Comprehensive Mathematical Functions and Sequence Analysis Study Guide

Recursive Sequences and Iterative Processes

A recursive sequence is a set of numbers where each term is generated based on the previous term using a specific rule or formula. In one exercise, a sequence is defined by the recursive formula $f(n+1) = -2f(n)$. This formula indicates that to find the next term in the sequence, the current term must be multiplied by a constant factor of $-2$.

In the specific application provided, the first term of the sequence is given as $f(1) = -1.5$. To calculate the second term ($n=1$), the formula is applied as follows: $f(1+1) = -2 \times f(1)$ $f(2) = -2 \times (-1.5)$ $f(2) = 3$ Thus, the term immediately following $-1.5$ in this geometric recursive sequence is $3$.

Another example of a recursive sequence involves a subtraction rule. Given the function $f(n+1) = f(n) - 2$ and an initial value of $f(1) = 10$, we can find subsequent terms through iteration. To find $f(3)$, we must first determine $f(2)$. Since $f(2) = f(1) - 2$, substituting the known value yields $f(2) = 10 - 2 = 8$. Following this, we find $f(3) = f(2) - 2$, which results in $f(3) = 8 - 2 = 6$.

Modeling Free Fall and Terminal Heights

Physical phenomena, such as a penny dropped from bleachers, can be modeled using function tables that track height over time. In a provided data set representing the function $h(t)$, where $h$ is the height in meters and $t$ is the time in seconds, the following values were recorded:

• At $t = 0\,s$, height $h(t) = 2\,m$

• At $t = 0.1\,s$, height $h(t) = 1.951\,m$

• At $t = 0.2\,s$, height $h(t) = 1.804\,m$

• At $t = 0.3\,s$, height $h(t) = 1.559\,m$

• At $t = 0.4\,s$, height $h(t) = 1.216\,m$

• At $t = 0.5\,s$, height $h(t) = 0.775\,m$

• At $t = 0.6\,s$, height $h(t) = 0.236\,m$

• At $t = 0.7\,s$, height $h(t) = -0.401\,m$

• At $t = 0.8\,s$, height $h(t) = -1.136\,m$

The objective is to determine when the penny hits the ground, which corresponds to the point where the height $h(t) = 0$. By observing the table, the height is positive at $0.6\,seconds$ ($0.236\,m$) and becomes negative at $0.7\,seconds$ ($-0.401\,m$). Because $0.236$ is closer to zero than $-0.401$, the time it takes for the penny to hit the ground, rounded to the nearest tenth of a second, is approximately $0.6\,seconds$.

Linear Relationships and Intercept Analysis

Linear graphs represent relationships between two variables where the rate of change is constant. In a scenario evaluating the amount of money Carly has left ($y$) after purchasing packs of baseball cards ($x$), the graph illustrates a downward trend. The $y$-axis represents "Money Left (in $)" and the $x$-axis represents "Packs of Cards Bought."

To find out how much money Carly has before making any purchases, one must identify the $y$-intercept of the graph, which is the value of $y$ when $x = 0$. Looking at the coordinate $(0, 20)$, it is clear that Carly starts with $\$20$. Each subsequent purchase reduces this total, following a linear path toward the $x$-axis.

Arithmetic Sequences and Common Differences

An arithmetic sequence is characterized by a common difference between successive terms. This difference is found by subtracting any term from the term that follows it. For the sequence $9, -1, -11, -21, \dots$, we can calculate the difference ($d$) as: $d = -1 - 9 = -10$ $d = -11 - (-1) = -10$ $d = -21 - (-11) = -10$ Because the value added to each term to get the next is consistently $-10$, the common difference is defined as $-10$.

Proportional Relationships and Unit Rates

Graphs of proportional relationships pass through the origin $(0,0)$ and have a constant ratio. In the case of calories in strawberries, a graph relates the "Number of Strawberries" ($x$) to the "Number of Calories" ($y$). Key points on this graph include $(3, 9)$, $(4, 12)$, $(5, 15)$, and $(8, 24)$.

By calculating the unit rate ($\frac{y}{x}$), we find that $\frac{9}{3} = 3$, $\frac{12}{4} = 3$, and $\frac{15}{5} = 3$. This demonstrates that each individual strawberry contains exactly $3$ calories. Similarly, in a graph of eraser costs, the points $(2, 1.00)$, $(4, 2.00)$, $(6, 3.00)$, and $(8, 4.00)$ show a unit rate of $\$0.50$ per eraser. Based on this proportionality, seven erasers would cost $7 \times 0.50 = \$3.50$.

Function Notation and Variable Interpretation

Function notation like $C(F)$ is used to express how one variable depends on another. In the context of temperature conversion, the formula for Celsius is provided as $C(F) = \frac{5}{9}(F - 32)$, where $F$ is the temperature in degrees Fahrenheit. If a temperature is recorded as $76.1^{\circ}F$, the expression $C(76.1)$ represents the result of converting that specific Fahrenheit value into degrees Celsius.

Mathematical equations can also be rewritten in function notation by isolating the dependent variable. For the equation $x - y = 3$, we solve for $y$ to establish the function $f(x)$. Subtracting $x$ from both sides gives $-y = -x + 3$, and multiplying by $-1$ yields $y = x - 3$. Therefore, in function notation, this is written as $f(x) = x - 3$.

Inverse Functions

An inverse function, denoted as $f^{-1}(x)$ or sometimes represented by a different letter like $h(x)$, effectively "undoes" the operation of the original function. To find the inverse of $f(x) = x + 3$, the variables are swapped and solved for $y$:

1. Start with $y = x + 3$

2. Swap variables: $x = y + 3$

3. Solve for $y$: $y = x - 3$ Thus, the inverse function is $h(x) = x - 3$.

Analyzing Graphs: Increasing Intervals and Minima

A function is considered increasing on an interval if, as the $x$-values increase, the $y$-values also increase. In a complex graph showing local extrema, the function increases between its local minimum and local maximum points. For a graph with a relative minimum at approximately $(-1.6, -5.6)$ and a relative maximum at $(0.8, 11.4)$, the function is increasing on the interval of $x$ values from $-1.6$ to $0.8$. Specific sub-intervals within this range where the function is increasing include $[-1.6, 0]$ and $[0, 0.8)$.

Local minima are the lowest points in a particular neighborhood of a graph. Determining the interval containing a local minimum involves identifying the $x$-range where the graph dips into a "valley." For a function graphed with a visible low point around $x = -4$, the interval $[-5, -3]$ would contain that local minimum.

Defining Functions and Sign Analysis

A set of ordered pairs is classified as a function if each input ($x$) corresponds to exactly one output ($y$). For a set containing points $(1, 1), (3, 4), (4, -2),$ and $(3, 2)$, the input $x = 3$ is associated with two different outputs ($4$ and $2$). To transform this set into a function, one of the pairs with the duplicate input must be removed—for instance, removing $(3, 4)$ would satisfy the vertical line test.

Sign analysis involves determining where a function is positive (f(x) > 0) or negative (f(x) < 0). Based on a graph with x-intercepts (roots) at approximately $-3, -1.1,$ and $0.9$, the function falls below the $x$-axis in two distinct regions. The function is negative (f(x) < 0) over the interval $(-\infty, -3)$ and again over the interval $(-1.1, 0.9)$.