Q: A savings account earns compound interest at 4% annually, compounded quarterly. If $$P = 1000$$, what is the amount after 2 years? (Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$)

$$A = 1000(1 + \frac{0.04}{4}$$)^{4 \times 2}$$ = 1000(1.01)^8$ \approx 1082.86$$

Question 1

A company models its profit with the quadratic function $$f(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. What is the maximum profit, and at how many units sold does it occur?

Accepted Answer

Maximum profit is $$f(4) = 26 at$$ $$x = 4$$ units.

Question 2

Given the polynomial $$f(x) = -3x^5$ + 2x^3$ - x + 7$$, what does the Leading Coefficient Test predict about the end behavior of the graph?

Accepted Answer

Falls to the right and rises to the left.

Question 3

The graph of $$f(x) = (x-2)^4$(x+1)$$ crosses or touches the x-axis at which points, and what is the behavior at each zero?

Accepted Answer

Touches at $$x=2 ($$even multiplicity), crosses at $$x=-1 ($$odd multiplicity).

Question 4

A rational function is given by $$f(x) = \frac{x^2$ - 4}{x^2$ - x - 6}. $$What are the vertical asymptotes?

Accepted Answer

At $$x = 2$$ and $$x = -3$$

Question 5

For the rational function $$f(x) = \frac{3x^2$ + 5}{2x^2$ - 7}$$, what is the horizontal asymptote?

Accepted Answer

$$y = \frac{3}{2}$$

Question 6

A biologist models population growth with $$f(x) = 5 \cdot 2^x. $$What is the value of $$f(3)$$, and what does it represent?

Accepted Answer

$$f(3) = 40$$, representing the population after 3 time periods.

Question 7

A savings account earns compound interest at 4% annually, compounded quarterly. If $$P = 1000$$, what is the amount after 2 years? (Use $$A = P(1 + \frac{r}{n}$$)^{nt}$$)

Accepted Answer

$$A = 1000(1 + \frac{0.04}{4}$$)^{4 	imes 2}$$ = 1000(1.01)^8$ \approx 1082.86$$

Question 8

Convert the logarithmic equation $$\log_3$ x = 4 to $$exponential form.

Accepted Answer

$$3^4 = x$

Question 9

What is the domain of $$f(x) = \ln(x-5)$$?

Accepted Answer

$$x \u003e 5$$

Question 10

Expand $$\log_2$(8x^3$$)$ using properties of logarithms.

Accepted Answer

$$\log_2$ 8 + 3\log_2$ x = 3 + 3\log_2$ x$$

Question 11

Condense $$2\log_5$ x + \log_5$ y - \log_5$ z$$ into a single logarithm.

Accepted Answer

$$\log_5$ \frac{x^2$ y}{z}$$

Question 12

Using the change-of-base formula, rewrite $$\log_7$ 50 in $$terms of common logarithms.

Accepted Answer

$$\frac{\log 50}{\log 7}$$

Question 13

A rational function $$f(x) = \frac{x^3$ + 2x^2$}{x^2$ - 1}$$ has what type of asymptote?

Accepted Answer

A slant asymptote, since the degree of the numerator is one more than the denominator.

Question 14

If $$f(x) = \frac{2x}{x^2$ - 4}$$, what is the x-intercept?

Accepted Answer

At $$x = 0$$

Question 15

A chemical reaction follows $$y = 100e$$^{-0.5t}$$, where y$ is the concentration and t$ is time in hours. What is the concentration after 2 hours?

Accepted Answer

$$y = 100e$$^{-1}$$ \approx 36.79$$

Question 16

Which of the following is NOT a property of logarithms?

Accepted Answer

$$\log_b (M + N) = \log_b M + \log_b N$$

Question 17

For the quadratic function $$f(x) = 2(x-3)^2$ + 5$$, what is the vertex and does the parabola open up or down?

Accepted Answer

Vertex at $$(3,5)$$, opens up.

Question 18

A rational function $$f(x) = \frac{2x^2$ + 3x + 1}{x^2$ - 4x + 4}$$ has a hole at which x-value?

Accepted Answer

At $$x = 2$$, since the denominator and numerator both have a factor of $$(x-2).$$

Quadratic, Polynomial, Rational, Exponential, and Logarithmic Functions: Key Concepts and Properties

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