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A Peakscor full-length test in progress: Reading and Writing, Test A, Module 1 of 4, question 7 of 27, with the passage, four answer choices, a highlighter, a 24:18 timer and Flag, Back and Next buttons.

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  2. 2

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    Come back tomorrow and climb. Progress shows your accuracy and your strongest skills.

Built on the digital SAT blueprint.

Every question is tagged by section, domain, skill and difficulty, so you can practice exactly what you need. Full tests run like test day: two Reading & Writing modules, a break, then two Math modules.

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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.

Test day

What test day looks like.

Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

  1. Reading & Writing · Module 1

    27 questions32 min

  2. Reading & Writing · Module 2

    27 questions32 min

    Adapts to how you did on Module 1.

  3. Break

    10 min

  4. Math · Module 1

    22 questions35 min

  5. Math · Module 2

    22 questions35 min

    Adapts to how you did on Module 1.

98 questions · 2 hr 14 min of testing, plus the break

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FAQ

Questions, answered.

The Daily, plus 36 practice questions every day: 9 easy, 9 medium, 9 hard and 9 challenging, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too, as are the 20-question diagnostic and PEAK Survivor’s first map. No card needed.

Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test, and every game map and level. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.

98 questions across four modules, in the same order as the digital SAT: 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 are built from harder material in every module.

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.

A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.

Every day at midnight Central Time. You get a fresh 9 easy, 9 medium, 9 hard and 9 challenging.

No. Peakscor is an independent SAT practice tool, not affiliated with the College Board. Scores shown are estimates.

Your next point starts today.

Free, no card.

Nonlinear functions · Lesson 5

Absolute value, radical and rational functions

Absolute value. The graph of f(x)=a∣x−h∣+kf(x) = a|x - h| + k is a V with its vertex at (h,k)(h, k), read the same way as vertex form of a quadratic. If aa is positive the V opens up and kk is the minimum value; if aa is negative it opens down and kk is the maximum. The two sides are straight lines with slopes aa and −a-a.

Square root. You can't take the square root of a negative number, so inside\sqrt{\text{inside}} needs inside ≥0\ge 0. For g(x)=2x−6+1g(x) = \sqrt{2x - 6} + 1, solve 2x−6≥02x - 6 \ge 0: the domain is x≥3x \ge 3. The graph starts at (3,1)(3, 1) and rises to the right, getting less steep. The domain depends on what is under the root, not on whether xx itself is positive.

Rational functions are fractions with xx in the denominator. They are undefined wherever the denominator is 0. If that factor doesn't cancel with the numerator, the graph has a vertical asymptote there: it shoots up or down beside a vertical line it never touches. For f(x)=2x−1x+3f(x) = \frac{2x - 1}{x + 3}, the vertical asymptote is x=−3x = -3. (If the same factor is in the numerator and cancels, that xx is still excluded from the domain, but the graph has a hole there instead of an asymptote.)

A rational function can also level off toward a horizontal asymptote as xx gets very large or very negative. When the top and bottom are both linear, ax+bcx+d\frac{ax + b}{cx + d}, the horizontal asymptote is y=acy = \frac{a}{c}, the ratio of the xx-coefficients. For f(x)=1x−2+1f(x) = \frac{1}{x - 2} + 1, the asymptotes are x=2x = 2 and y=1y = 1.

The graph of y = \frac{1}{x - 2} + 1 with its asymptotes−3−2−11234567−4−3−2−11234560xy(3, 2)
The graph of y=1x−2+1y = \frac{1}{x - 2} + 1 with its asymptotes Two separate branches. A dashed vertical line at x = 2 and a dashed horizontal line at y = 1 are the asymptotes. The right branch passes through (3, 2). The left branch crosses the y-axis between y = 0 and y = 1, below the dashed line, and crosses the x-axis at (1, 0). Both branches bend toward the dashed lines without touching them.

Worked example Easy

The graph of y = f(x)−3−2−1123456789−6−5−4−3−2−1123456xy(3, -4)(-1, 0)(7, 0)(0, -1)
The graph of y=f(x)y = f(x) A V-shaped graph opening upward with its vertex, the lowest point, at (3, -4). It crosses the x-axis at (-1, 0) and (7, 0) and the y-axis at (0, -1).

The graph of y=f(x)y = f(x) is shown. Which equation defines ff?

  1. f(x)=∣x+3∣−4f(x) = |x + 3| - 4
  2. f(x)=∣x−4∣−3f(x) = |x - 4| - 3
  3. f(x)=∣x−3∣+4f(x) = |x - 3| + 4
  4. f(x)=∣x−3∣−4f(x) = |x - 3| - 4Answer

How to solve it

  1. The vertex is (3,−4)(3, -4), so h=3h = 3 and k=−4k = -4 in ∣x−h∣+k|x - h| + k.
  2. That gives f(x)=∣x−3∣−4f(x) = |x - 3| - 4.
  3. Check the intercepts: f(0)=3−4=−1f(0) = 3 - 4 = -1, f(−1)=4−4=0f(-1) = 4 - 4 = 0 and f(7)=4−4=0f(7) = 4 - 4 = 0.

Why each choice is right or wrong

  • A. Incorrect. This reads the sign of hh backward. ∣x+3∣|x + 3| puts the vertex at x=−3x = -3.
  • B. Incorrect. This puts 4 inside the absolute value and 3 outside, the reverse of what the graph needs. ∣x−4∣−3|x - 4| - 3 has its vertex at (4,−3)(4, -3), but the graph's vertex is (3,−4)(3, -4): the number inside gives the xx-coordinate (3) and the number outside gives the yy-coordinate (−4-4).
  • C. Incorrect. This puts the vertex at (3,4)(3, 4), above the xx-axis. The constant outside is read as written, so a vertex at y=−4y = -4 needs −4-4.
  • D. Correct. The vertex (3,−4)(3, -4) gives ∣x−3∣−4|x - 3| - 4, and it matches the intercepts (0,−1)(0, -1), (−1,0)(-1, 0) and (7,0)(7, 0).

Worked example Medium

g(x)=2x−6+1g(x) = \sqrt{2x - 6} + 1

What is the domain of the function gg?

  1. All real numbers xx such that x≥−3x \ge -3
  2. All real numbers xx such that x≥0x \ge 0
  3. All real numbers xx such that x≥3x \ge 3Answer
  4. All real numbers xx such that x≥6x \ge 6

How to solve it

  1. The expression under the square root must not be negative: 2x−6≥02x - 6 \ge 0.
  2. Add 6: 2x≥62x \ge 6. Divide by 2: x≥3x \ge 3.
  3. Check: g(3)=0+1=1g(3) = \sqrt{0} + 1 = 1 is defined, and g(2)=−2+1g(2) = \sqrt{-2} + 1 is not.

Why each choice is right or wrong

  • A. Incorrect. This moves the −6-6 to the other side without changing its sign, writing 2x≥−62x \ge -6, and gets x≥−3x \ge -3. Adding 6 to both sides gives 2x≥62x \ge 6. At x=0x = 0, 2x−6=−62x - 6 = -6, so g(0)g(0) is undefined.
  • B. Incorrect. The restriction is on the expression under the root, not on xx. At x=1x = 1, 2x−6=−42x - 6 = -4, so g(1)g(1) is undefined even though xx is positive.
  • C. Correct. 2x−6≥02x - 6 \ge 0 gives x≥3x \ge 3.
  • D. Incorrect. This stops at 2x≥62x \ge 6 and reads off the 6 without dividing by 2. At x=4x = 4, 2x−6=22x - 6 = 2, so g(4)g(4) is defined even though 4<64 \lt 6.

Worked example Hard

The graph of y = f(x) and its asymptotes−6−5−4−3−2−112345−4−3−2−112345678xy(0, 3)(-2, 1)
The graph of y=f(x)y = f(x) and its asymptotes Two branches with a dashed vertical asymptote at x = -1 and a dashed horizontal asymptote at y = 2. The right branch is above y = 2 and crosses the y-axis at (0, 3). The left branch is below y = 2, passes through (-2, 1), and crosses the x-axis between x = -2 and x = -1.

The graph of the rational function ff is shown. Which of the following could define ff?

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

How to solve it

  1. The vertical asymptote x=−1x = -1 means the denominator is 0 at x=−1x = -1: the denominator is x+1x + 1. That rules out the first choice.
  2. The horizontal asymptote y=2y = 2 means the ratio of the xx-coefficients is 2. Of the remaining choices, only 2x+3x+1\frac{2x + 3}{x + 1} has 21=2\frac{2}{1} = 2.
  3. Check the points: f(0)=31=3f(0) = \frac{3}{1} = 3 and f(−2)=−1−1=1f(-2) = \frac{-1}{-1} = 1. Both match.

Why each choice is right or wrong

  • A. Incorrect. The denominator x−1x - 1 is zero at x=1x = 1, so the vertical asymptote would be x=1x = 1, not x=−1x = -1.
  • B. Incorrect. This matches the vertical asymptote and even the point (0,3)(0, 3), but its horizontal asymptote is y=11=1y = \frac{1}{1} = 1, and f(−2)=1−1=−1f(-2) = \frac{1}{-1} = -1, not 1.
  • C. Incorrect. The horizontal asymptote would be y=3y = 3, and f(0)=2f(0) = 2, not 3. This swaps the numerator's coefficients.
  • D. Correct. The denominator gives the vertical asymptote x=−1x = -1, the ratio 21\frac{2}{1} gives the horizontal asymptote y=2y = 2, and the function passes through (0,3)(0, 3) and (−2,1)(-2, 1).

Trap Reading the asymptote from the wrong part

The vertical asymptote comes from the zero of the denominator, with its sign: x+1x + 1 in the denominator gives x=−1x = -1. A zero of the numerator (where the denominator isn't also 0) is an xx-intercept, not an asymptote. And the horizontal asymptote of ax+bcx+d\frac{ax + b}{cx + d} is ac\frac{a}{c}, not a ratio of the constants.

Trap Domain of a square root

The domain of 2x−6\sqrt{2x - 6} is not x≥0x \ge 0 and not x≥6x \ge 6. Set the whole expression under the root ≥0\ge 0 and solve it completely: x≥3x \ge 3.

Desmos When Desmos is faster

For a square root, find where the domain starts by hand: the graph starts where the expression under the root is 0, so for 2x−6+1\sqrt{2x - 6} + 1 it starts at x=3x = 3. Then graph the function in Desmos to confirm. Desmos draws the graph only where it is defined, so the curve should begin at x=3x = 3 and continue to the right.

Desmos doesn't draw asymptotes for you. Near a vertical asymptote you see the curve shoot up or down, and far out to the sides you see it level off. To test a guess, type the line, such as x = -1 or y = 2, and see whether the curve hugs it without touching.

To match an equation to a graph printed in the question, type each choice and check its asymptotes and a labeled point.

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Privacy Policy

Last updated October 7, 2026

Peakscor is an independent SAT® and AP® practice website at peakscor.com, built and run by its student founders in Houston, Texas. This policy explains what we collect, why, and what you can do about it. If anything here is unclear, email support@peakscor.com.

The short version

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Last updated October 5, 2026

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We work hard to make Peakscor accurate and reliable, but it is provided “as is.” Questions or explanations may sometimes contain mistakes, and the site may sometimes be unavailable.

10. Limit of liability

To the extent the law allows, Peakscor is not liable for indirect or consequential losses, and our total liability for any claim is limited to the amount you paid us in the 12 months before the claim, or $50 if you paid nothing.

11. Ending access

You can stop using Peakscor at any time. We may suspend accounts that break these terms. If we ever shut Peakscor down, we will refund the unused part of any paid subscription.

12. Governing law

These terms are governed by the laws of the State of Texas.

13. Changes

We may update these terms. We will change the date at the top, and tell you on the site about any important change before it takes effect.

14. Contact

support@peakscor.com

How our questions are made

Last updated October 5, 2026

Every question, passage, explanation and lesson on Peakscor is original SAT-style practice. None of it is an official SAT® or AP® question, and we are not affiliated with the College Board.

Our rules

  • We never copy or closely paraphrase College Board questions, or questions from any other publisher.
  • We never give College Board questions or materials to AI tools, and we never use them as examples to imitate.
  • The structure of the digital SAT is public: its sections, question types, skills and timing. We follow that structure, but the content inside it is ours.

How a question gets written

  • Math: most questions come from templates we wrote. Each template builds a new question and computes its answer, so the key is always checked by the math itself.
  • Reading and Writing: we use AI tools to help draft passages and questions, modeled only on our own original sample questions. We check questions before they go live, and we audit the question bank regularly for wrong answer keys, questions with two defensible answers, and difficulty labels that are off.
  • Lessons: every Learn question is solved blind by a separate reviewer who doesn’t see the answer key. If the reviewer disagrees with the key, or finds two defensible answers, the question is fixed or cut before it goes live.
  • AP: historical documents are public domain (for example, the U.S. Constitution) or our own summaries, labeled as summaries. For real past exam questions, we link to the College Board’s AP Central site instead of copying them.

Scores

Scores on Peakscor are estimates based on how you do on our practice questions. They are not official scores and don’t predict an exact result.

Found a problem?

If a question looks wrong, or you think something is too close to published material, email support@peakscor.com. We reply within one business day and take reported content down while we review it.