Q: Which of the following best describes the end behavior of the polynomial $$f(x) = -3x^4$ + 2x^2$ - 5$$?

As $$x \to \infty$$, $$f(x) \to -\infty$$; as $$x \to -\infty$$, $$f(x) \to -\infty$$

Q: An investment of $$70 increases in value at a continuous rate of 4.2$$\%$ per year. What is its value in 4 years?

$$70e^{0.042 \times 4}$$ \approx 83.32$$

Q: A town starts with $$10,000$$ residents and grows at a continuous rate of $$5\%$$ per year. What is the population after $$8$$ years?

$$10,000e$$^{0.05 \times 8}$$ \approx 14,918$$

Question 1

A ball is thrown and its height in meters, $$h$$, as a function of the horizontal distance $$d (in $$meters) is given by $$h(d) = -0.04d^2$ + 0.74d + 1.8. $$What is the maximum height the ball reaches?

Accepted Answer

$$h_{max} = 7.625$$ meters

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 8x + 7$$, what are the coordinates of its vertex?

Accepted Answer

$$(2, -1)$$

Question 3

Which of the following best describes the end behavior of the polynomial $$f(x) = -3x^4$ + 2x^2$ - 5$$?

Accepted Answer

As $$x 	o \infty$$, $$f(x) 	o -\infty$$; as $$x 	o -\infty$$, $$f(x) 	o -\infty$$

Question 4

Suppose $$Q = 120e$$^{0.07t}$$. What is the continuous growth rate k$ and the percent growth rate per year?

Accepted Answer

$$k = 0.07$$, percent growth rate = $$7\%$$

Question 5

An investment of $$70 increases in value at a continuous rate of 4.2$$\%$ per year. What is its value in 4 years?

Accepted Answer

$$70e^{0.042 	imes 4}$$ \approx 83.32$$

Question 6

Which of the following is the correct formula for the half-life $$T_{1/2} of a $$radioactive substance that decays at a continuous rate $$k$$?

Accepted Answer

$$T_{1/2} = \frac{\ln 2}{k}$$

Question 7

Given the exponential function $$f(x) = 0.85^x$$, which statement is true about its graph?

Accepted Answer

It is a decreasing function with horizontal asymptote $$y = 0$$

Question 8

Suppose $$1000 is $$invested in an account yielding interest at a nominal rate of $$8\%$$ per year. Find the effective annual yield if the interest is compounded quarterly.

Accepted Answer

$$8.24\%$$

Question 9

Rewrite $$\log 0.001$$ using exponents.

Accepted Answer

$$\log 0.001 = -3$$

Question 10

Order the numbers $$e, e$$^{1/3}$$, 2^2$, 6^1$, 10^1$ from smallest to largest.

Accepted Answer

$$e^{1/3}$$, $$2^2$$, e, $$6^1$$, $$10^1$$

Question 11

A town starts with $$10,000$$ residents and grows at a continuous rate of $$5\%$$ per year. What is the population after $$8$$ years?

Accepted Answer

$$10,000e$$^{0.05 	imes 8}$$ \approx 14,918$$

Question 12

Given $$Q = 290(1.827)^t$$, find the starting value $$a$$, the growth factor $$b$$, and the continuous growth rate $$k.$$

Accepted Answer

$$a = 290$$, $$b = 1.827$$, $$k = \ln(1.827) \approx 0.602$$

Question 13

Which of the following is the correct conversion of $$Q = 120(1.07)^t to $$the form $$Q = ae$$^{kt}$$?

Accepted Answer

$$Q = 120e$$^{0.0677t}$$

Question 14

If $$y = \log_{10}(x)$$, what is $$x in $$terms of $$y$$?

Accepted Answer

$$x = 10^y$$

Question 15

Suppose $$Q = 5000e$$^{0.06t}$$ and Q$$ = $$5400e^{0.05t}$$, both compounded continuously. When will the smaller investment catch up to the larger one?

Accepted Answer

$$t = \frac{\ln(5400/5000)}{0.06 - 0.05} \approx 7.83$$ years

Question 16

Which of the following is the correct expression for the exponential function with initial value $$a$$ and percent change $$r$$ per period?

Accepted Answer

$$Q = a(1 + r)^t$$

Question 17

Given the function $$f(x) = 2x^2$ - 4x + 1$$, determine if the parabola opens upward or downward and find its axis of symmetry.

Accepted Answer

Opens upward; axis of symmetry $$x = 1$$

Question 18

If $$y = \ln(x)$$, solve for $$x in $$terms of $$y.$$

Accepted Answer

$$x = e^y$$

Precalculus Study Guide: Quadratic, Polynomial, Exponential, and Logarithmic Functions

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