AP Calculus AB Quiz: Sketching Slope Fields

What this quiz covers

This quiz focuses on Sketching Slope Fields, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Calculus AB.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Slopes depend only on $x$; positive for $x>0$, negative for $x<0$, very steep near $x=0$. (correct answer)
2. Slopes depend only on $y$; positive for $y>0$, negative for $y<0$.
3. Slopes are $0$ along $x=0$ and increase with $|x|$.
4. Slopes are $0$ along $y=x$ and symmetric about that line.
5. All slopes are positive and constant.

Explanation: This question asks for the slope field matching dy/dx = 1/x (with x ≠ 0), where slopes depend only on x. The function is undefined at x = 0, creating a vertical asymptote. For x > 0, slopes are positive with 1/x > 0. For x < 0, slopes are negative with 1/x < 0. Slopes become very steep (large magnitude) as x approaches zero from either side. As |x| increases, slopes flatten toward zero but never reach zero. Choice B incorrectly suggests slopes depend on y, but 1/x involves only the x-variable. For rational functions with vertical asymptotes, identify where the denominator equals zero and analyze the sign and behavior on either side.

Question 2

1. Slopes depend only on $x$; same in each vertical column.
2. Slopes are $0$ along both axes; positive in quadrants I and III, negative in II and IV; steeper farther from axes. (correct answer)
3. Slopes are $0$ along $y=x$; positive above and negative below.
4. Slopes are always negative and get steeper as $|x|$ increases.
5. Slopes are $0$ along $y=0$ and $y=1$; positive between them.

Explanation: This question asks for the slope field matching dy/dx = xy, where slopes depend on both x and y coordinates. Slopes are zero when xy = 0, which occurs along both coordinate axes (x = 0 or y = 0). In quadrant I (x > 0, y > 0), both factors are positive, creating positive slopes. In quadrant III (x < 0, y < 0), both factors are negative, making their product positive. In quadrants II and IV, the factors have opposite signs, yielding negative slopes. Slopes become steeper as points move farther from the axes since |xy| increases. Choice C incorrectly claims slopes are zero along y = x rather than along the axes. For products like xy, find where each factor equals zero and analyze the sign in each quadrant.

Question 3

1. Slopes depend only on $y$; nonnegative everywhere; $0$ along $y=0$; steeper as $|y|$ increases. (correct answer)
2. Slopes depend only on $x$; nonnegative everywhere; $0$ along $x=0$.
3. Slopes are positive for $y>0$ and negative for $y<0$.
4. Slopes are $0$ along $y=\pm 1$ and negative between them.
5. Slopes are undefined along $y=0$.

Explanation: This question asks for the slope field matching dy/dx = √|y|, where slopes depend only on y. All points in the same horizontal row share identical slopes. Slopes are zero only when |y| = 0, so only along y = 0. The square root function ensures all slopes are nonnegative. For y ≠ 0, slopes are positive and increase as |y| increases, since √|y| grows with distance from the y-axis. The absolute value makes the slope pattern symmetric about the x-axis. Choice C incorrectly suggests slopes can be negative, but √|y| ≥ 0 always. For expressions involving absolute values and square roots, note the nonnegativity and symmetry properties that arise.

Question 4

1. Slopes are $0$ along $y=-x$; sign changes across that line; for fixed $(x+y)$, slopes flatten as $|x|$ increases. (correct answer)
2. Slopes are $0$ along $y=x$; sign changes across that line; for fixed $(x-y)$, slopes flatten as $|x|$ increases.
3. Slopes depend only on $y$ and flatten as $|y|$ increases.
4. Slopes depend only on $x$ and are always positive.
5. Slopes are undefined along $x=0$.

Explanation: This question involves sketching the slope field for dy/dx = (x + y)/(1 + x²), where slopes depend on both variables. Slopes are zero when x + y = 0, so along the line y = -x. Above this line where y > -x, slopes are positive since x + y > 0. Below this line where y < -x, slopes are negative since x + y < 0. For any fixed value of (x + y), slopes flatten as |x| increases since the denominator 1 + x² grows. Choice B incorrectly identifies y = x rather than y = -x as the zero-slope line. For rational functions with numerators of the form ax + by, set the numerator to zero to find the zero-slope line.

Question 5

1. Slopes depend only on $x$; $0$ at $x=k\pi$; periodic in vertical columns.
2. Slopes depend only on $y$; $0$ at $y=k\pi$; periodic in horizontal rows. (correct answer)
3. Slopes are $0$ along $y=\sin x$ and change sign across the curve.
4. Slopes are always positive and flatten as $|y|$ increases.
5. Slopes are $0$ along $y=x$ and positive above.

Explanation: This question requires sketching the slope field for dy/dx = sin y, where slopes depend only on y. All points in the same horizontal row share identical slopes since the equation involves only y. Slopes are zero when sin y = 0, which occurs at y = kπ for integer k. Slopes are positive when sin y > 0 and negative when sin y < 0, following the periodic pattern of sine. The slope pattern repeats every 2π units in the y-direction, creating horizontal stripes of alternating slope signs. Choice A incorrectly suggests slopes depend on x, but sin y involves only the y-variable. For trigonometric equations in y, the slope pattern creates periodic horizontal bands.

Question 6

1. Slopes are $0$ along $y=\pm x$; positive where $|x|>|y|$ and negative where $|y|>|x|$. (correct answer)
2. Slopes are $0$ along $y=x^2$; positive above, negative below.
3. Slopes depend only on $x$; always nonnegative.
4. Slopes depend only on $y$; always nonpositive.
5. Slopes are undefined along $y=\pm x$.

Explanation: This question requires sketching the slope field for dy/dx = (x² - y²)/2, where slopes depend on both variables. Slopes are zero when x² - y² = 0, so x² = y² or y = ±x. These two lines y = x and y = -x divide the plane into four regions. Slopes are positive when x² > y² (where |x| > |y|) and negative when x² < y² (where |y| > |x|). The diagonal lines create a pattern where slopes are positive in the regions farther from the y-axis and negative in regions closer to the y-axis. Choice B incorrectly suggests a parabolic zero-slope curve rather than the linear boundaries y = ±x. For expressions like x² - y², factor as (x-y)(x+y) or note where |x| = |y| to find zero-slope lines.

Question 7

1. Slopes are $0$ along the line $y=-x$; slopes increase moving upward.
2. Slopes are $0$ along the line $y=x$; slopes positive below $y=x$ and negative above. (correct answer)
3. Slopes depend only on $y$; equal slopes across each horizontal row.
4. Slopes depend only on $x$; equal slopes down each vertical column.
5. All slopes are positive; segments tilt upward everywhere.

Explanation: This question requires sketching a slope field for the differential equation dy/dx = x - y. The slope at any point (x, y) equals x - y, so slopes are zero when x = y (along the line y = x). When x > y (below the line y = x), slopes are positive since x - y > 0. When x < y (above the line y = x), slopes are negative since x - y < 0. Choice A incorrectly suggests slopes are zero along y = -x rather than y = x. The key strategy is to identify where the right-hand side equals zero to find lines of horizontal tangents, then determine the sign pattern on either side.

Question 8

1. Slopes depend only on $y$; $0$ along $y=0$; sign matches $y$; slopes flatten as $|y|$ increases. (correct answer)
2. Slopes depend only on $x$; $0$ along $x=0$; sign matches $x$.
3. Slopes are $0$ along $y=\pm 2$ and undefined at $y=0$.
4. Slopes are always positive and increase with $|y|$.
5. Slopes are $0$ along $y=x$ and undefined along $y=-x$.

Explanation: This question requires sketching the slope field for dy/dx = y/(y² + 4), where slopes depend only on y. All points in the same horizontal row share identical slopes. Slopes are zero when y = 0 (along the x-axis). The sign of slopes matches the sign of y: positive for y > 0 and negative for y < 0. The denominator y² + 4 is always positive (≥ 4), so it doesn't affect the sign but does affect magnitude. Slopes are largest near y = 0 and flatten as |y| increases. Choice B incorrectly suggests slopes depend on x, but the equation involves only y. For rational functions y/g(y) where g(y) > 0, slopes have the same sign as y and are maximized where the denominator is minimized.

Question 9

1. Slopes are $0$ along $y=0$ and along vertical lines where $\cos x=0$; sign depends on both $y$ and $x$. (correct answer)
2. Slopes are $0$ along $x=0$ and along horizontal lines where $\cos y=0$.
3. Slopes depend only on $x$ and are periodic; never zero.
4. Slopes depend only on $y$ and are periodic; never zero.
5. Slopes are $0$ along $y=x$ and change sign across it.

Explanation: This question involves sketching the slope field for dy/dx = y cos x, where slopes depend on both variables. Slopes are zero when either factor equals zero: y = 0 (x-axis) or cos x = 0 (vertical lines at x = π/2 + kπ). The sign depends on both factors: slopes are positive when y and cos x have the same sign, negative when they have opposite signs. Above the x-axis (y > 0), slopes follow the sign of cos x, while below the x-axis (y < 0), slopes have opposite sign to cos x. Choice D incorrectly suggests slopes depend only on y, missing the crucial x-dependence through cos x. For products involving trigonometric functions, identify where each factor equals zero and analyze the combined sign pattern.

Question 10

1. Slopes are $0$ along $y=-x$; positive above and negative below.
2. Slopes are $0$ along $y=x$; positive above $y=x$ and negative below. (correct answer)
3. Slopes depend only on $x$; $0$ along $x=0$.
4. Slopes depend only on $y$; $0$ along $y=0$.
5. All slopes are positive in quadrant IV.

Explanation: This question involves sketching the slope field for dy/dx = y - x, where slopes depend on both variables. Slopes are zero when y - x = 0, so along the line y = x. Above this line where y > x, slopes are positive since y - x > 0. Below this line where y < x, slopes are negative since y - x < 0. The line y = x serves as the boundary between positive and negative slope regions, with slopes becoming steeper as points move farther from this line. Choice A incorrectly identifies y = -x as the zero-slope line, but the equation requires y = x. For linear combinations like y - x, set the expression equal to zero to find the dividing line, then analyze signs on either side.

Question 11

1. Slopes are $0$ along $y=5$; negative above $y=5$ and positive below; same within each horizontal row. (correct answer)
2. Slopes are $0$ along $x=5$; negative to the right and positive to the left.
3. Slopes are $0$ along $y=x$; negative above and positive below.
4. Slopes are $0$ along $y=0$ and $y=5$; positive between them and negative outside.
5. All slopes are constant and negative everywhere.

Explanation: This question asks for the slope field matching dy/dx = -(y - 5), a cooling model where temperature y approaches ambient temperature 5. The slope depends only on y, creating identical slopes across each horizontal row. Slopes are zero when -(y - 5) = 0, so y = 5. When y > 5, we have (y - 5) > 0, making -(y - 5) < 0 (negative slopes). When y < 5, we have (y - 5) < 0, making -(y - 5) > 0 (positive slopes). Choice D incorrectly suggests equilibria at both y = 0 and y = 5, but only y = 5 gives zero slope. For cooling/heating problems, identify the ambient temperature where slope equals zero, then check signs above and below that value.

Question 12

1. Slopes are $0$ along $y=2$; undefined along $x=-1$; sign changes across both lines. (correct answer)
2. Slopes are $0$ along $x=-1$; undefined along $y=2$.
3. Slopes depend only on $x$; undefined at $x=-1$; positive for $x>-1$.
4. Slopes depend only on $y$; $0$ along $y=2$; positive above.
5. Slopes are $0$ along $y=-x$ and undefined along $y=x$.

Explanation: This question asks for the slope field matching dy/dx = (y - 2)/(x + 1) where defined. The function is undefined when x + 1 = 0, so along x = -1. Slopes are zero when y - 2 = 0, so along y = 2. These two perpendicular lines divide the plane into regions with different slope characteristics. The line y = 2 provides horizontal tangents while x = -1 represents a vertical asymptote where slopes become infinite. The sign of slopes depends on the signs of both numerator and denominator factors. Choice B incorrectly reverses which line has zero slopes versus undefined slopes. For rational functions, identify the zeros of numerator and denominator separately to understand the complete slope behavior.

Question 13

1. Slopes are $0$ along $y=x^2$; positive above the parabola and negative below. (correct answer)
2. Slopes are $0$ along $x=y^2$; positive to the right and negative to the left.
3. Slopes depend only on $x$ and are always nonnegative.
4. Slopes depend only on $y$ and are $0$ along $y=0$.
5. Slopes are $0$ along $y=-x$; positive above and negative below.

Explanation: This question requires sketching the slope field for dy/dx = y - x², where slopes depend on both variables. Slopes are zero when y - x² = 0, so y = x² (along the parabola). Above the parabola where y > x², slopes are positive since y - x² > 0. Below the parabola where y < x², slopes are negative since y - x² < 0. The parabolic curve y = x² serves as the boundary between positive and negative slope regions. Choice E incorrectly suggests slopes are zero along y = -x, but the equation requires y = x² for zero slope. When analyzing dy/dx = g(x,y), set the right-hand side equal to zero to find curves of horizontal tangents, then test signs on either side.

Question 14

1. Slopes depend only on $x$; always positive; steeper for smaller $x$ and flatten as $x$ increases. (correct answer)
2. Slopes depend only on $y$; always positive; steeper for smaller $y$.
3. Slopes are $0$ along $x=0$ and negative for $x>0$.
4. Slopes alternate sign periodically as $x$ increases.
5. Slopes are $0$ along $y=e^{-x}$ and change sign across the curve.

Explanation: This question requires sketching the slope field for dy/dx = e^(-x), where slopes depend only on x. All points in the same vertical column share identical slopes. The exponential function e^(-x) is always positive, so slopes are always positive throughout the plane. Slopes are largest when x is smallest (moving leftward) and approach zero as x increases (moving rightward). At x = 0, the slope equals 1, and slopes decay exponentially for positive x while growing exponentially for negative x. Choice B incorrectly suggests slopes depend on y, but e^(-x) involves only the x-variable. For exponential functions, note they never reach zero but can approach zero or infinity depending on the sign of the exponent.

Question 15

1. Slopes are $0$ along $y=-2$ and $y=1$; positive for $y>1$ and $y<-2$, negative for $-2<y<1$. (correct answer)
2. Slopes are $0$ along $x=-2$ and $x=1$; positive between them.
3. Slopes depend only on $x$; $0$ at $x=\pm 1$.
4. Slopes are $0$ along $y=0$ and $y=3$; positive between.
5. Slopes are $0$ along $y=x$ and change sign across it.

Explanation: This question involves sketching the slope field for dy/dx = (y - 1)(y + 2), where slopes depend only on y. All points in the same horizontal row share identical slopes. Slopes are zero when either factor equals zero: y = 1 or y = -2. For y > 1, both factors are positive, creating positive slopes. For -2 < y < 1, the first factor is negative while the second is positive, creating negative slopes. For y < -2, both factors are negative, making their product positive. Choice D incorrectly identifies y = 0 and y = 3 as equilibrium points rather than y = -2 and y = 1. For factored expressions, find where each factor equals zero, then analyze the sign of the product in each interval.

Question 16

1. Slopes depend only on $y$; always positive, largest near $y=0$, flattening as $|y|$ increases. (correct answer)
2. Slopes depend only on $x$; always positive, largest near $x=0$.
3. Slopes are $0$ on $y=0$ and negative elsewhere.
4. Slopes are positive above $y=0$ and negative below $y=0$.
5. Slopes are $0$ along $y=x$ and change sign across it.

Explanation: This question involves sketching the slope field for dy/dx = 1/(1 + y²), where slopes depend only on y. All points in the same horizontal row share identical slopes. The denominator 1 + y² is always positive, so slopes are always positive regardless of y-value. Slopes are largest when the denominator is smallest, which occurs at y = 0 where the slope equals 1. As |y| increases, the denominator grows, causing slopes to flatten toward zero but never reaching zero. Choice D incorrectly suggests slopes can be negative, but the fraction is always positive. For rational functions with positive denominators, slopes maintain the same sign throughout and are largest where the denominator is minimized.

Question 17

1. Slopes are constant along diagonals $y-x=\text{constant}$ only.
2. Slopes are $0$ along $y=x$; slopes are undefined along $y=-x$; sign changes across these lines. (correct answer)
3. Slopes are $0$ along $y=-x$; undefined along $y=x$.
4. Slopes depend only on $x$ and are undefined at $x=0$.
5. Slopes depend only on $y$ and are undefined at $y=0$.

Explanation: This question involves sketching the slope field for dy/dx = (y - x)/(y + x) where defined. The function is undefined when y + x = 0, so along the line y = -x. Slopes are zero when y - x = 0, so along the line y = x. These two lines divide the plane into regions with different slope characteristics. The lines y = x and y = -x serve as boundaries where the slope behavior changes, with y = x giving horizontal tangents and y = -x representing undefined slopes. Choice C incorrectly reverses which line has zero slopes versus undefined slopes. For rational functions, separately find where the numerator equals zero (giving zero slopes) and where the denominator equals zero (giving undefined slopes).

Question 18

1. Slopes are $0$ along $y=1$ and along $x=0$; sign changes across both. (correct answer)
2. Slopes are $0$ only along $y=1$; same in every vertical column.
3. Slopes are $0$ only along $x=0$; same in every horizontal row.
4. Slopes are $0$ along $y=x$ and $y=-x$.
5. Slopes are always positive and get steeper as $|y|$ increases.

Explanation: This question requires sketching the slope field for dy/dx = x(1 - y), where slopes depend on both variables. Slopes are zero when either factor equals zero: x = 0 (y-axis) or 1 - y = 0 (y = 1). These create two lines where slopes are horizontal. The sign depends on both factors: slopes are positive when x and (1 - y) have the same sign, and negative when they have opposite signs. Above y = 1, we have (1 - y) < 0, while below y = 1, we have (1 - y) > 0. Choice B incorrectly suggests only one zero-slope line, missing the y-axis. For products of linear factors, find where each factor equals zero to identify all zero-slope lines, then analyze signs in each region.

Question 19

1. Slopes depend only on $x$ and repeat periodically in vertical columns.
2. Slopes depend only on $y$ and repeat periodically in horizontal rows; $0$ at $y=\frac{\pi}{2}+k\pi$. (correct answer)
3. Slopes are $0$ along $y=x$ and change sign across it.
4. Slopes are always positive and increase with $y$.
5. Slopes are always negative and decrease with $y$.

Explanation: This question involves sketching the slope field for dy/dx = cos y, where slopes depend only on y. All points in the same horizontal row share identical slopes since the equation involves only y. Slopes are zero when cos y = 0, which occurs at y = π/2 + kπ for integer k. Slopes are positive when cos y > 0 and negative when cos y < 0, following the periodic pattern of cosine. Maximum positive slopes occur at y = 2kπ, while maximum negative slopes occur at y = π + 2kπ. Choice A incorrectly suggests slopes depend on x, but cos y involves only the y-variable. For trigonometric equations in y, the slope pattern repeats periodically in horizontal strips.

Question 20

1. Slopes depend only on $x$; vertical columns share the same tilt.
2. Slopes are $0$ along $x=0$ and get steeper as $|x|$ increases.
3. Slopes are $0$ on the horizontal lines $y=0$ and $y=2$; positive for $0<y<2$, negative for $y>2$. (correct answer)
4. Slopes are $0$ on the vertical lines $x=0$ and $x=2$; positive for $0<x<2$.
5. Slopes are $0$ along $y=-x$; positive above that line and negative below.

Explanation: This question involves sketching a slope field for dy/dx = y(2 - y), a logistic differential equation with carrying capacity 2. The slope depends only on y, so all points in the same horizontal row share identical slopes. Slopes are zero when y(2 - y) = 0, which occurs at y = 0 and y = 2. For 0 < y < 2, both factors are positive, making slopes positive. For y > 2, we have y > 0 but (2 - y) < 0, making slopes negative. Choice A incorrectly states slopes depend on x, but this equation shows y-dependence only. To analyze logistic models, find the equilibrium points where growth rate equals zero, then check the sign between equilibria.