Q: For the polynomial function $$f(x) = (x-3)^2$(x+2)$$, what are the x-intercepts and their behavior?

$$x = 3$$ \text{ (touches)},\ x = -2$ \text{ (crosses)}

Question 1

Which of the following functions is a polynomial?

Accepted Answer

$$f(x) = 3x^2$ + 2x + 1$$

Question 2

For the polynomial function $$f(x) = (x-3)^2$(x+2)$$, what are the x-intercepts and their behavior?

Accepted Answer

$$x = 3$$ 	ext{ (touches)},\ x = -2$ 	ext{ (crosses)}

Question 3

For the given rational function, find the equation of the horizontal asymptote (if applicable): $$f(x) = \frac{5x}{2x^2$ + 7}$$

Accepted Answer

$$y = 0$$

Question 4

For the following function, write the equation for the vertical asymptote: $$f(x) = \frac{x+8}{x+11}$$

Accepted Answer

$$x = -11$$

Question 5

Does the function $$f(x) = -\frac{3}{x+2}$$ involve a reflection, and if so, over which axis?

Accepted Answer

Yes, reflection over the x-axis

Question 6

Which of the following is the parent function of the exponential function $$f(x) = 5$$^{(x-2)}$$ + 1$$?

Accepted Answer

$$f(x) = 5^x$$

Question 7

Evaluate $$e^4$$ using a calculator and round to the nearest hundredth.

Accepted Answer

54.60

Question 8

Express the following equation in exponential form and solve for $$x$$: $$\log_2$ x = -4$$

Accepted Answer

$$x = 1/16$$

Question 9

What is the domain of the logarithmic function $$g(x) = \log_4$(x-3)$$?

Accepted Answer

$$x \u003e 3$$

Question 10

Condense the given equation to a single logarithm whose coefficient is 1, and evaluate without using a calculator if possible: $$\frac{1}{3}(\log_3$ a + \log_3$ b) - \log_3$(a+b)$$

Accepted Answer

$$\log_3$\left(\frac{(ab$$)^{1/3}$$}{a+b}\right)$$

Question 11

Consider the one-to-one function $$f(x) = 14^x + 8. $$Determine the equation of its inverse function.

Accepted Answer

$$f^{-1}(x$$) = \log_{14}(x-8)$$

Question 12

Determine two angles, between $$-2\pi$$ and $$2\pi$$, where the terminal side of each angle passes through the origin and point $$Y on $$the unit circle as shown. Express your answer in radians.

Accepted Answer

$$-7\pi/4,\ 3\pi/4$$

Question 13

Which of the following pairs of angles are coterminal?

Accepted Answer

60 degrees and 420 degrees

Question 14

If an angle measures 45 degrees, what is its supplement?

Accepted Answer

135 degrees

Question 15

Determine the arc length intercepted by the given central angle and radius of the circle. Express the answer to two decimal places. $$r = 4.56\ 	ext{ft},\ 	heta = \frac{7\pi}{8}\ 	ext{rad}$$

Accepted Answer

28.58 ft

Question 16

In a right triangle, if the tangent of angle $$	heta is 1$$, what is the measure of angle $$	heta in $$degrees?

Accepted Answer

45 degrees

Question 17

A 30-60-90 triangle has a hypotenuse of 14. What is the length of the short leg?

Accepted Answer

$$7$$ units

Question 18

Solve for $$x if$$ $$\cos(x) = \sin(45\degree).$$

Accepted Answer

$$x = 60$$ degrees

Question 19

A right triangle has legs of lengths 6 and 8 units. What is the length of the hypotenuse?

Accepted Answer

10 units

Question 20

Which of the following is the exact value of the given trigonometric function? $$\csc 45\degree$$

Accepted Answer

$$\sqrt{2}$$

Question 21

If the coordinates of a point on the unit circle are $$(1/2,\ \sqrt{3}/2)$$, what is the sine of the corresponding angle?

Accepted Answer

$$\sqrt{3}/2$$

Question 22

What is the tangent of 30 degrees?

Accepted Answer

$$\sqrt{3}/3$$

Question 23

Evaluate the following trigonometric expression: $$7 	an 45\degree + 2 \cos 60\degree - \cot^2$ 225\degree$$

Accepted Answer

7

Question 24

Using the unit circle, what is the cotangent of 45 degrees?

Accepted Answer

1

Question 25

What is the period of the function $$y = \sin(3x)$$?

Accepted Answer

$$2\pi/3$$

Question 26

Rewrite the function $$y = \sin(4x - \pi) to $$clearly show the phase shift and period.

Accepted Answer

$$y = \sin(4(x - \pi/4))$$

Question 27

What happens to the values of the cosecant function as it approaches its asymptotes?

Accepted Answer

The values approach infinity

Question 28

How does the period of the tangent function affect its graph compared to the sine function?

Accepted Answer

The tangent graph repeats every $$\pi$$ units, while the sine graph repeats every $$2\pi$$ units.

Precalculus Study Guide: Functions, Exponents, Logarithms, and Trigonometry

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