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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

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  5. Math · Module 2

    22 questions35 min

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

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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.

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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 2

Quadratic functions in three forms

A quadratic function has x2x^2 as its highest power, and its graph is a parabola. If the coefficient of x2x^2, called aa, is positive, the parabola opens up and its vertex is the minimum point. If aa is negative, it opens down and the vertex is the maximum point.

The parabola is symmetric about a vertical line through the vertex, the axis of symmetry. Two points at the same height are mirror images across it, so the axis is exactly halfway between them.

The same quadratic can be written three ways, and each form shows something different. Standard form f(x)=ax2+bx+cf(x) = ax^2 + bx + c shows the yy-intercept, (0,c)(0, c). The vertex is at x=−b2ax = -\frac{b}{2a}; plug that in to get the yy-coordinate. Vertex form f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k shows the vertex, (h,k)(h, k), and so the maximum or minimum value, kk. Factored form f(x)=a(x−p)(x−q)f(x) = a(x - p)(x - q) shows the xx-intercepts, (p,0)(p, 0) and (q,0)(q, 0), and the vertex is halfway between them, at x=p+q2x = \frac{p + q}{2}.

Watch the signs. In (x−3)2(x - 3)^2 the vertex has h=3h = 3, and in (x+3)2(x + 3)^2 it has h=−3h = -3. In (x+1)(x−5)(x + 1)(x - 5) the zeros are −1-1 and 5.

The minimum value of ff is a yy-value, kk. The minimum occurs at an xx-value, hh. Read which one the question asks for.

Which form shows what
FormExampleShows directly
Standard: ax2+bx+cax^2 + bx + cx2−2x−3x^2 - 2x - 3yy-intercept (0,−3)(0, -3); vertex at x=−b2a=1x = -\frac{b}{2a} = 1
Vertex: a(x−h)2+ka(x - h)^2 + k(x−1)2−4(x - 1)^2 - 4Vertex (1,−4)(1, -4), so the minimum value is −4-4
Factored: a(x−p)(x−q)a(x - p)(x - q)(x+1)(x−3)(x + 1)(x - 3)xx-intercepts (−1,0)(-1, 0) and (3,0)(3, 0)
The graph of y = x^2 - 2x - 3, which is also y = (x - 1)^2 - 4 and y = (x + 1)(x - 3)−3−2−112345−5−4−3−2−11234560xy(1, -4)(-1, 0)(3, 0)
The graph of y=x2−2x−3y = x^2 - 2x - 3, which is also y=(x−1)2−4y = (x - 1)^2 - 4 and y=(x+1)(x−3)y = (x + 1)(x - 3) An upward-opening parabola with vertex (1, -4), the minimum point. It crosses the x-axis at (-1, 0) and (3, 0) and the y-axis at (0, -3). The axis of symmetry is the line x = 1.

Worked example Easy

f(x)=2(x+3)2−7f(x) = 2(x + 3)^2 - 7

What are the coordinates of the vertex of the graph of y=f(x)y = f(x) in the xyxy-plane?

  1. (3,−7)(3, -7)
  2. (−3,−7)(-3, -7)Answer
  3. (−7,−3)(-7, -3)
  4. (3,7)(3, 7)

How to solve it

  1. The function is in vertex form, a(x−h)2+ka(x - h)^2 + k.
  2. Write (x+3)(x + 3) as (x−(−3))(x - (-3)), so h=−3h = -3.
  3. The constant outside is k=−7k = -7.
  4. The vertex is (−3,−7)(-3, -7). Check: f(−3)=2(0)2−7=−7f(-3) = 2(0)^2 - 7 = -7, and every other input makes 2(x+3)22(x + 3)^2 positive, so −7-7 is the lowest value.

Why each choice is right or wrong

  • A. Incorrect. This reads hh as +3+3. In vertex form the parentheses hold x−hx - h, so x+3x + 3 means h=−3h = -3.
  • B. Correct. (x+3)2=(x−(−3))2(x + 3)^2 = (x - (-3))^2, so h=−3h = -3, and k=−7k = -7. The vertex is (−3,−7)(-3, -7).
  • C. Incorrect. This swaps the coordinates. The number inside the parentheses gives the xx-coordinate and the number outside gives the yy-coordinate.
  • D. Incorrect. This flips both signs. Only the sign of hh is reversed when you read it from (x+3)(x + 3). The kk-value is read as written, −7-7.

Worked example Medium

The graph of y = f(x)−3−2−11234567−10−8−6−4−224680xy(-1, 0)(5, 0)(0, -5)(2, -9)
The graph of y=f(x)y = f(x) An upward-opening parabola. It crosses the x-axis at (-1, 0) and (5, 0), crosses the y-axis at (0, -5), and has its lowest point at (2, -9).

The graph of the quadratic function ff is shown. Which equation defines ff?

  1. f(x)=(x−1)(x+5)f(x) = (x - 1)(x + 5)
  2. f(x)=(x+1)(x−5)f(x) = (x + 1)(x - 5)Answer
  3. f(x)=−(x+1)(x−5)f(x) = -(x + 1)(x - 5)
  4. f(x)=(x+2)2−9f(x) = (x + 2)^2 - 9

How to solve it

  1. The xx-intercepts are −1-1 and 5, so the factored form is f(x)=a(x+1)(x−5)f(x) = a(x + 1)(x - 5).
  2. Use the yy-intercept to find aa: f(0)=a(1)(−5)=−5af(0) = a(1)(-5) = -5a, and the graph shows f(0)=−5f(0) = -5, so a=1a = 1.
  3. So f(x)=(x+1)(x−5)f(x) = (x + 1)(x - 5). Check the vertex: halfway between −1-1 and 5 is x=2x = 2, and f(2)=(3)(−3)=−9f(2) = (3)(-3) = -9, which matches (2,−9)(2, -9).

Why each choice is right or wrong

  • A. Incorrect. This reverses the signs of the zeros. (x−1)(x+5)(x - 1)(x + 5) is zero at x=1x = 1 and x=−5x = -5, but the graph crosses at −1-1 and 5.
  • B. Correct. (x+1)(x−5)(x + 1)(x - 5) is zero at −1-1 and 5, gives f(0)=−5f(0) = -5, and opens upward, matching every feature of the graph.
  • C. Incorrect. The zeros are right, but the negative sign makes the parabola open downward, with f(0)=5f(0) = 5. The graph opens upward and crosses the yy-axis at −5-5.
  • D. Incorrect. This reads the vertex's xx-coordinate with the wrong sign. (x+2)2−9(x + 2)^2 - 9 has its vertex at (−2,−9)(-2, -9) and crosses the xx-axis at 1 and −5-5. It does give f(0)=−5f(0) = -5, which is why only checking the yy-intercept isn't enough.

Worked example Hard

f(x)=−2x2+12x−7f(x) = -2x^2 + 12x - 7

What is the maximum value of the function ff defined above?

  1. −61-61
  2. 33
  3. 1111Answer
  4. 6565

How to solve it

  1. a=−2a = -2 is negative, so the parabola opens down and the vertex is the maximum point.
  2. The vertex is at x=−b2a=−122(−2)=3x = -\frac{b}{2a} = -\frac{12}{2(-2)} = 3.
  3. The maximum value is the yy-coordinate there: f(3)=−2(9)+12(3)−7=−18+36−7=11f(3) = -2(9) + 12(3) - 7 = -18 + 36 - 7 = 11.

Why each choice is right or wrong

  • A. Incorrect. This drops the minus sign in −b2a-\frac{b}{2a} and uses x=−3x = -3: f(−3)=−18−36−7=−61f(-3) = -18 - 36 - 7 = -61. The vertex is at x=3x = 3, and a parabola's maximum can't be lower than its yy-intercept, −7-7.
  • B. Incorrect. 3 is where the maximum occurs, the xx-coordinate of the vertex. The maximum value is f(3)f(3).
  • C. Correct. The vertex is at x=3x = 3, and f(3)=11f(3) = 11. Because the parabola opens down, 11 is the greatest value of ff.
  • D. Incorrect. This multiplies −2-2 by 3 before squaring, (−2⋅3)2=36(-2 \cdot 3)^2 = 36, which gives 36+36−7=6536 + 36 - 7 = 65. Square 3 first, then multiply by −2-2: −2(9)=−18-2(9) = -18.

Trap Reading the wrong sign or the wrong coordinate

In a(x−h)2+ka(x - h)^2 + k, only hh is read with its sign reversed: (x+3)2(x + 3)^2 gives h=−3h = -3, while kk is read as written. And the minimum or maximum value is kk, a yy-value, while the xx-value hh is where it occurs. Both slips are common wrong choices.

Trap Matching only one feature of a graph

When you pick an equation for a graph, a wrong choice may match the yy-intercept but not the zeros, or the zeros but not the direction it opens. Check at least two features, such as both zeros and the yy-intercept.

Desmos When Desmos is faster

Type the function, such as y = -2x^2 + 12x - 7. Click the parabola and Desmos marks points of interest, including the vertex and the intercepts, as gray dots. Click a dot to see its coordinates, here (3,11)(3, 11). You may need to zoom out to see the whole parabola.

To match an equation to a graph printed in the question, type each choice and compare its vertex, zeros and yy-intercept with the points on the printed graph.

For vertex form, reading (h,k)(h, k) by eye is faster than typing. Desmos helps most with standard form, where the vertex takes a computation.

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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.

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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.