Question 1

For the quadratic function $$y = a x^2$ - 18x + 13$$, determine the values of $$a so $$that the function has no x-intercepts.

Accepted Answer

$$[-8/13, \infty)$$

Question 2

Identify the graph of the rational function whose domain is $$(-\infty, 5) \cup (5, \infty).$$

Accepted Answer

Graph with a vertical asymptote at $$x=5$$ and two branches, one in quadrant II and one in quadrant IV.

Question 3

For the polynomial $$f(x) = x^4$ - 3x^3$ + 6x^2$ - 12x + 10$$, is there any real zero greater than 3?

Accepted Answer

No

Question 4

Given the rational function $$f(x) = \frac{11x - 121}{x + 17}$$, identify its vertical and horizontal asymptotes.

Accepted Answer

Vertical asymptote: $$x = -17$$, Horizontal asymptote: $$y = -11$$

Question 5

Consider the function $$f(x) = \frac{2}{x}. $$Calculate its average rate of change from $$x_1 = 16$ to x_2$ = 25.$$

Accepted Answer

$$-\frac{1}{4}$$

Question 6

Identify whether the function $$f(x) = 3x^4$ is even or odd, and describe its symmetry.

Accepted Answer

Even, Symmetric about y-axis

Question 7

Find the quotient of the following polynomials using synthetic division: $$(x^4 - 1296$$) \div (x - 6)$$.

Accepted Answer

$$x^3$$ + $$6x^2 - 36x + 216$

Question 8

Determine the end behavior of the graph of $$f(x) = -x^6$ + x^4$ using the Leading Coefficient Test.

Accepted Answer

Graph falls to the left and right

Question 9

Plot the function $$f(x) = (7/8)^x on $$the rectangular coordinate system. Which graph best represents this function?

Accepted Answer

Graph showing exponential decay, passing through $$(0,1)$$ and $$(1,7/8)$$

Question 10

Find $$\log_{23} 29 in $$terms of $$M$$ and $$N$$, if $$\log 23 = M$$ and $$\log 29 = N.$$

Accepted Answer

$$N/M$$

Question 11

Graph the equations $$f(x) = 2^x$$ and $$g(x) = \log_2$ x on $$the same coordinate system. Which graph correctly shows both functions?

Accepted Answer

Graph with $$f(x) as an $$increasing exponential curve and $$g(x) as a $$logarithmic curve passing through $$(1,0)$$

Question 12

Determine $$f(-1/2)$$ for $$f(x) = 15^x. $$Approximate the answer to the nearest thousandth.

Accepted Answer

0.258

Question 13

Draw the graph of the function $$f(x) = \left(\frac{1}{8}\$$right)^{-x}$$.

Accepted Answer

Graph showing exponential growth, passing through $$(0,1)$$ and $$(1,8)$$

Question 14

Which of the following is the logarithmic form of $$9^y = 450$$?

Accepted Answer

$$\log_9$ 450 = y$$

Question 15

Find $$x$$ for the equation $$5|\ln x| - 25 = 0.$$

Accepted Answer

$$x = 1/e^5$ 	ext{ or } e^5$

Question 16

A biologist models the population of bacteria in a lab as $$P(t) = 500 \cdot 2^t$$, where $$t is in $$hours. How many bacteria will there be after 3 hours?

Accepted Answer

$$4000$$

Question 17

A chemical reaction follows the decay model $$C(t) = 100e$$^{-0.2t}$$, where C$ is the concentration in mg/L and t$ is time in hours. What is the concentration after 5 hours? Round to the nearest mg/L.

Accepted Answer

$$37$$

Question 18

A company’s revenue is modeled by $$R(x) = 100x - x^2$, where x$ is the number of units sold. What is the maximum revenue?

Accepted Answer

$$2500$$

Question 19

A medical researcher uses the function $$T(x) = 98.6 + 0.5x to $$model a patient’s temperature $$T in $$degrees Fahrenheit after $$x$$ hours of treatment. If the temperature after 4 hours is 100.6°F, what was the initial temperature?

Accepted Answer

$$98.6^\circF$$

College Algebra: Key Concepts and Problem Types

Guida di studio - Domande di pratica

Study guide - Flashcard