Algebra and Functions Quiz

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Questions and Answers

What is the number of bacteria present after 10 hours if the growth is modeled by B(t) = 140e^(0.535t)?

29,485 (correct)

14,000

70,000

140

Which of the following best describes the graph behavior of the function g based on the provided data points?

Decreasing and quadratic

Increasing and quadratic

Increasing and linear

Decreasing and linear (correct)

What is the outcome of the composition f(g(0)) if f(x) = x^2 + 2 and g(x) = x - 5?

2 (correct)

-3

Which equation represents the secant line for the function f on the interval -2 ≤ x ≤ 4?

y = -2x + 3 Signup and view all the answers

For the function h(x) = (2x - 5)^3, what are the functions f and g such that h(x) = f(g(x))?

f(x) = x^3, g(x) = 2x - 5 Signup and view all the answers

What is the inverse of the function y = 5 - x^2, and is this inverse a function?

y = ±√(5 - x), No Signup and view all the answers

What is the range of the function?

(-∞, 7/4) U (7/4, 3) U (3, ∞) Signup and view all the answers

Which of the following represents the correct horizontal asymptote for the function?

y = 3 Signup and view all the answers

What is the vertical asymptote of the function?

x = 2 Signup and view all the answers

For which value of x does the function have a zero?

x = 1/3 Signup and view all the answers

What is the domain of the function?

(-∞, -2) U (-2, 2) U (2, ∞) Signup and view all the answers

What formula represents the volume of the box formed from the cardboard?

V(x) = x(8 - 2x)(11 - 2x) Signup and view all the answers

What is the prediction for the cost of an item in the year 2019 using the provided linear model?

14.69 Signup and view all the answers

Which of the following describes the end behavior of the function as x approaches negative infinity?

f(x) → 3 Signup and view all the answers

What is the average rate of change of the function f(x) = x^2 - 3x from x₁ = 2 to x₂ = 7?

10 Signup and view all the answers

At what point does the function f(x) = 4 - x^2 have a constant rate of change?

At the vertex Signup and view all the answers

For the polynomial p(x) = (x - 3i)(x + 3i)(x - 4)², what is the degree of the polynomial?

4 Signup and view all the answers

What are the extrema of the function y = x^4 - 4x^3 + 3x?

Maximum: (0.554, 1.076); Minimum: (2.912, -18.13) Signup and view all the answers

What is the solution set for the inequality x^3 + 5x^2 - 9x < 45?

(-10, -5) U (-3, 3) Signup and view all the answers

Identify the zeros and their multiplicities for the function f(x) = (x – 2)^2(x + 3).

x = 2 (multiplicity 2), x = -3 (multiplicity 1) Signup and view all the answers

What is the maximum volume of the given object?

60,013 in^3 Signup and view all the answers

What value of x corresponds to the maximum volume?

1.525 in Signup and view all the answers

What is the number of elk when t = 25 years?

402.5 Signup and view all the answers

What is the horizontal asymptote of the elk population model?

780 Signup and view all the answers

What is the 31st term of the sequence defined by a(n) = 22 - 3(n - 1)?

-68 Signup and view all the answers

Which of the following is the correct explicit formula for the geometric sequence with a₁ = 12 and r = -1/2?

a_n = 12(-1/2)^(n - 1) Signup and view all the answers

What is the general rule for the arithmetic sequence where a₁ = 20 and d = 3?

a_n = 20 + 3(n - 1) Signup and view all the answers

What is the exponential function representing the bacteria culture after 8 hours, starting with 3000 bacteria and increasing by 30% every hour?

N(t) = 3000(1.3)^t Signup and view all the answers

Which of the following indicates the behavior of f(x) = -3(2)^x as x approaches positive and negative infinity?

lim(x→-∞) f(x) = -∞, lim(x→∞) f(x) = 1 Signup and view all the answers

Study Notes

AP Precalculus Review Notes

Increasing/Decreasing Intervals: Identify intervals where a function is increasing or decreasing. Example: For f(x) = (x - 3)² - 4, the function is increasing for x > 3 and decreasing for x < 3.
Positive/Negative Intervals: Determine intervals where a function is positive (above the x-axis) or negative (below the x-axis). Example: For f(x), the function is positive for (-∞, 1) and (5,∞) and negative for (1.5, 5).
Evaluating a Function: Substitute the given input value (x-value) into the function to find the output (y-value). Example: Given f(x) = 4x² - 3x + 2, f(-3) is 47.
Average Rate of Change (AROC): Calculate the change in the function's value over a specific interval divided by the change in the input values. Example: For f(x) = x² - 1 over [-2, 4] AROC is 2.
Secant Line Equation: Find the equation of the line connecting two points on a graph. Example: The secant line equation crossing (2, -2) and (7, 28) is y = 6x - 14.
Average Rate of Change of a Function (from x₁ to x₂): Given a function f(x) and two x-values, x₁ and x₂, the average rate of change is calculated using (f(x₂) - f(x₁))/(x₂ - x₁). Example: AROC from x = 2 and x = 7 is 10, on a given function.
Table of Average Rate of Change: Given specific intervals and their corresponding Average Rate Of Change for a given function. Example: For y = 4 - x², the average rate of change across given intervals is listed.
Difference in Average Rates of Change: Find the difference between the average rates of change across two intervals of interest. Example: The difference between average rates of change of a given function, is 2.
Constant Rate of Change: Points where the rate of change doesn't change consistently; Usually found at vertices. Example: The rate of change is constant at the vertex.
Equation for Average Rate of Change: A linear equation y = -2x. represents average rate of change versus x value.
Polynomial with Real Coefficients: Identify roots and multiplicity and construct a polynomial with real coefficients. Example: A polynomial with degree 4 and zero values: -3i, and 4 (multiplicity 2) is p(x) = (x²- 8x² +25x² - 72x +144).
Determining Even/Odd Functions (Algebraically): Test for even or odd functions by substituting -x into the function and comparing it with f(x) or -f(x); Even functions f(-x)=f(x) and odd functions f(-x)=-f(x).
Solving Inequalities with Sign Analysis: Analyze inequalities to locate solution intervals. Example: Solve x³ + 5x² - 9x < 45 by determining intervals where the function is positive or negative.
Extrema (Maximum/Minimum): Find maximum and minimum values using technology.. Example: Given y = x⁴ − 4x³ + 3x, the max/min values for x values are determined via a calculator.
Domain: The set of possible input values (x-values) for a function, including restrictions from variables.
Vertical Asymptotes: Line that the graph of a function approaches but never touches as x approaches a certain value. Example: In a given function, a vertical asymptote is found at x = 2.
Horizontal Asymptotes: A line that the graph of a function approaches as x approaches positive or negative infinity. Example: In a given function, a horizontal asymptote is found at y = 3.
Hole Coordinates: Point where a function appears to have a hole but it is not defined at that specific x value usually found by factoring and canceling common factors. Example: A hole at (-2, 7/4).
End Behavior: Describe the behavior of a function as x approaches positive or negative infinity using limit notation. Example: limit (x → ∞) and limit (x→ -∞ ) are evaluated.
Zeros of a function: Points where a function intersects the x-axis or when y=0. Example; The function zeros in the given function are found to be solved.
Binomial Theorem: Expand binomial expressions. Example: (2x − 1)⁴ = 16x⁴ − 32x³ + 24x² − 8x + 1 is expanded to evaluate a given binomial.
Inequalities using sign analysis: Determine solution intervals for inequalities.
Linear Regression: Create linear models given data. Example: Find a linear regression equation from various points, to evaluate a certain input
Arithmetic Sequences: Determine the terms or formulas in a sequence. Example: Find a general term for a given arithmetic with a₁, and d.
Geometric Sequences: Find general or individual terms of a given sequence. Example: Determine an for the specified geometric sequence.
Exponential Growth/Decay: Identify the type based on the growth/decay factor or rate.
End Behavior of a function (limits): Determine the behavior as x-values go to ∞ or –∞. Example: Given limit x→ −∞ and limit x→∞ values are evaluated graphically.
Maximum Volume of a Box: Find the maximum volume of a box formed by cutting squares from a larger sheet of cardboard. Example given 8 in * 11 in cardboard, the max volume of the box is found for a given size square.
Horizontal Asymptotes: A horizontal line that the graph of a function approaches as x approaches positive or negative infinity.
Graphing rational functions: Graphing functions with asymptotes, intercepts, and holes.
Describing the graph of function k: (from a given graph): Determine the key features of function k, such as increasing/decreasing, concavity, and its behavior around specific x-values or regions in a graph.
Average rate of change over an interval (for a function g): Find the change in the function's y-values over a specific x-interval in a graph.

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Test your understanding of algebraic functions and their properties with this quiz. Questions cover topics such as function behavior, composition, inverse functions, asymptotes, and more. Get ready to apply your knowledge to different scenarios and formulas.