Comprehensive Study Guide: Linear Functions, Arithmetic Sequences, Domain and Range, and Scattergpuw. Plots

Linear Functions and Function Notation

• Overview of Function Notation

  • Linear equations can be written in function notation by replacing the dependent variable $y$ with $f(x)$, read as "$f$ of $x$".

  • Function notation explicitly identifies oni I’m $x$ as the independent variable (input) and $f(x)$ as the dependent variable (output).

• Converting Linear Equations to Function Notation

  • Problem 1: Rewrite $y = 3x + 7.5$ in function notation.

  • Expression: $f(x) = 3x + 7.5$

  • Problem 2: Rewrite $8x + 2y = 3$ in function notation.

  • Step 1: Isolate the $y$-term: $2y = -8x + 3$

  • Step 2: Divide all terms by $2$: $y = -4x + 1.5$

  • Step 3: Replace $y$ with $f(x)$: $f(x) = -4x + 1.5$

• Evaluating Linear Functions

  • Given the function $f(x) = -2.5x + 7$:

  • Finding $f(3)$:

    • Substitute $x = 3$ into the function: $f(3) = -2.5(3) + 7$

    • Simplify: $f(3) = -7.5 + 7 = -0.5$

  • Finding $x$ when $f(x) = 7$:

    • Set the function output to $7$: $7 = -2.5x + 7$

    • Subtract $7$ from both sides: $0 = -2.5x$

    • Divide by $-2.5$: $x = 0$

Intercepts, Zeros, and Solutions of Linear Functions

• Definitions and Terminology

  • $x$-Intercept: The point where a graph crosses or touches the $x$-axis. At this point, the $y$-value is always zero ($y = 0$).

  • Alternative Names for $x$-Intercepts:

  • Solutions

  • Roots

  • Zeros

  • $y$-Intercept: The point where a graph crosses or touches the $y$-axis. At this point, the $x$-value is always zero ($x = 0$).

• Procedures for Finding Intercepts

  • To find the $x$-intercept: Set $y$ (or $f(x)$) equal to zero ($0$) and solve the resulting equation for $x$.

  • To find the $y$-intercept: Set $x$ equal to zero ($0$) and evaluate the function for $y$ (or $f(0)$).

• Algebraic Calculation Examples

  • Function 1: $f(x) = 2x + 3$

  • $x$-intercept: Set $f(x) = 0 \rightarrow 0 = 2x + 3 \rightarrow 2x = -3 \rightarrow x = -\frac{3}{2} = -1.5$. Ordered pair: $\left(-\frac{3}{2}, 0\right)$.

  • $y$-intercept: Set $x = 0 \rightarrow f(0) = 2(0) + 3 = 3$. Ordered pair: $(0, 3)$.

  • Function 2: $f(x) = -5x$

  • $x$-intercept: Set $f(x) = 0 \rightarrow 0 = -5x \rightarrow x = 0$. Ordered pair: $(0, 0)$.

  • $y$-intercept: Set $x = 0 \rightarrow f(0) = -5(0) = 0$. Ordered pair: $(0, 0)$.

  • Function 3: $f(x) = -x + 3$

  • $x$-intercept: Set $f(x) = 0 \rightarrow 0 = -x + 3 \rightarrow x = 3$. Ordered pair: $(3, 0)$.

  • $y$-intercept: Set $x = 0 \rightarrow f(0) = -(0) + 3 = 3$. Ordered pair: $(0, 3)$.

• Graphing Functions and Determining Zeros

  • Function: $f(x) = \frac{1}{2}x - 2$

  • Zero calculation: Set $f(x) = 0 \rightarrow \frac{1}{2}x - 2 = 0 \rightarrow \frac{1}{2}x = 2 \rightarrow x = 4$.

  • Zero of the equation: $x = 4$

  • Function: $f(x) = \frac{4}{3}x + 4$

  • Zero calculation: Set $f(x) = 0 \rightarrow \frac{4}{3}x + 4 = 0 \rightarrow \frac{4}{3}x = -4 \rightarrow x = -3$.

  • Zero of the equation: $x = -3$

  • Function: $f(x) = 4x - 4$

  • Zero calculation: Set $f(x) = 0 \rightarrow 4x - 4 = 0 \rightarrow 4x = 4 \rightarrow x = 1$.

  • Zero of the equation: $x = 1$

Domain and Range of Discrete and Continuous Relations

• Discrete Domain and Range

  • Definition: Discrete data consists of disconnected, individual points (such as ordered pairs, lists, tables, or scattered points on a graph).

  • Domain: The set of all $x$-values (inputs). Listed from least to greatest inside curly braces $\{\}$, with duplicate values omitted.

  • Range: The set of all $y$-values (outputs). Listed from least to greatest inside curly braces $\{\}$, with duplicate values omitted.

• Discrete Examples and Practice Problems

  • Mapping Diagram Example:

  • Inputs ($x$): $\{5, 8, 10, 12, 14\}$

  • Outputs ($y$): Arrows point from $5 \rightarrow 6$, $8 \rightarrow 11$, $10 \rightarrow 11$, $12 \rightarrow 11$, $14 \rightarrow 6$

  • Is it a function? Yes, because every input $x$ maps to exactly one output $y$.

  • Domain: $\{5, 8, 10, 12, 14\}$

  • Range: $\{6, 11\}$

  • Table Example 1:

  • Values: $x \in \{-2, -1, 0, 1, 2\}$, $f(x) \in \{4, 1, 0, 1, 5\}$

  • Domain: $\{-2, -1, 0, 1, 2\}$

  • Range: $\{0, 1, 4, 5\}$

  • Table Example 2:

  • Values: $x \in \{0, 2, 4, 6\}$, $y \in \{5, 11, 17, 23\}$

  • Domain: $\{0, 2, 4, 6\}$

  • Range: $\{5, 11, 17, 23\}$

  • Ordered Pairs Example 1: $\{(1, 2), (-1, 2), (3, 4), (-3, 4)\}$

  • Domain: $\{-3, -1, 1, 3\}$

  • Range: $\{2, 4\}$

  • Ordered Pairs Example 2: $\{(1, 8), (5, 3), (7, 6), (2, 2), (8, 4), (3, 9), (5, 7)\}$

  • Is it a function? No, because the input $x = 5$ maps to two different outputs ($3$ and $7$).

  • Domain: $\{1, 2, 3, 5, 7, 8\}$

  • Range: $\{2, 3, 4, 6, 7, 8, 9\}$

  • Scatter Graph of Discrete Points:

  • Plotted points: $(-4, -2)$, $(-2, 1)$, $(-2, 2)$, $(-1, 0)$, $(3, -2)$, $(3, 1)$

  • Is it a function? No, because inputs $x = -2$ and $x = 3$ each map to multiple outputs.

  • Domain: $\{-4, -2, -1, 3\}$

  • Range: $\{-2, 0, 1, 2\}$

  • Discrete Word Problem (Pumpkin Patch):

  • Riley picks $1$ to $3$ pumpkins. Cost function: $p(x) = 12x + 20$, where $x$ is the number of pumpkins picked and $p(x)$ is total cost.

  • Evaluation:

    • For $x = 1$: $p(1) = 12(1) + 20 = 32$

    • For $x = 2$: $p(2) = 12(2) + 20 = 44$

    • For $x = 3$: $p(3) = 12(3) + 20 = 56$

  • Domain: $\{1, 2, 3\}$

  • Range: $\{32, 44, 56\}$

• Continuous Domain and Range

  • Definition: Continuous data consists of connected lines, curves, or rays that encompass all real values within specified boundaries.

  • Reading Continuous Graphs:

  • Domain: Read the graph from left to right to identify the lowest and highest $x$--values used.

  • Range: Read the graph from bottom to top to identify the lowest and highest $y$--values used.

• Notations for Continuous Data:

  • Inequality Notation: Uses inequality symbols ($<, \le, >, \ge$).

  • Open circles ($\circ$) use strict inequalities ($<$ or $>$).

  • Closed circles ($\bullet$) use non-strict inequalities ($\le$ or $\ge$).

  • Interval Notation: Uses brackets and parentheses to define numerical intervals.

  • Parentheses ( ) are used for open circles ($\circ$) and for infinite boundaries ($\infty$ or $-\infty$).

  • Square brackets [ ] are used for closed circles ($\bullet$).

  • Format: [least value, greatest value]

• Continuous Examples and Practice Problems

  • Problem 1 (Line Segment):

  • Closed circle at $(-4, -2)$ and open circle at $(3, 5)$.

  • Domain (Inequality): $-4 \le x < 3$

  • Domain (Interval Notation): $[-4, 3)$

  • Range (Inequality): $-2 \le y < 5$

  • Range (Interval Notation): $[-2, 5)$

  • Problem 2 (Ray Extending Left and Down):

  • Closed endpoint at $(3, 5)$ extending infinitely left towards $x \rightarrow -\infty$ and down towards $y \rightarrow -\infty$.

  • Domain (Inequality): $x \le 3$

  • Domain (Interval Notation): $(-\infty, 3]$

  • Range (Inequality): $y < 5$

  • Range (Interval Notation): $(-\infty, 5)$

  • Problem 3 (Bounded Curve):

  • Closed boundaries from $x = -4$ to $x = 4$ and $y = 2$ to $y = 8$

  • Domain (Inequality): $-4 \le x \le 4$

  • Domain (Interval Notation): $[-4, 4]$

  • Range (Inequality): $2 \le y \le 8$

  • Range (Interval Notation): $[2, 8]$

  • Problem 4 (Ray Extending Up and Right):

  • Closed endpoint at $(4, 10)$ extending infinitely right ($x \rightarrow \infty$) and up ($y \rightarrow \infty$).

  • Domain (Inequality): $x \ge 4$

  • Domain (Interval Notation): $[4, \infty)$

  • Range (Inequality): $y \ge 10$

  • Range (Interval Notation): $[10, \infty)$

  • Problem 5 (Vertical Line):

  • Vertical line passing through $x = 4$, extending infinitely in both vertical directions.

  • Domain (Inequality): $x = 4$

  • Domain (Interval Notation): $[4, 4]$ or $\{4\}$

  • Range (Inequality): All real numbers ($-\infty < y < \infty$)

  • Range (Interval Notation): $(-\infty, \infty)$

  • Problem 6 (Symmetrical Bounded Region):

  • Region bounded between $x = -8$ and $x = 8$, and $y = -8$ and $y = 8$

  • Domain (Inequality): $-8 \le x \le 8$

  • Domain (Interval Notation): $[-8, 8]$

  • Range (Inequality): $-8 \le y \le 8$

  • Range (Interval Notation): $[-8, 8]$

  • Problem 7 (Continuous Ray starting at $(0, 1)$):

  • Starts at $(0, 1)$ passing through $(2, 2)$, $(4, 3)$, $(6, 4)$ into quadrant I.

  • Is it a function? Yes, passes the vertical line test.

  • Domain: $[0, \infty)$ or $x \ge 0$

  • Range: $[1, \infty)$ or $y \ge 1$

Arithmetic Sequences

• Core Concepts and Definitions

  • Arithmetic Sequence: A sequence of numbers in which the common difference, $d$, between any two consecutive terms is constant.

  • Common Difference ($d$): The constant value added to or subtracted from each term to get the next term ($d = a_n - a_{n-1}$).

• Subscript Notation vs. Function Notation

  • Subscript Notation ($a_n$):

  • $a_n$ represents the term value.

  • $n$ represents the term number/position ($1^{\text{st}}, 2^{\text{nd}}, 3^{\text{rd}}, \dots$).

  • Example Sequence ($3, 10, 17, 24, \dots$): $a_1 = 3$, $a_2 = 10$, $a_3 = 17$.

  • Function Notation ($f(n)$):

  • $f(n)$ represents the term value.

  • $n$ represents the term number/position.

  • Example Sequence ($15, 22, 29, 36, \dots$): $f(1) = 15$, $f(2) = 22$, $f(3) = 29$.

• Identifying Arithmetic Sequences

  • Sequence $2, 4, 7, 11, \dots$: No (differences are $+2, +3, +4$; not constant).

  • Sequence $1, 4, 7, 10, \dots$: Yes, common difference $d = 3$

  • Sequence $-4, -8, -12, -16, \dots$: Yes, common difference $d = -4$

• Extending Sequences (Finding Next Three Terms)

  • Sequence $6, 10, 14, 18, \dots$: Common difference $d = 4$. Next three terms: $22, 26, 30$

  • Sequence $15, 9, 3, -3, \dots$: Common difference $d = -6$. Next three terms: $-9, -15, -21$

  • Sequence $-7, -2, 3, 8, \dots$: Common difference $d = 5$. Next three terms: $13, 18, 23$

• Explicit Formulas for Arithmetic Sequences

  • Definition: An explicit formula allows finding any term ($a_n$) in a sequence directly without calculating previous terms. Inputting the term position $n$ outputs the term value.

  • General Explicit Formula:     $a_n = a_1 + (n - 1)d$

  • $a_n$ = $n^{\text{th}}$ term

  • $a_1$ = first term

  • $n$ = term number/position

  • $d$ = common difference

  • Explicit Formula Practice Problems:

  • Problem 1: Given sequence $7, 13, 19, 25, \dots$

    • Identify parameters: $a_1 = 7$, $d = 6$

    • Formula derivation: $a_n = 7 + (n - 1)6 = 7 + 6n - 6 = 6n + 1$

    • Simplified explicit formula: $a_n = 6n + 1$

    • Find the $23^{\text{rd}}$ term ($a_{23}$): $a_{23} = 6(23) + 1 = 138 + 1 = 139$

  • Problem 2: Given sequence $-13, -11, -9, -7, \dots$

    • Table values: $(1, -13), (2, -11), (3, -9), (4, -7)$

    • Identify parameters: $a_1 = -13$, $d = 2$

    • Formula derivation: $a_n = -13 + (n - 1)2 = 2n - 15$

    • Simplified explicit formula: $a_n = 2n - 15$

    • Find the $17^{\text{th}}$ term ($a_{17}$): $a_{17} = 2(17) - 15 = 34 - 15 = 19$

  • Problem 3: Given sequence $11, 2, -7, -16, -25, \dots$

    • Identify parameters: $a_1 = 11$, $d = -9$

    • Formula derivation: $a_n = 11 + (n - 1)(-9) = -9n + 20$ (or $y = -9x + 11$ relative to indexing)

    • Simplified explicit formula: $a_n = -9n + 11$

    • Find the $10^{\text{th}}$ term ($a_{10}$): $a_{10} = -9(10) + 11 = -79$

  • Problem 4: Given sequence $12.5, 10, 7.5, 5, 2.5, \dots$

    • Identify parameters: $a_1 = 12.5$, $d = -2.5$

    • Simplified explicit formula: $a_n = -2.5n + 12.5$

    • Find the $25^{\text{th}}$ term ($a_{25}$): $a_{25} = -2.5(25) + 12.5 = -62.5 + 12.5 = -50$

  • Problem 5 (Sticker Collection Word Problem):

    Dwas

  • Problem 6 (Converting Recursive to Explicit):

    • Recursive formula given: $a_n = a_{n-1} + 3.2$, $a_1 = 6$

    • Simplified explicit formula: $a_n = 3.2n + 2.8$

    • Find the $16^{\text{th}}$ term ($a_{16}$): $a_{16} = 3.2(16) + 2.8 = 51.2 + 2.8 = 54$

• Recursive Formulas for Arithmetic Sequences

  • Definition: A recursive formula defines each term of a sequence using the previous term ($a_{n-1}$ or $f(n-1)$) along with the common difference ($d$). A recursive formula must always state the initial term ($a_1$ or $f(1)$).

  • General Form:

  • Subscript notation: $a_n = a_{n-1} + d$, with given $a_1$

  • Function notation: $f(n) = f(n - 1) + d$, with given $f(1)$

  • Recursive Formula Practice Problems:

  • Problem 1: Given recursive formula $a_n = a_{n-1} + 6$ with $a_1 = 9$.

    • First term: $a_1 = 9$

    • Second term: $a_2 = 9 + 6 = 15$

    • Third term: $a_3 = 15 + 6 = 21$

    • Fourth term: $a_4 = 21 + 6 = 27$

    • First four terms: $9, 15, 21, 27$

  • Problem 2: Given recursive formula $f(n) = f(n - 1) - 10$ with $f(1) = 12$.

    • First term: $f(1) = 12$

    • Second term: $f(2) = 12 - 10 = 2$

    • Third term: $f(3) = 2 - 10 = -8$

    • Fourth term: $f(4) = -8 - 10 = -18$

    • First four terms: $12, 2, -8, -18$

  • Problem 3: Write a recursive formula in subscript notation for $16, 12, 8, 4, \dots$

    • Identify parameters: $a_1 = 16$, $d = -4$

    • Recursive formula: $a_n = a_{n-1} - 4$; $a_1 = 16$

  • Problem 4: Write a recursive formula in subscript notation for $-20, -15, -10, -5, \dots$

    • Identify parameters: $a_1 = -20$, $d = 5$

    • Recursive formula: $a_n = a_{n-1} + 5$; $a_1 = -20$

Scatter Plots, Correlation, and Causation

• Scatter Plots and Types of Correlation

  • Scatter Plot: A graph in a coordinate plane in which two sets of numerical data are plotted as ordered pairs $(x, y)$.

  • Correlation (Association): The relationship between two variables or events. It can be described by its strength (strong, weak, or none) and direction (positive or negative).

  • Positive Correlation: As $x$ increases, $y$ increases.

  • Negative Correlation: As $x$ increases, $y$ decreases.

  • No Correlation: $x$ and $y$ have no observable linear relationship.

• Fundamental Rule: CORRELATION DOES NOT IMPLY CAUSATION.

• Scatter Plot Analysis Examples

  • Example 1: Exercise and Calories Burned

  • Graph variables: Time in minutes ($x$) vs. Calories Burned ($y$)

* Correlation type: **Positive**
* Meaning in situation: As the amount of exercise time increases, the number of calories burned increases.

• Example 2: Library Fines and Books Borrowed

  • Graph variables: Books Borrowed ($x$) vs. Library Fine in dollars ($y$)

* Correlation type: **No correlation**
* Meaning in situation: The number of books borrowed does not relate to the size of the library fine.

• Example 3: Weight-Lifting

  • Graph variables: Weight in pounds ($x$) vs. Repetitions ($y$)

* Correlation type: **Negative**
* Meaning in situation: The heavier the weight lifted, the fewer repetitions a person can complete.

• Example 4: Car Dealership Revenue

  • Graph variables: Year '99 to '08 ($x$) vs. Revenue ($y$)

* Correlation type: **Positive**
* Meaning in situation: Over time, as years progress, more revenue is made.

• Correlation vs. Causation Analysis

  • Causation: A direct relationship between two events where one event directly causes or produces the other event.

  • Scenario 1: Exercise and Calories Burned per Minute

  • Relationship: Causation (Positive)

  • Explanation: Exercising directly causes the body to expend energy and burn calories.

  • Scenario 2: Christmas Tree Sales and Employee Sick Days

  • Relationship: Correlation (Positive)

  • Explanation: Selling Christmas trees does not cause employees to become sick. Cold winter weather in December independently causes both an increase in tree sales and an increase in winter illnesses.

  • Scenario 3: Outside Temperature and Hot Chocolate Sales

  • Relationship: Causation (Negative)

  • Explanation: As the outside temperature decreases, the cold weather directly causes people to buy warm drinks like hot chocolate.

  • Scenario 4: Time Spent Watching TV and School Grades

  • Relationship: Correlation (Negative)

  • Explanation: Watching TV itself does not directly alter grades, but increased TV time decreases study time, which leads to declining school performance.

  • Scenario 5: Ice Cream Sales and Sunglasses Sales

  • Relationship: Correlation (Positive)

  • Explanation: Buying ice cream does not cause people to buy sunglasses. Warm, sunny weather independently causes higher demand for both items.

  • Scenario 6: Hot Chocolate Sales and Heating Bills

  • Relationship: Correlation (Positive)

  • Explanation: Buying hot chocolate does not cause heating bills to rise. Cold weather independently causes both hot chocolate consumption and increased heating system usage.

  • Scenario 7: Ice Cream Sales and Electric/Cooling Bills

  • Case Study: A study revealed a positive correlation between ice cream sales and cooling bills. Kayla claims this means ice cream sales cause electric bills to increase.

  • Agreement: Disagree

  • Explanation: While a positive correlation exists, eating ice cream does not increase electricity usage. Hot summer weather causes people to run air conditioning units higher and also eat more ice cream.