AP Style MCQ Unit 2 Notes

Topic 2.1: Change in Arithmetic and Geometric Sequences

Geometric Sequence Expressions: Problems 1-4 involve identifying the expression for the $n^{th}$ term of a geometric sequence based on a graph.

Topic 2.2: Change in Linear and Exponential Functions

Problem 5: An arithmetic sequence represents the number of seats in consecutive rows of a theater. The fourth row has 30 seats, and the eighth row has 54 seats. Find the number of seats in the tenth row.
Problem 6: Given an arithmetic sequence $sn$ with $s0 = 12$ and $s4 = -12$ , find the value of $s2$ .
Problems 7 & 8: Determine whether a sequence is arithmetic or geometric based on the constancy of successive terms' differences or proportional change.
Problem 9: Determine the value of $s_n$ in an arithmetic sequence given selected values.
Problem 10: Find the value of a geometric sequence with given conditions.
Problem 11: Model the number of students earning a qualifying score on an AP math exam using an arithmetic sequence. In year 3, the number was 52, and in year 6, it was 61. Find the function that gives the number of students in year $n$ .
Problem 12: Model the number of infected computers using a geometric sequence. On day 6, 750 computers were infected, and on day 10, 3200 computers were infected. Find the number of infected computers on day 14.

Topic 2.3: Exponential Functions

Problems 13-19: Determine the end behavior of various exponential functions.
- End behavior considers what happens to the function's value as $x$ approaches positive or negative infinity.
Problems 20-23: Given a table of values for an exponential function $f(x) = ab^x$ , determine if the function demonstrates exponential growth or decay based on the values of $a$ and $b$ .
- Exponential growth occurs when a > 0 and b > 1.
- Exponential decay occurs when a > 0 and 0 < b < 1.
Problems 24-29: Determine whether an exponential function $f(x) = ab^x$ is increasing or decreasing and whether its graph is concave up or concave down.
- If a > 0 and b > 1, the function is increasing and concave up.
- If a > 0 and 0 < b < 1, the function is decreasing and concave up.
Problems 30-32: Determine if an exponential function is increasing or decreasing at an increasing or decreasing rate.

Topic 2.4: Exponential Function Manipulation

Problems 36-40: Identify equivalent forms of given exponential functions through algebraic manipulation.
Problem 41: Given a function $f(x)$ that is a horizontal translation of an exponential function $g(x)$ , find an equivalent form for $f(x)$ that expresses it as a vertical dilation of $g(x)$ .
Problem 42: Given a function $f(x)$ that is a vertical dilation of an exponential function $g(x)$ , find an equivalent form for $f(x)$ that expresses it as a horizontal translation of $g(x)$ .
Problem 43: The graph of a function $f(x)$ is a horizontal dilation of $g(x)$ , and $g(x)$ is equivalent to a given expression. Find a possible expression for $f(x)$ .

Topic 2.5: Exponential Function Context and Data Modeling

Problem 44: Given a table of values for an increasing function $f(x)$ representing the number of cars stuck in traffic, determine the best model (linear, exponential, or logarithmic) for the data.
Problem 45: Model the number of cars traveling along a highway using a function $f(t)$ . The total number of cars increases by a given percentage each hour, and at a specific time, the total number of cars is known. Find an expression for $f(t)$ in terms of minutes.
Problem 46: Model the amount of Radon-222 remaining after $d$ days using the function $f(d) = A0 (1/2)^{\frac{d}{3.8}}$ , where $A0$ is the initial amount. Find a function that models the amount remaining after $w$ weeks.
Problem 47: Model the number of residents subscribing to a newspaper with exponential decay. The number decreases by a given percentage each year, and in a specific year, the number is known. Find an expression for the number of subscribers in terms of months.
Problem 48: Model the value of an investment using a function that increases by a given percentage each month. At a specific time, the investment value is known. Find an expression for the investment value in terms of years.
Problem 49: Given a table of values for a decreasing function, determine which model best applies to the situation.

Topic 2.6: Competing Function Model Validation

Problems 50-53: Analyze residual plots to determine the appropriateness of linear, quadratic, and exponential regression models. Understand that a random scatter of residuals indicates an appropriate model, while a pattern suggests the model is not appropriate.
Problem 54: Estimate the residual for a data point given a data set and its linear regression model.
Problem 55: Determine whether the residual of a point is positive or negative based on whether the model produces an overestimate or underestimate at that point.
Problem 56: Compare the residuals for different points in a data set and determine which point has a greater error in the model.

Topic 2.7: Composition of Functions

Problems 57-61: Evaluate composite functions using tables and graphs. For example, if $h(x) = f(g(x))$ , find $h(a)$ given values or graphs of $f(x)$ and $g(x)$ .
Problems 62-63: Find the expression for a composite function given expressions for the individual functions. For example, find $f(g(x))$ given $f(x)$ and $g(x)$ .

Topic 2.8: Inverse Functions

Problem 64: Given the graph of a piecewise defined function $f$ , identify the graph of its inverse $f^{-1}$ .
Problems 65-66: Given the graph of a piecewise defined function $f$ , determine the domain or minimum value of its inverse $f^{-1}$ .
Problem 67: Given a table of values for an increasing function $f$ , find the value of $f^{-1}(a)$ for a given $a$ .
Problems 68-71: Evaluate inverse functions using tables.
Problem 72: Determine the properties of the inverse function $f^{-1}$ (increasing/decreasing, concave up/down) given the properties of the original function $f$ .
- If $f$ is increasing and concave up, then $f^{-1}$ is also increasing but concave down.
Problem 73: Evaluate $f^{-1}(a)$ given the graph of a piecewise defined function $f$ .
Problem 74: Given that the graph of $f(x)$ is a transformation of $g(x)$ and a point $(a, b)$ is on the graph of $g(x)$ , determine which point is on the graph of $f^{-1}(x)$ .
Problem 75: Find an expression for the inverse of a rational function.
Problem 76: Determine whether the inverse of a function can be constructed and, if so, the domain or range of the inverse.

Topic 2.9: Logarithmic Expressions

Problems 77-78: Evaluate logarithmic functions.
Problem 79: Find the input value that yields a given output value for a logarithmic function.
Problem 80: Determine for which values of $x$ the graphs of two logarithmic functions intersect.

Topic 2.10: Inverses of Exponential Functions

Problem 81: Determine the inverse of an exponential function and its domain.
Problem 82: Given an exponential function $f(x) = ab^x$ and two points on its graph, determine which statements about the graph are true.
Problems 83-85: Determine whether a function is best modeled by an exponential or logarithmic function based on how the input and output values change.

Topic 2.11: Logarithmic Functions

Problems 86-88: Determine whether a logarithmic function is increasing or decreasing and whether its graph is concave up or concave down.
Problems 89-90: Evaluate limit statements about the graph of a logarithmic function.
Problems 91-92: Describe the end behavior of logarithmic functions.
Problem 93: Identify the graph of a function given its end behavior.

Topic 2.12: Logarithmic Function Manipulation

Problems 94-101: Manipulate logarithmic expressions using properties of logarithms to find equivalent expressions.
Problems 102-103: Determine how the graphs of two logarithmic functions are related (horizontal or vertical translation/dilation).
Problems 104-105: Simplify logarithmic expressions using properties of logarithms.

Topic 2.13: Exponential and Logarithmic Equations and Inequalities

Problems 106-111: Find the points of intersection of the graphs of exponential and logarithmic functions.
Problems 112-115: Solve exponential and logarithmic equations and inequalities.
Problems 116-117: Find the points of intersection of the graphs of exponential and logarithmic functions involving more complex expressions.
Problems 118-119: Find the $x$ -coordinate of the point of intersection of the graphs of logarithmic functions with different bases.
Problems 120-123: Solve conditions where a logarithmic function must satsify certain conditions to find out the range of the inputs that satisfy the condition
Problems 124-125: Given two functions and the function with its inverse, find out the x value, and range where is satsifies certain conditions for values larger or smaller than some constat.

Topic 2.14: Logarithmic Function Context and Data Modeling

Problem 126: Given a table of data representing the population of fish in a lake, use a logarithmic regression to predict the population at a specific time.
Problem 127: Given a table of values for a function, determine whether the function is best modeled by an exponential or logarithmic model.
Problem 128: Define transformations on a logarithmic function.
Problem 129: Model the rates for input and outputs and state when one rate is larger than the other one.
Problem 130: Describe the graph of the inverse of a function.
- If a function doubles its output each time the input values increase the it is a geometric function (think of $2^x$ ), which it's inverse if logarithmic.
Problem 131: Explain how well a log regression model fits a set of data
Problem 132: Given an exponential function that's value is continuously decaying, at what time will it reach a certain minimum?

Topic 2.15: Semi-log Plots

Problem 133: Describe how data will appear on a semi-log plot.
- Data that exhibit exponential behavior will appear linear when graphed on a semi-log plot with the vertical axis logarithmically scaled
Problem 134: Relate a graph to it's semi log graph
Problem 135: Relate Exponential and log functions through semi log data plots
Problems 136-137: Relate table graphs to their semi log scaled forms
Problem 138: Determine the semi log graph of a certain data set.
Problem 139: Determine concavity of a semi log function graphed in cartesian product
Problem 140: Connect linearity of semi log functions to cartesian graphs
Problem 141: Connect constants on semi log graph slopes and y intercepts to cartesian constants.
Problem 142: Determine an appropriate function given semi log regressions

To solve the first problem which involves identifying the expression for the $n^{th}$ term of a geometric sequence based on a graph, follow these steps:

Understand the Definition of a Geometric Sequence: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
- The general form of a geometric sequence is given by the formula: $an = a1 imes r^{(n-1)}$ where:
 - $a_n$ is the $n^{th}$ term,
 - $a_1$ is the first term,
 - $r$ is the common ratio,
 - $n$ is the term position.
Analyze the Graph: Look at the graph provided in the problem. You should be able to identify:
- The first term of the sequence (value at $n=1$ ).
- The values of subsequent terms.
Determine the Common Ratio: Calculate the ratio between consecutive terms of the sequence. To find the common ratio $r$ , divide the value of the second term by the value of the first term:
$r = \frac{a2}{a1}$
Write the Expression: Using the first term and the common ratio, insert these values into the general formula $an = a1 imes r^{(n-1)}$ to write the expression for the $n^{th}$ term.
Verify: Check if the terms generated by your expression match the values shown in the graph to ensure your expression is correct.