Q: Rewrite the quadratic function $$f(x) = x^2$ + 2x - 3 in $$vertex form and state the vertex $$(h, k). Is $$the vertex a minimum or maximum?

$$f(x) = (x + 1)^2$ - 4$$,\ \text{vertex}:\ (-1, -4),\ \text{minimum}$

Q: Which of the following best describes the end behavior of the rational function $$f(x) = \frac{x}{x^2$ + 5x - 36}$$?

As $$x \to \pm\infty$$, $$f(x) \to 0$$

Question 1

Rewrite the quadratic function $$f(x) = x^2$ + 2x - 3 in $$vertex form and state the vertex $$(h, k). Is $$the vertex a minimum or maximum?

Accepted Answer

$$f(x) = (x + 1)^2$ - 4$$,\ 	ext{vertex}:\ (-1, -4),\ 	ext{minimum}$

Question 2

A ball is thrown in the air from the top of a stage. Its height in meters above the ground as a function of time in seconds is $$h(t) = -4.9t^2$ + 25t + 8. $$How long does it take the ball to reach its maximum height?

Accepted Answer

$$t = 2.55$$ seconds

Question 3

Given 200 feet of fencing to build a rectangular corral, what dimensions maximize the enclosed area?

Accepted Answer

Length = 50 ft, Width = 50 ft

Question 4

Which of the following best describes the end behavior of the rational function $$f(x) = \frac{x}{x^2$ + 5x - 36}$$?

Accepted Answer

As $$x 	o \pm\infty$$, $$f(x) 	o 0$$

Question 5

Which of the following exponential functions represents exponential decay?

Accepted Answer

$$y = \left(\frac{1}{2}\right)^x$$

Question 6

Evaluate $$f(x) = 2(5)^x at$$ $$x = -3. $$Round your answer to four decimal places.

Accepted Answer

$$0.0160$$

Question 7

Find a formula for the exponential function that passes through the points $$(0, 6)$$ and $$(3, 750).$$

Accepted Answer

$$f(x) = 6(5)^x$$

Question 8

Which of the following is the correct logarithmic form of the equation $$19^x = y$$?

Accepted Answer

$$\log_{19}(y) = x$$

Question 9

Expand the logarithmic expression $$\ln(3ab \cdot 5c) as $$much as possible.

Accepted Answer

$$\ln(3) + \ln(a) + \ln(b) + \ln(5) + \ln(c)$$

Question 10

Condense the expression $$\ln(7) + \ln(x) - \ln(y) to a $$single logarithm.

Accepted Answer

$$\ln\left(\frac{7x}{y}\right)$$

Question 11

Solve for $$x$$: $$4^{-3x-2}$$ = $$4^{-x}.$$

Accepted Answer

$$x = -1$$

Question 12

A population of bacteria doubles every hour. If the culture started with 10 bacteria, which equation represents the population as a function of time $$t (in $$hours)?

Accepted Answer

$$A(t) = 10e$$^{\ln(2) t}$$

Question 13

The isotope Carbon-14 has a half-life of 5,730 years. A scientist has a sample of 295 grams. Which equation represents the amount of Carbon-14 left after $$t$$ years?

Accepted Answer

$$A(t) = 295e$$^{-0.000121t}$$

Question 14

Which of the following is the correct product rule for logarithms?

Accepted Answer

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

Question 15

Given the function $$f(x) = \log_3$(x)$$, what is the $$x-$$intercept?

Accepted Answer

$$(1, 0)$$

Question 16

Which of the following best describes the domain of the parent logarithmic function $$y = \log_b(x)$$?

Accepted Answer

$$(0, \infty)$$

Question 17

Which transformation describes the graph of $$g(x) = 3$$^{-x}$$ + 4$$ compared to $$f(x) = 3^x$$?

Accepted Answer

Reflected about the $$y-$$axis and shifted up 4 units

Question 18

Which of the following is the correct power rule for logarithms?

Accepted Answer

$$\log_b(m^n) = n \cdot \log_b(m)$$

Precalculus Unit 2 Practice Problems – Step-by-Step Study Guidance

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