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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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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.

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

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Nonlinear functions · Lesson 4

Polynomial zeros, factors and end behavior

A zero of a function pp is an input aa with p(a)=0p(a) = 0. On the graph, a zero is an xx-intercept, the point (a,0)(a, 0).

Zeros and factors go together: **x−ax - a is a factor of p(x)p(x) exactly when p(a)=0p(a) = 0**. So the factor (x+2)(x + 2) means a zero at x=−2x = -2, and a factor (2x+3)(2x + 3) means a zero where 2x+3=02x + 3 = 0, at x=−32x = -\frac{3}{2}. Going the other way, if p(5)=0p(5) = 0, then x−5x - 5 is a factor. More generally, p(a)p(a) is the remainder when p(x)p(x) is divided by x−ax - a.

Crossing or touching. If a factor appears once, such as (x+2)(x + 2), the graph crosses the xx-axis there. If it appears squared, such as (x−1)2(x - 1)^2, the graph touches the axis and turns back, because a square is never negative and doesn't change sign. In general an even power touches and an odd power crosses.

End behavior. For large positive or negative xx, the term with the highest power wins. Multiply the leading terms of the factors to get it: (x+2)(x−1)2(x + 2)(x - 1)^2 starts with x3x^3. With an odd degree, the two ends go opposite ways: if the leading coefficient is positive, the graph falls on the left and rises on the right. With an even degree, both ends go the same way: both up if the leading coefficient is positive, both down if it is negative.

The yy-intercept is p(0)p(0). In factored form, set x=0x = 0 in every factor and multiply.

The graph of y = (x + 2)(x - 1)^2−3−2−1123−8−6−4−224680xy(-2, 0)(-1, 4)(0, 2)(1, 0)
The graph of y=(x+2)(x−1)2y = (x + 2)(x - 1)^2 A cubic curve that rises from the lower left, crosses the x-axis at (-2, 0), reaches a high point at (-1, 4), passes through (0, 2), comes down to touch the x-axis at (1, 0) without crossing, and then rises to the upper right.

Worked example Easy

p(x)=(x−4)(2x+3)(x+1)p(x) = (x - 4)(2x + 3)(x + 1)

Which of the following is an xx-intercept of the graph of y=p(x)y = p(x) in the xyxy-plane?

  1. (−4,0)(-4, 0)
  2. (32,0)\left(\frac{3}{2}, 0\right)
  3. (0,−12)(0, -12)
  4. (−32,0)\left(-\frac{3}{2}, 0\right)Answer

How to solve it

  1. An xx-intercept is where p(x)=0p(x) = 0, which happens when one of the factors is 0.
  2. x−4=0x - 4 = 0 gives x=4x = 4. 2x+3=02x + 3 = 0 gives x=−32x = -\frac{3}{2}. x+1=0x + 1 = 0 gives x=−1x = -1.
  3. The xx-intercepts are (4,0)(4, 0), (−32,0)\left(-\frac{3}{2}, 0\right) and (−1,0)(-1, 0). Only (−32,0)\left(-\frac{3}{2}, 0\right) is a choice.

Why each choice is right or wrong

  • A. Incorrect. This reads the zero from x−4x - 4 with the wrong sign. x−4=0x - 4 = 0 at x=4x = 4, so the intercept is (4,0)(4, 0).
  • B. Incorrect. This flips the sign when solving 2x+3=02x + 3 = 0. Subtracting 3 gives 2x=−32x = -3, so x=−32x = -\frac{3}{2}.
  • C. Incorrect. (0,−12)(0, -12) is the yy-intercept: p(0)=(−4)(3)(1)=−12p(0) = (-4)(3)(1) = -12. An xx-intercept has a yy-coordinate of 0.
  • D. Correct. 2x+3=02x + 3 = 0 when x=−32x = -\frac{3}{2}, so p(−32)=0p\left(-\frac{3}{2}\right) = 0 and the graph crosses the xx-axis there.

Worked example Medium

The graph of y = f(x)−4−3−2−112345−20−15−10−55101520250xy(-3, 0)(0, 12)(1, 0)(4, 0)
The graph of y=f(x)y = f(x) A curve that comes up from the lower left, crosses the x-axis at (-3, 0), reaches a high point of about 20.7 near x = -1.4, comes down through (0, 12) to cross the x-axis at (1, 0), reaches a low point of about -12.6 near x = 2.7, turns back up, and crosses again at (4, 0) before rising to the upper right.

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

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

How to solve it

  1. The graph crosses the xx-axis at −3-3, 1 and 4, so the factors are (x+3)(x + 3), (x−1)(x - 1) and (x−4)(x - 4), each to an odd power.
  2. The graph falls on the left and rises on the right, so the degree is odd and the leading coefficient is positive.
  3. (x+3)(x−1)(x−4)(x + 3)(x - 1)(x - 4) fits: its leading term is x3x^3, and f(0)=(3)(−1)(−4)=12f(0) = (3)(-1)(-4) = 12 matches the yy-intercept.

Why each choice is right or wrong

  • A. Incorrect. This reverses the sign of every zero. (x−3)(x+1)(x+4)(x - 3)(x + 1)(x + 4) is zero at 3, −1-1 and −4-4.
  • B. Incorrect. The zeros are right, but the negative sign flips the graph: it would rise on the left, fall on the right, and cross the yy-axis at −12-12.
  • C. Correct. Zeros at −3-3, 1 and 4, a positive leading term x3x^3 (falls left, rises right), and f(0)=12f(0) = 12 all match the graph.
  • D. Incorrect. The squared factor would make the graph touch the axis at x=1x = 1 and turn back, but the graph crosses there. This function also has degree 4, so both ends would go up, and f(0)=−12f(0) = -12.

Trap Sign of the zero

The factor (x+5)(x + 5) gives the zero −5-5, not 5. For a factor like (3x−2)(3x - 2), solve 3x−2=03x - 2 = 0: the zero is 23\frac{2}{3}, not 2. The choice with the sign flipped is a likely trap.

Trap Ignoring how the graph meets the axis

Two equations can have the same zeros and still have different graphs. If the graph touches the xx-axis and turns back, that factor is squared (an even power). If it crosses, the power is odd. Then check the ends of the graph to confirm the sign of the leading coefficient.

Desmos When Desmos is faster

Type the polynomial, such as y = (x - 4)(2x + 3)(x + 1), and click the curve. Desmos marks the xx-intercepts as gray dots; click one to see its coordinates. A zero like −32-\frac{3}{2} shows as −1.5-1.5.

To test whether x−3x - 3 is a factor of a given p(x)p(x), define p(x) in Desmos and type p(3). If it shows 0, x−3x - 3 is a factor. Any other value is the remainder.

For a graph printed in the question, type each choice and compare where it crosses or touches the xx-axis and which way its ends point.

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