Q: For the exponential function $$y = a b^x$$ with $$a \u003e 0$$ and $$0 \u003c b \u003c 1$$, which statement is true?

It is decreasing and concave up, with $$y \to 0 as$$ $$x \to \infty$$

Question 1

Which of the following is the equation of a linear function $$f(x)$$ such that $$f(0) = 5$$ and $$f(x)$$ increases by 3 for every increase of 1 in $$x$$?

Accepted Answer

$$f(x) = 3x + 5$$

Question 2

Given an exponential function $$f(x)$$ such that $$f(0) = 5$$ and $$f(x) is $$multiplied by 3 for each increase of 1 in $$x$$, which equation represents $$f(x)$$?

Accepted Answer

$$f(x) = 5 	imes 3^x$$

Question 3

A linear function $$f$$ satisfies $$f(3) = 16$$ and $$f(5) = 4. $$What is the equation for $$f(x)$$?

Accepted Answer

$$f(x) = -6x + 34$$

Question 4

Which quadratic function has zeros at $$x = -2$$ and $$x = 3$$, and satisfies $$f(0) = 6$$?

Accepted Answer

$$f(x) = 2(x + 2)(x - 3)$$

Question 5

The secant line for the graph of $$y = \log x$$ over the interval $$[1, 100]$$ has which slope?

Accepted Answer

$$\dfrac{2 - 0}{99}$$

Question 6

An exponential function models the growth of a population over time $$t (in $$years), with initial value $$P_0$$ and a growth rate of 5% per year. Which equation correctly represents this model?

Accepted Answer

$$P(t) = P_0$ 	imes 1.05^t$$

Question 7

The graph of $$y = \dfrac{5}{2} - 2x is$$:

Accepted Answer

Decreasing with slope $$-2$$ and $$y-$$intercept $$\dfrac{5}{2}$$

Question 8

For the exponential function $$y = a b^x$$ with $$a \u003e 0$$ and $$0 \u003c b \u003c 1$$, which statement is true?

Accepted Answer

It is decreasing and concave up, with $$y 	o 0 as$$ $$x 	o \infty$$

Question 9

The graph of $$y = \log_b x$$ for $$b \u003e 1 is$$:

Accepted Answer

Increasing and concave down, with $$x-$$intercept at $$x = 1$$ and vertical asymptote at $$x = 0$$

Question 10

The function $$y = 6x^2$ - 12x + 5 is a $$parabola that is concave up. What are the coordinates of its vertex?

Accepted Answer

$$(1, -1)$$

Question 11

What is the value of $$\log_2$ 16$$?

Accepted Answer

$$4$$

Question 12

What is the value of $$\log_4$ 4$$?

Accepted Answer

$$1$$

Question 13

What is the value of $$\log_2$ (-4)$$?

Accepted Answer

Undefined

Question 14

Simplify $$\log_4$ \left( \dfrac{1}{16} \right).$$

Accepted Answer

$$-2$$

Question 15

Express $$\log_2$ 64$$ using the change of base formula to base 4.

Accepted Answer

$$\dfrac{\log_4$ 64}{\log_4$ 2}$$

Question 16

Simplify $$e^{\ln 5x^2}.$$

Accepted Answer

$$5x^2$$

Question 17

Simplify $$\ln(\log 10^7$).$$

Accepted Answer

$$\ln 7$$

Question 18

Solve for $$x$$: $$3^x = 12.$$

Accepted Answer

$$x = \log_3$ 12$$

Question 19

Solve for $$x$$: $$\ln(x - 1) = 3.$$

Accepted Answer

$$x = e^3$ + 1$$

Question 20

If a population grows exponentially at 5% per year, how many years does it take to double? (Express your answer in terms of natural logarithms.)

Accepted Answer

$$t = \dfrac{\ln 2}{0.05}$$

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

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