Algebra and Functions Quiz

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.