Mixed Deck — All AP PRECALCULUS Topics Flashcards

For which values of x in [0, 2π) is cos x > 1/2?

Answer: 0 ≤ x < π/3 or 5π/3 < x < 2π

cos x = 1/2 at x = π/3 and x = 5π/3. Cosine is above 1/2 near x = 0, so the solution is [0, π/3) together with (5π/3, 2π).

Where is the hole in the graph of f(x) = (x² + x − 6)/(x² − 4)?

Answer: (2, 5/4)

Factoring gives f(x) = (x + 3)(x − 2)/((x − 2)(x + 2)). The factor (x − 2) cancels, so the hole is at x = 2. The simplified function (x + 3)/(x + 2) gives y = 5/4 there.

Let f(x) = 2x + 3 and g(x) = x². What is the value of (f ∘ g)(2)?

Answer: 11

(f ∘ g)(2) = f(g(2)) = f(4) = 2(4) + 3 = 11. The inner function g is applied first.

What is the domain of f(x) = ln(5 − 2x)?

Answer: x < 5/2

The argument of a logarithm must be strictly positive, so 5 − 2x > 0, which gives x < 5/2.

Which function is the inverse of h(x) = eˣ + 1?

Answer: h⁻¹(x) = ln(x − 1)

Set y = eˣ + 1, subtract 1 to get y − 1 = eˣ, then take ln: x = ln(y − 1). Swapping variables gives h⁻¹(x) = ln(x − 1).

What is the value of arcsin(sin(5π/6))?

Answer: π/6

sin(5π/6) = 1/2, and arcsin returns the angle in [−π/2, π/2] whose sine is 1/2, which is π/6.

Let f(x) = (x + 1)²(x − 3)³. Which describes the behavior of the graph of f at x = 3?

Answer: It crosses the x-axis and flattens near the zero

The zero x = 3 has odd multiplicity 3, so the graph crosses the axis, and because the multiplicity is greater than 1 it flattens near the intercept.

On an interval, a function's average rates of change over consecutive equal-length subintervals are all positive and increasing. What does this indicate about the graph of the function on that interval?

Answer: Increasing and concave up

Positive rates of change mean the function is increasing, and rates that grow larger mean the graph is concave up.

A student fits an exponential model to a data set and plots the residuals. Which observation about the residual plot best supports the model being appropriate?

Answer: The residuals are scattered randomly above and below zero with no pattern.

A residual is the actual value minus the predicted value. Random scatter around zero with no pattern shows the model captures the trend in the data.

Let g(x) = −3·2^x + 5. Which statement about g is true?

Answer: g has a horizontal asymptote at y = 5 and decreases without bound as x → ∞.

The vertical shift of +5 places the horizontal asymptote at y = 5, as x → −∞. The negative coefficient reflects the graph, so g(x) → −∞ as x → ∞.

Which rectangular equation is equivalent to the polar equation r = 6 cos θ?

Answer: (x - 3)² + y² = 9

Multiplying by r gives r² = 6r cos θ, so x² + y² = 6x. Completing the square gives (x - 3)² + y² = 9.

What is the range of y = sec x?

Answer: (−∞, −1] ∪ [1, ∞)

Since sec x = 1/cos x and |cos x| ≤ 1 (with cos x ≠ 0), |sec x| ≥ 1. The range is therefore (−∞, −1] ∪ [1, ∞).

What is the exact value of log₄ 32?

Answer: 5/2

Write both as powers of 2: 32 = 2⁵ and 4 = 2². Then log₄ 32 = 5/2 by the change of base formula, or because 4^(5/2) = 2⁵ = 32.

What is the greatest possible number of relative extrema (turning points) of a polynomial function of degree 6?

Answer: 5

A polynomial of degree n has at most n − 1 turning points, so a degree 6 polynomial has at most 5.

For an exponential function f(x) = a·b^x with b > 0, f(1) = 10, f(3) = 40 and f(5) = 160. What is f(2)?

Answer: 20

Outputs are multiplied by 4 over each 2-unit step, so b² = 4 and b = 2. Then a = 10/2 = 5, and f(2) = 5·2² = 20.

Which expression is equivalent to (x² + 5x + 7)/(x + 2) for x ≠ −2 and shows its slant asymptote?

Answer: x + 3 + 1/(x + 2)

Dividing gives quotient x + 3 and remainder 1, because (x + 3)(x + 2) = x² + 5x + 6. So the expression equals x + 3 + 1/(x + 2).

Consider f(x) = (x − 2)/((x − 2)(x + 3)). Which statement correctly describes the graph of f?

Answer: A vertical asymptote at x = −3 and a hole at x = 2

The factor (x − 2) cancels, which leaves a hole at x = 2. The factor (x + 3) remains in the denominator, which gives a vertical asymptote at x = −3.

The point (4, 5) lies on the graph of y = f(x). Which point must lie on the graph of g(x) = −f(x − 3) + 2?

Answer: (7, −3)

Replacing x with x − 3 shifts the graph right 3, giving x = 7; the negative reflects the y-value to −5 and +2 raises it to −3.

For g(x) = −2 cos(x) + 5, which pair gives the midline and the amplitude?

Answer: Midline y = 5, amplitude 2

The midline is y = d = 5. The amplitude is |a| = |−2| = 2, so the graph oscillates between 3 and 7.

For the polar function r = 2 + 4 sin θ, which values of θ in [0, 2π) give r = 0, so the graph passes through the pole?

Answer: 7π/6 and 11π/6

Setting 2 + 4 sin θ = 0 gives sin θ = -1/2, which occurs at θ = 7π/6 and 11π/6.