Free Printable AP Precalculus Formulas and Justifications Review

1.

The common ratio of a geometric sequence, $g_n$ , is given by which of the following formulas?

a)

A. $r=\frac{g_n-g_k}{n-k}$

b)

B. $r=\frac{g_n+g_k}{n+k}$

c)

C. $r=\frac{g_n}{g_{n+1}}$

d)

D. $r=\frac{g_{n+1}}{g_n}$

2.

What is the end behavior of the function $f\left(x\right)=ab^x$ where a = 2 and b = 3?

a)

$\lim_{x\rightarrow-\infty}f\left(x\right)=0\ and\ \lim_{x\rightarrow\infty}f\left(x\right)=\infty$

b)

$\lim_{x\rightarrow-\infty}f\left(x\right)=\infty\ and\ \lim_{x\rightarrow\infty}f\left(x\right)=0$

c)

$\lim_{x\rightarrow-\infty}f\left(x\right)=\infty\ and\ \lim_{x\rightarrow\infty}f\left(x\right)=\infty$

d)

$\lim_{x\rightarrow-\infty}f\left(x\right)=0\ and\ \lim_{x\rightarrow\infty}f\left(x\right)=0$

3.

A model is justified as appropriate for a data set if the residual plot:

a)

A. Has a clearly defined pattern

b)

B. Has large values

c)

C. Has small values

d)

D. Has no pattern

4.

For the expression $\frac{b^m}{b^n}$ which of the following is correct?

a)

A. $b^{\left(m+n\right)}$

b)

B. $b^{mn}$

c)

C. $b^{\left(m-n\right)}$

d)

D. $\frac{1}{b^n}$

5.

For the expression $\left(b^m\right)^n$ , which of the following is correct?

a)

A. $b^{\left(m+n\right)}$

b)

B. $b^{mn}$

c)

C. $b^{\left(m-n\right)}$

d)

D. $\frac{1}{b^n}$

6.

For the expression log(xy), which of the following is correct?

a)

A. log x + log y

b)

B. log x - log y

c)

C. n log x

d)

D. $\frac{\log x}{\log y}$

7.

For the expression $\log\left(\frac{x}{y}\right)$ which of the following is correct?

a)

A. log x + log y

b)

B. log x - log y

c)

C. n log x

d)

D. $\frac{\log x}{\log y}$

8.

For the expression $\log\left(x^n\right)$ , which of the following is correct?

a)

A. log x + log y

b)

B. log x - log y

c)

C. n log x

d)

D. $\frac{\log x}{\log y}$

9.

Inverse functions are reflections over

a)

the line y=x

b)

The y-axis

c)

The x-axis

d)

The origin

10.

Which of the following are equal to x select all that apply

a)

$f\left(f^{-1}\left(x\right)\right)$

b)

$f^{-1}\left(f\left(x\right)\right)$

c)

$\left(f\left(f\left(x\right)\right)\right)$

d)

$f\left(y\right)$

11.

Residual=

a)

Actual

b)

Error

c)

Actual-Predicted

d)

Predicted Value

12.

If a+bi is a zero of a polynomial function f, which of the following is also a zero of f?

a)

b+ai

b)

b−ai

c)

a−bi

d)

0

13.

If the input values of a function change proportionally, over equal length output intervals, then the function is

a)

Linear

b)

Quadratic

c)

Exponential

d)

Logarithmic

14.

Where a polynomial function switches between increasing and decreasing, or at the included endpoint, the polynomial function will have

a)

Extrema

b)

Point of Inflection

c)

Zero

d)

Asymptote

15.

A periodic relationship occurs when:

a)

the input values demonstrate a repeating pattern over successive equal-length intervals

b)

the output values demonstrate a repeating pattern over successive equal-length intervals

c)

the output values do not demonstrate a repeating pattern over successive equal-length intervals

d)

the input values do not demonstrate a repeating pattern over successive equal-length intervals

16.

The period of a function is the:

a)

length of the y-values that is takes for the function to complete one cycle

b)

length of the x-values that is takes for the function to complete one cycle

c)

frequency of the x-values that is takes for the function to complete one cycle

d)

frequency of the y-values that is takes for the function to complete one cycle

17.

The radian measure of an angle in standard position is:

a)

the ratio of the length of the arc of a circle to the diameter of that same circle.

b)

height on the unit circle.

c)

For a unit circle the radian measure is the same as the length of the arc.

d)

The degree measure of any coterminal angle.

18.

On the unit circle sin θ=

a)

$\frac{1}{x}$

b)

$\frac{y}{x}$

c)

y

d)

$\frac{x}{y}$

19.

On the unit circle cos θ=

a)

x

b)

y

c)

$\frac{1}{x}$

d)

$\frac{1}{y}$

20.

On the unit circle tan θ=

a)

x

b)

y

c)

$\frac{x}{y}$

d)

$\frac{y}{x}$

21.

On the unit circle cot θ=

a)

1/y

b)

x/y

c)

1/x

d)

y/x

22.

On the unit circle csc θ=

a)

$\frac{1}{y}$

b)

$\frac{x}{y}$

c)

$\frac{1}{x}$

d)

$\frac{y}{x}$

23.

On the unit circle sec θ=

a)

$\frac{1}{x}$

b)

$\frac{1}{y}$

c)

$\frac{x}{y}$

d)

$\frac{y}{x}$

24.

For a sinusoidal function the midline is

a)

A. a

b)

B. Half of the distance between the minimum and maximum values

c)

C. Halfway between the minimum and maximum values.

d)

D. the reciprocal of the period

25.

For a sinusoidal function the amplitude is

a)

A. the reciprocal of the period

b)

B. the average of the maximum and minimum

c)

C. the volume

d)

D. is the distance from the midline to the maximum

26.

For a sinusoidal function f(x) = A sin [B(x + C)] + D the period is

a)

A. $\frac{C}{B}$

b)

B. $\frac{\pi}{B}$

c)

C. $\frac{2\pi}{B}$

d)

D. B

27.

For a sinusoidal function f(x) = A sin [B(x + C)] + D the frequency is

a)

A. |B|

b)

B. $\frac{2\pi}{B}$

c)

C. $\frac{B}{2\pi}$

d)

D. $\frac{C}{B}$

28.

For a sinusoidal function f(x) = A sin [B(x + C)] + D the phase shift is

a)

A. |C|

b)

B. $-\frac{C}{B}$

c)

C. $\frac{\left|C\right|}{B}$

d)

D. -C

29.

The graph of y = f(X) = a tan[b(x+ c)] + d is a vertical dilation of the graph of y = tan(x) by a factor of

a)

A. a

b)

B. b

c)

C. d

d)

D. -d

30.

The graph of y = f(x) = a tan[b(x + c)] + d has a period of

a)

A. $-\frac{c}{b}$

b)

B. $\frac{2\pi}{b}$

c)

C. $\frac{b}{2\pi}$

d)

D. $\frac{\pi}{\left|b\right|}$

31.

The domain of arcsin x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left[0,\pi\right]$

32.

The range of arcsin x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left[0.\pi\right]$

33.

The range of arccos x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left[0.\pi\right]$

34.

The domain of arccos x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left[0.\pi\right]$

35.

The domain of arctan x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left(-\infty,\propto\right)$

36.

The range of arctan x is

a)

A. $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$

b)

B. [-1,1]

c)

C. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$

d)

D. $\left(-\infty,\propto\right)$

37.

The range of tan^-1x is:

a)

(−∞, ∞)

b)

(−π/2, π/2)

c)

[−π/2, π]

d)

[0, π]

38.

sin²θ + cos²θ =

a)

1

b)

cos 2θ

c)

sin 2θ

d)

−1

39.

1 + tan²θ =

a)

sec²θ

b)

−1

c)

csc²θ

d)

sec θ

40.

1 + cot²θ =

a)

−1

b)

sec²θ

c)

csc²θ

d)

sec θ

41.

cos^-1(x) =

a)

$\sin^{-1}\left(\sqrt[]{y}\right)$

b)

$\sin^{-1}\left(\sqrt[]{1+x^2}\right)$

c)

$\sin^{-1}\left(\sqrt[]{1-x^2}\right)$

d)

$\tan^{-1}\left(\sqrt[]{1+x^2}\right)$

42.

When changing from rectangular to polar coordinates $r^2$ =

a)

$x^2+y^2$

b)

( tan θ )

c)

( r sin θ )

d)

( r cos θ )

43.

For the expression ( a + bi ), which of the following is the correct transformation into polar form?

a)

( r sin θ + i r cos θ )

b)

( r cos θ + i r sin θ )

c)

( r cos θ - i r sin θ )

d)

( i r sin θ - i r cos θ )

44.

The distance between ( r = f(θ) ) and the origin is increasing if:

a)

( r = f(θ) ) is increasing

b)

( r = f(θ) ) is negative and decreasing

c)

( r = f(θ) ) is decreasing

d)

( r = f(θ) ) is positive and decreasing

45.

If ( f ) is an odd function, then:

a)

is not an even function

b)

( f(-x) = -f(x) )

c)

( f(-x) = f(x) )

d)

is graphically symmetric about the x-axis

46.

Calculate the value of $\sec\left(\frac{\pi}{3}\right)$ .

a)

$\frac{1}{2}$

b)

$\frac{2\sqrt[]{3}}{3}$

c)

2

d)

1

47.

Find the value of $\tan\left(\frac{\pi}{2}\right)$ .

a)

$\pi$

b)

0

c)

undefined

d)

1

48.

Compute $\cos\left(\frac{5\pi}{4}\right)$ .

a)

$\frac{\sqrt[]{2}}{2}$

b)

-1

c)

$-\frac{\sqrt[]{2}}{2}$

d)

1/2

49.

Determine the value of $\sin\left(\frac{11\pi}{6}\right)$ .

a)

$\frac{\sqrt[]{3}}{2}$

b)

$\frac{1}{2}$

c)

$-\frac{\sqrt[]{3}}{2}$

d)

$-\frac{1}{2}$

50.

$\cos\left(A+B\right)$

a)

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

b)

cos(A + B) = cos(A) + sin(A)sin(B)

c)

cos(A + B) = cos(A) - sin(B)

d)

cos(A + B) = sin(A)sin(B) + cos(A)cos(B)

51.

which of the following is not equal to $\cos\left(2x\right)$

a)

$\sin^2x-\cos^2x$

b)

$\cos^2x-\sin^2x$

c)

$1-2\sin^2x$

d)

$2\cos^2x-1$

52.

$\sin^{-1}\left(-\frac{\sqrt[]{3}}{2}\right)$

a)

$-\frac{\pi}{3}$

b)

$\frac{5\pi}{3}$

c)

$\frac{\pi}{3}$

d)

$-\frac{\pi}{6}$

53.

If a rational function f has a hole located at the point with coordinates $\left(a,L\right)$ then

a)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=L$

b)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=\infty$

c)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=0$

d)

$f\left(a\right)=L$

54.

If a rational function f has a vertical asymptote at x=a then

a)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=L$

b)

$\lim_{x\rightarrow a^-}f\left(x\right)=\pm\infty\ or\lim_{x\rightarrow a^+}f\left(x\right)=\pm\infty$

c)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=0$

d)

$\lim_{x\rightarrow a^-}f\left(x\right)=\lim_{x\rightarrow a^+}f\left(x\right)=\infty$

55.

Rational functions have slant asymptotes when

a)

the degree of the denominator is exactly one greater than the degree of the numerator.

b)

the degree of the numerator is equal to the degree of the denominator.

c)

the degree of the numerator is exactly one greater than the degree of the denominator.

d)

the degree of the numerator is greater than the degree of the denominator.

56.

When the average rate of change over equal length input-value intervals is decreasing,

a)

The function is linear.

b)

The function is concave down.

c)

The function is concave up.

d)

The function has a constant rate of change.

57.

What is the change of base formula?

a)

$\log_b(a)=\frac{\log_k(b)}{\log_k(a)}\$ .

b)

$\log_b(a)=\frac{\log_b(k)}{\log_a(k)}\$

c)

$\log_b(a)=\frac{\log_k(b)}{\log_a(k)}\$

d)

$\log_b(a)=\frac{\log_k(a)}{\log_k(b)}\$

58.

If the input values of a function change proportionally, over equal length output intervals, then the function is

a)

exponential

b)

logarithmic

c)

quadratic

d)

linear

59.

Where a polynomial function switches between increasing and decreasing, or at the included endpoint, the polynomial function will have

a)

global maximum or minimum

b)

constant value

c)

local maximum or minimum

d)

linear behavior

60.

What will you make on the AP Exam?

a)

A. 2

b)

B. 3

c)

C. 4

d)

D. 5

AP Precalculus Formulas and Justifications Review

Total questions: 60

Worksheet time: 30mins

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