SAT Math · Algebra

Linear functions and slope on the SAT

A linear function changes by the same amount every step, so it can be written f(x)=mx+bf(x) = mx + b. The slope mm is the rate of change: rise over run, or the "per" number in a word problem. The intercept bb is the starting value, the output when x is 0. SAT questions ask you to find m and b from two points, a table or a graph, and to say what they mean.

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The key idea

Slope is change in output divided by change in input. Take any two points on the line, subtract in the same order on top and bottom:

m=y2−y1x2−x1f(x)=mx+bm = \frac{y_2 - y_1}{x_2 - x_1} \qquad f(x) = mx + b

In a word problem, the slope always has units of "output per input": dollars per hour, centimeters per week, gallons per minute. The intercept has the output's units: dollars, centimeters, gallons.

Three ways the SAT writes the same line
FormLooks likeWhat you can read off
Slope-intercepty=mx+by = mx + bSlope mm, y-intercept bb
Point-slopey−y1=m(x−x1)y - y_1 = m(x - x_1)Slope mm, a point (x1,y1)(x_1, y_1)
StandardAx+By=CAx + By = CSlope −AB-\frac{A}{B}, y-intercept CB\frac{C}{B}, x-intercept CA\frac{C}{A}

Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals, so their product is −1-1: a slope of 23\frac{2}{3} pairs with −32-\frac{3}{2}.

Worked examples

Example 1: the line through two points

Problem Find the equation of the line through (2,5)(2, 5) and (6,17)(6, 17).

  1. Find the slope. Subtract in the same order on top and bottom.
    m=17−56−2=124=3m = \frac{17 - 5}{6 - 2} = \frac{12}{4} = 3
  2. Put one point into y=3x+by = 3x + b to find b. Using (2,5)(2, 5):
    5=3(2)+b5 = 3(2) + b
  3. Solve for b.
    b=−1b = -1
  4. Check with the other point: 3(6)−1=173(6) - 1 = 17. It works.

Answer y=3x−1y = 3x - 1

Example 2: build the function from a table, then explain it

Problem A candle burns at a steady rate. After 1 hour it is 21 cm tall, after 3 hours 15 cm, and after 6 hours 6 cm. Write the height h(t)h(t) after t hours and say what each number means.

  1. Slope from the first two rows: change in height over change in time.
    m=15−213−1=−3m = \frac{15 - 21}{3 - 1} = -3
  2. Make sure the third row fits the same rate: from hour 3 to hour 6 the height drops 9 cm in 3 hours, which is also −3-3 per hour.
  3. The intercept is the height at t=0t = 0, not at t=1t = 1. Go back one hour from 21 cm by adding 3.
    h(0)=21+3=24h(0) = 21 + 3 = 24
  4. So h(t)=−3t+24h(t) = -3t + 24. The candle started 24 cm tall and loses 3 cm each hour. It burns out when h(t)=0h(t) = 0.
    −3t+24=0-3t + 24 = 0

Answer h(t)=−3t+24h(t) = -3t + 24: it starts at 24 cm, shrinks 3 cm per hour, and burns out after 8 hours.

Example 3: a perpendicular line from standard form

Problem Line k is perpendicular to the line 4x−6y=94x - 6y = 9 and passes through (4,−1)(4, -1). What is the y-intercept of line k?

  1. Slope of the given line is −AB-\frac{A}{B}, with A=4A = 4 and B=−6B = -6.
    m=−4−6=23m = -\frac{4}{-6} = \frac{2}{3}
  2. Perpendicular slope: flip it and change the sign.
    mk=−32m_k = -\frac{3}{2}
  3. Use y=−32x+by = -\frac{3}{2}x + b with the point (4,−1)(4, -1).
    −1=−32(4)+b-1 = -\frac{3}{2}(4) + b
  4. So −1=−6+b-1 = -6 + b.
    b=5b = 5

Answer The y-intercept is 5, at the point (0,5)(0, 5).

Example 4 (SAT-hard): skip the equation

Problem For a linear function f, f(2)=11f(2) = 11 and f(5)=20f(5) = 20. What is f(10)−f(4)f(10) - f(4)?

  1. Slope from the two given points.
    m=20−115−2=3m = \frac{20 - 11}{5 - 2} = 3
  2. For a line, a change in output is always slope times change in input. The inputs 10 and 4 are 6 apart.
    f(10)−f(4)=3⋅6=18f(10) - f(4) = 3 \cdot 6 = 18
  3. Check by building f: f(x)=3x+5f(x) = 3x + 5, so f(10)=35f(10) = 35 and f(4)=17f(4) = 17, and 35−17=1835 - 17 = 18.

Answer 18

Common mistakes

  • Mixing the subtraction order. 17−52−6\frac{17 - 5}{2 - 6} gives the wrong sign. Fix: whichever point goes first on top goes first on the bottom too.
  • Calling A the slope in Ax+By=CAx + By = C. The slope of 4x−6y=94x - 6y = 9 is 23\frac{2}{3}, not 4. Fix: solve for y, or use −AB-\frac{A}{B}.
  • Using the first table row as the starting value. If the table starts at t=1t = 1, that row is not the intercept. Fix: the intercept is the output when the input is exactly 0.
  • Forgetting the sign on a perpendicular slope. The reciprocal of 23\frac{2}{3} is 32\frac{3}{2}, but perpendicular needs −32-\frac{3}{2}. Fix: check that the two slopes multiply to −1-1.
  • Explaining the slope with the wrong units. In h(t)=−3t+24h(t) = -3t + 24, 3 is centimeters per hour, not hours per centimeter. Fix: say "output units per one input unit".

Quick methods

Practice

5 SAT-style questions

  1. What is the slope of the line through (−3,4)(-3, 4) and (5,−8)(5, -8)?

    1. −32-\frac{3}{2}
    2. −23-\frac{2}{3}
    3. 32\frac{3}{2}
    4. −6-6
    Show answer

    Answer: −32-\frac{3}{2}

    Rise over run: −8−45−(−3)=−128=−32\frac{-8 - 4}{5 - (-3)} = \frac{-12}{8} = -\frac{3}{2}. The run is 8, not 2: subtracting a negative adds. −23-\frac{2}{3} puts run over rise.

  2. What is the slope of the line 3x+5y=303x + 5y = 30?

    1. −35-\frac{3}{5}
    2. 35\frac{3}{5}
    3. −53-\frac{5}{3}
    4. 33
    Show answer

    Answer: −35-\frac{3}{5}

    Solve for y: 5y=−3x+305y = -3x + 30, so y=−35x+6y = -\frac{3}{5}x + 6. The 3 is the x coefficient in standard form, not the slope.

  3. The water in a tank, in gallons, t minutes after a pump starts is W(t)=850−25tW(t) = 850 - 25t. What is the best interpretation of 25?

    1. The tank starts with 25 gallons.
    2. The pump removes 25 gallons each minute.
    3. The tank is empty after 25 minutes.
    4. The pump adds 25 gallons each minute.
    Show answer

    Answer: The pump removes 25 gallons each minute.

    25 is multiplied by t, so it is a rate, and its minus sign means the water goes down. The tank starts with 850 gallons, and it empties after 850÷25=34850 \div 25 = 34 minutes.

  4. Which line is perpendicular to y=4x−7y = 4x - 7?

    1. y=4x+7y = 4x + 7
    2. y=−4x+1y = -4x + 1
    3. y=−14x+3y = -\frac{1}{4}x + 3
    4. y=14x−7y = \frac{1}{4}x - 7
    Show answer

    Answer: y=−14x+3y = -\frac{1}{4}x + 3

    Perpendicular slopes multiply to −1-1: 4⋅(−14)=−14 \cdot (-\frac{1}{4}) = -1. y=4x+7y = 4x + 7 is parallel. A slope of −4-4 changes only the sign, and 14\frac{1}{4} only flips the number. You need both changes.

  5. Student-produced response: a linear function p has p(1)=−2p(1) = -2 and p(6)=18p(6) = 18. What is p(0)p(0)?

    Show answer

    Answer: -6

    Slope: 18−(−2)6−1=4\frac{18 - (-2)}{6 - 1} = 4. Going from x=1x = 1 back to x=0x = 0 subtracts one slope: −2−4=−6-2 - 4 = -6. So p(x)=4x−6p(x) = 4x - 6.

Frequently asked questions

Is slope the same as rate of change?

For a line, yes. The slope is the rate of change, and it is the same between any two points. For curves like parabolas the rate keeps changing, which is one way to tell a linear function from a nonlinear one in a table.

How do I find the slope from standard form?

For Ax+By=CAx + By = C, the slope is −AB-\frac{A}{B}. If you do not trust the shortcut, solve for y: subtract Ax from both sides and divide by B. The number in front of x is the slope.

What does the y-intercept mean in a word problem?

It is the value of the output when the input is 0: the starting amount, the flat fee, or the height at time zero. Say it with units, like "the plan costs $12 before any texts are sent".

How do I tell if two lines are parallel or perpendicular?

Write both in y=mx+by = mx + b form and compare slopes. Equal slopes with different intercepts mean parallel. Slopes that multiply to −1-1 mean perpendicular. Equal slopes and equal intercepts mean it is the same line.

Sources

  1. College Board: SAT Math, Algebra skills, accessed October 1, 2026

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  • “Why is the starting height 24 and not 21?”
  • “What does the slope mean in this problem, in plain words?”
  • “Give me a table and let me find the equation myself.”
  • “I keep getting the sign of the slope wrong. Can you watch me do one?”

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