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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.

Unlimited practice with no daily limit, plus every full-length test and Challenge test. Pro is $20 a month or $180 a year. Checkout isn’t open yet, so nothing can be bought today.

98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

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Linear functions · Lesson 1

From a table or graph to an equation

To write f(x)=mx+bf(x) = mx + b you need two numbers: the slope mm and the intercept bb. A table or a graph gives you points. Your job is to turn them into those two numbers.

Slope from two points. For points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2), m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}: the change in output over the change in input, or rise over run. Subtract in the same order on top and bottom. From (1,4)(1, 4) to (5,16)(5, 16): m=16−45−1=124=3m = \frac{16 - 4}{5 - 1} = \frac{12}{4} = 3.

**In a table, check the step in xx.** If xx goes 0, 2, 4 and f(x)f(x) goes 3, 7, 11, the output rises 4 from row to row, but each row is 2 units of xx, so m=42=2m = \frac{4}{2} = 2.

The intercept. If a point has x=0x = 0, its yy-value is bb. Otherwise, substitute any point into y=mx+by = mx + b and solve: with m=3m = 3 and the point (1,4)(1, 4), 4=3(1)+b4 = 3(1) + b, so b=1b = 1. A graph described in words gives the same information. A line that crosses the yy-axis at (0,5)(0, 5) has b=5b = 5. An xx-intercept at (6,0)(6, 0) is just another point, where the output is 0.

Check with both points. Substitute each given point into your equation. A choice built from a wrong slope often still passes through the point used to find bb, so only the other point catches it.

Worked example Easy

The table shows three values of the linear function ff.

xxf(x)f(x)
03
27
411

Which equation defines ff?

  1. f(x)=4x+3f(x) = 4x + 3
  2. f(x)=12x+3f(x) = \frac{1}{2}x + 3
  3. f(x)=2x+3f(x) = 2x + 3Answer
  4. f(x)=2x+7f(x) = 2x + 7

How to solve it

  1. Find the slope from two rows. From (0,3)(0, 3) to (2,7)(2, 7): m=7−32−0=42=2m = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2.
  2. The row with x=0x = 0 gives the intercept: f(0)=3f(0) = 3, so b=3b = 3.
  3. So f(x)=2x+3f(x) = 2x + 3. Check the third row: 2(4)+3=112(4) + 3 = 11.

Why each choice is right or wrong

  • A. Incorrect. 4 is the change in f(x)f(x) from one row to the next, but xx goes up by 2 in each row. The slope is 42=2\frac{4}{2} = 2. This function gives f(2)=11f(2) = 11, not 7.
  • B. Incorrect. This divides the change in xx by the change in f(x)f(x), 24\frac{2}{4}, which is run over rise. Slope is the change in output over the change in input: 42=2\frac{4}{2} = 2.
  • C. Correct. From (0,3)(0, 3) to (2,7)(2, 7), the slope is 7−32−0=2\frac{7 - 3}{2 - 0} = 2, and f(0)=3f(0) = 3 gives b=3b = 3. Check: f(4)=2(4)+3=11f(4) = 2(4) + 3 = 11.
  • D. Incorrect. This uses f(2)=7f(2) = 7 as the intercept. The intercept is the output when x=0x = 0, which the table gives as 3. This function gives f(0)=7f(0) = 7.

Worked example Medium

The graph of the linear function ff in the xyxy-plane is a line that passes through the points (−2,9)(-2, 9) and (4,−3)(4, -3).

Which equation defines ff?

  1. f(x)=−2x+5f(x) = -2x + 5Answer
  2. f(x)=2x+13f(x) = 2x + 13
  3. f(x)=−12x+8f(x) = -\frac{1}{2}x + 8
  4. f(x)=−2x+9f(x) = -2x + 9

How to solve it

  1. Slope: m=−3−94−(−2)=−126=−2m = \frac{-3 - 9}{4 - (-2)} = \frac{-12}{6} = -2.
  2. Substitute one point into y=−2x+by = -2x + b. Using (−2,9)(-2, 9): 9=−2(−2)+b=4+b9 = -2(-2) + b = 4 + b, so b=5b = 5.
  3. So f(x)=−2x+5f(x) = -2x + 5. Check the other point: −2(4)+5=−3-2(4) + 5 = -3. Both points fit.

Why each choice is right or wrong

  • A. Correct. The slope is −3−94−(−2)=−2\frac{-3 - 9}{4 - (-2)} = -2, and 9=−2(−2)+b9 = -2(-2) + b gives b=5b = 5. Both points fit: f(−2)=9f(-2) = 9 and f(4)=−3f(4) = -3.
  • B. Incorrect. This subtracts in opposite orders on top and bottom, −3−9−2−4=2\frac{-3 - 9}{-2 - 4} = 2, then finds bb from 9=2(−2)+b9 = 2(-2) + b. The line passes through (−2,9)(-2, 9) but gives f(4)=21f(4) = 21, not −3-3.
  • C. Incorrect. This puts the change in xx over the change in yy, 6−12=−12\frac{6}{-12} = -\frac{1}{2}, then finds bb from 9=−12(−2)+b9 = -\frac{1}{2}(-2) + b. The line passes through (−2,9)(-2, 9) but gives f(4)=6f(4) = 6, not −3-3.
  • D. Incorrect. The slope is right, but 9 is the yy-value at x=−2x = -2, not at x=0x = 0, so it isn't the intercept. This function gives f(−2)=13f(-2) = 13, not 9.

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