
$g (x) = 3 (16 x - 17)$ . What is the $y$ -coordinate of the $y$ -intercept of $y = g (x) - 5$ ?
1. A) $- 56$
2. B) $- 51$
3. C) $- 22$
4. D) $- 17$
the question
$g (x) = 3 (16 x - 17), y = g (x) - 5$
May US 2024: $y$ -intercept of $g (x) - 5$ .

🧑‍🏫 Rule 10 · A function built from anotherstep 1 of 4

1
📋 The question · SAT · March 2024 · Math 800 Book Q54
A table gives $g (- 25) = 4$ , $g (- 9) = 0$ , $g (15) = 6$ , where $g (x) = \frac{f ( x )}{x + 5}$ and $f$ is linear. What is the $y$ -coordinate of the $y$ -intercept of $y = f (x)$ ?
(type the answer)
the question
$g (x) = \frac{f ( x )}{x + 5}$
March 2024: a table gives $g$ ; we need the $y$ -intercept of $f$ .

Real question 3SAT · October US 2024 · Math 800 Book Q151

$figure from the test$

The graph of $y = f (x) - 11$ is shown. Which equation defines the linear function $f$ ?

🧑‍🏫 Rule 11 · Slope is the rate, intercept is the startstep 1 of 5

1
📋 The question · SAT · December 2023 · Math 800 Book Q37
Harper deposited a fixed amount into her bank account each month. $f (t) = 50 + 55 t$ gives the amount in the account after $t$ monthly deposits. What is the best interpretation of 55?
1. A)With each monthly deposit, the amount in Harper’s bank account increased by $55
2. B)Harper made a total of 55 monthly deposits
3. C)Before any deposits, the account had $55
4. D)After 1 monthly deposit, the account had $55
the model
$f (t) = 50 + 55 t$
December 2023: Harper’s account after $t$ monthly deposits.

🧑‍🏫 Rule 12 · Reading f(a) = b in wordsstep 1 of 4

1
📋 The question · SAT · August US 2023 · Math 800 Book Q23
$w (t) = 3.1 t - 1.7$
The function $w$ models the weight, in pounds, of a pig $t$ months after it was born. Which statement is the best interpretation of $w (7) = 20$ ?
1. A)The pig weighed 20 pounds 7 months after it was born.
2. B)The pig weighed 7 pounds 20 months after it was born.
3. C)The pig’s weight increased by 20 pounds during the 7th month.
4. D)The pig’s weight increased by 7 pounds during the 20th month.
the question
$w (7) = 20$
August US 2023: $w (t)$ = pig’s weight $t$ months after birth.

🧑‍🏫 Rule 13 · Graphs of ax + by = c in a storystep 1 of 5

1
📋 The question · SAT · May 2024 · Math 800 Book Q80
$figure from the test$
A 600-foot row; each oak needs 40 ft and each pine 30 ft. $40 x + 30 y = 600$ models oak trees $x$ and pine trees $y$ . Which graph represents this relationship?
1. A)Graph A
2. B)Graph B
3. C)Graph C
4. D)Graph D
the question
$40 x + 30 y = 600$
May 2024: oak trees $x$ (40 ft each), pine trees $y$ (30 ft each) in a 600-ft row.

Trap · “each” vs “total”

55 t

is the total from

t

deposits; 55 alone is per deposit. Options love to swap them.

Why it works · why the intercept is the start

At the start nothing has happened yet: input 0. Plug 0 in and only

b

is left.

Real question 4SAT · May 2024 · Math 800 Book Q80

$figure from the test$

A botanist wants to plant certain types of oak trees and pine trees in a 600-foot-long row. Each oak tree requires 40 feet of space and each pine tree 30 feet. The equation $40 x + 30 y = 600$ models the possible number of oak trees, $x$ , and pine trees, $y$ . Which graph represents this relationship?

🧑‍🏫 Rule · Horizontal and vertical linesstep 1 of 4

1
horizontal
$y = c \Rightarrow m = 0$
Every point has the same height; $y$ never changes, so the rise is 0.

🧑‍🏫 Rule · Point-slope formstep 1 of 3

1
the form
$y - y_{1} = m (x - x_{1})$
When you know a slope and any point — no need to find $b$ first.

Take with you

▸Slope $= \frac{Δ y}{Δ x}$ , same order top and bottom. Standard form: $- \frac{A}{B}$ .
▸ $y$ -intercept: put $x = 0$ . $x$ -intercept: put $y = 0$ .
▸Slope + point: write $y = m x + b$ , plug the point, solve for $b$ .
▸Parallel = same slope; perpendicular = flip and change sign.
▸“First hour, then each additional” → first + rate × $(h - 1)$ .
▸ $f (a)$ : input in brackets, output outside. $f (x) + k$ moves up $k$ .
▸In a story: slope = amount per unit, intercept = starting amount; for $a x + b y = c$ , each intercept is “all of one kind”.

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