Chapter 4.1/4.2 Notes - College Algebra

Introduction to Linear Functions and Slope

• Definition of a Linear Function:

  • A linear function is a specific type of function whose graph forms a straight line.

  • The word "linear" is directly derived from the word "line."

  • The graph of a linear function is visually represented as a straight line with arrows on both ends to indicate that it extends infinitely in both directions.

  • Background grid lines (vertical and horizontal lines) on coordinate graphs are provided strictly for measurement and context; they are not part of the function itself.

• Primary Concepts and Terminology:

  • In algebra, as well as in advanced mathematics such as Calculus I and Calculus II, the most critical property of a linear function is its slope.

  • Synonymous terms for slope include:

  • Rate of change: Describes how rapidly the function output rises or falls relative to its input.

  • Gradient: An equivalent term used to describe the steepness or incline of the line.

  • Conceptual Metaphor: Slope can be conceptualized as walking on hills:

  • A gentle, shallow decline represents a small negative slope.

  • A steep decline where the drop is nearly vertical—requiring one to run to prevent falling—represents a large negative slope.

Formulas and Mathematical Definitions of Slope

• Word Formula:

  • $\text{Slope} = \frac{\text{Rise}}{\text{Run}}$

  • $\text{Rise}$ indicates vertical change (distance moving up or down).

  • $\text{Run}$ indicates horizontal change (distance moving left to right).

• Delta Notation Formula:

  • $\text{Slope} = \frac{\tau y}{\tau x}$

  • The symbol $\tau$ is the capital Greek letter Delta, representing "change in."

  • The expression represents the change in $y$ divided by the change in $x$

• Algebraic Two-Point Formula:

  • $m = \frac{y_2 - y_1}{x_2 - x_1}$

  • The letter $m$ is the standard variable designated for slope.

  • $(x_1, y_1)$ represents the coordinate pair of the first chosen point.

  • $(x_2, y_2)$ represents the coordinate pair of the second chosen point.

  • Subscripts ($1$ and $2$) serve exclusively to keep track of which coordinates belong to which point. Point order selection does not alter the calculated slope as long as order consistency is maintained in both numerator and denominator.

Distinction Between Points and Intervals

• Ambiguity of Parenthetical Notation:

  • The mathematical notation $(2, 8)$ uses two numbers separated by a comma within parentheses.

  • Depending on the context of the problem, this notation represents two completely different mathematical concepts:

  • Ordered Pair (Point): Indicates a specific location on a coordinate plane where the $x$-coordinate is $2$ and the $y$-coordinate is $8$

  • Interval: Indicates a bounded continuous set where a variable $x$ lies strictly between $2$ and $8$ (i.e., $2 < x < 8$).

  • Without surrounding contextual information, $(2, 8)$ cannot be distinguished as a point or an interval.

Step-by-Step Slope Calculation Examples

• Example 1: Calculating Slope from Graph Coordinates

  • Points identified on a graph: $(1, 5)$ and $(2, 8)$.

  • Assign point designations: $(x_1, y_1) = (1, 5)$ and $(x_2, y_2) = (2, 8)$.

  • Apply the slope formula:

  • $m = \frac{8 - 5}{2 - 1}$

  • $m = \frac{3}{1} = 3$

  • Interpretation of result:

  • The slope is $+3$, representing a steep positive incline and an increasing function.

  • For every $1$ unit increase in the $x$-value, the $y$-value increases by $3$ units.

• Example 2: Calculating Slope and Demonstrating Label Order Independence

  • Given points: $(-5, 3)$ and $(2, -1)$.

  • Calculation Method 1 (Assigning $(x_1, y_1) = (-5, 3)$ and $(x_2, y_2) = (2, -1)$):

  • $m = \frac{-1 - 3}{2 - (-5)}$

  • Numerator: $-1 - 3 = -4$

  • Denominator: $2 - (-5) = 2 + 5 = 7$

  • Slope: $m = -\frac{4}{7}$

  • Calculation Method 2 (Reversing point labels: $(x_1, y_1) = (2, -1)$ and $(x_2, y_2) = (-5, 3)$):

  • $m = \frac{3 - (-1)}{-5 - 2}$

  • Numerator: $3 - (-1) = 3 + 1 = 4$

  • Denominator: $-5 - 2 = -7$

  • Slope: $m = \frac{4}{-7} = -\frac{4}{7}$

  • Conclusion: Both assignments produce the exact same slope $m = -\frac{4}{7}$.

  • Interpretation: For every $7$ units increase in $x$, the $y$-value decreases by $4$ units.

• Example 3: Graph-Based Counting Shortcut

  • When determining slope directly from a line graph, select points with integer coordinates to avoid fractional calculations. Points containing $0$ as a coordinate simplify arithmetic.

  • Points chosen from graph: $(0, -2)$ and $(4, 0)$.

  • $m = \frac{0 - (-2)}{4 - 0} = \frac{2}{4} = \frac{1}{2}$

  • Counting Method:

  • Move from point $(0, -2)$ vertically up $1$ unit ($\text{rise} = 1$).

  • Move horizontally right $2$ units ($\text{run} = 2$).

  • $\text{Slope} = \frac{\text{rise}}{\text{run}} = \frac{1}{2}$

  • Caution: Downward movement yields a negative rise and requires a negative slope.

Fundamental Properties and Equations of Linear Functions

• Essential Properties:

  • A linear function is uniquely defined by two primary properties:

  1. Slope ($m$): The rate of change.

  2. $\text{y}$-intercept ($b$): The point where the graph crosses the vertical $y$ axis, occurring where $x = 0$

• Slope-Intercept Form:

  • $y = mx + b$ or $f(x) = mx + b$

  • $m$ represents the slope.

  • $b$ represents the $y$-intercept value.

  • Example: Given slope $m = \frac{1}{2}$ and $y$-intercept $b = 1$, the equation is $y = \frac{1}{2}x + 1$

  • Concept: This equation is a functional rule that computes the output $y$ for any chosen input $x$

• Structural Rule for Linear Equations:

  • An equation is linear if and only if it matches the format $\text{number} \times x + \text{number}$.

  • The variable $x$ must be raised strictly to the first power ($x^1$).

  • Equations containing $x^2$, $x^3$, $x^4$, or square roots ($\tau{x}$) are non-linear.

Parallel and Perpendicular Lines

• Parallel Lines:

  • Definition: Lines located in the same plane that possess identical slopes ($m_1 = m_2$) but different $y$-intercepts ($b_1 \neq b_2$).

  • Graphical appearance: Parallel lines increase or decrease at the exact same rate and never intersect.

  • Real-World Analogy:

  • Three individuals (Johnny, Susie, and Alex) sell items and earn a fixed rate of \n0.50 (or \n2.00) per item sold.

  • Starting bank accounts:

    • Johnny starts in debt by \n2.00 ($b = -2$).

    • Susie starts with \n0.00 ($b = 0$).

    • Alex starts with \n2.00 in savings ($b = 2$).

  • Resulting linear equations: $y = 0.5x - 2$, $y = 0.5x$, $y = 0.5x + 2$

  • Because all three have the exact same rate of change ($m = 0.5$), their graphs are parallel lines. Alex remains permanently \n2.00 ahead of Susie, and Susie remains permanently \n2.00 ahead of Johnny.

• Perpendicular Lines:

  • Definition: Lines that intersect at a right angle ($90^\tau$).

  • Slope Relationship: Slopes of perpendicular lines are negated reciprocals (negative flips) of each other.

  • Formula: $m_{\text{perpendicular}} = -\frac{1}{m}$

  • Example: If a line has slope $m = 3$, any line perpendicular to it has slope $m = -\frac{1}{3}$.

  • Worked Example:

  • Given line: $y = -\frac{5}{3}x + \frac{1}{3}$

  • Point to pass through: $(3, 5)$

  • Equation of Parallel Line through $(3, 5)$:

    • Retains identical slope: $m = -\frac{5}{3}$

    • Point-slope equation: $y - 5 = -\frac{5}{3}(x - 3)$

  • Equation of Perpendicular Line through $(3, 5)$:

    • Negated reciprocal slope: $m = -\tau-\frac{3}{5}\tau = \frac{3}{5}$

    • Point-slope equation: $y - 5 = \frac{3}{5}(x - 3)$

Graphing Linear Functions

• Graphing Step-by-Step Procedure:

  • Given function: $y = -\frac{3}{4}x - 4$

  • Identify key values: Slope $m = -\frac{3}{4}$, $y$-intercept $b = -4$ (point $(0, -4)$).

  • Step 1: Plot the $y$-intercept at $(0, -4)$.

  • Step 2: Use the slope to find additional points.

  • Interpretation A ($\text{rise} = -3$, $\text{run} = 4$): Move down $3$ units to $y = -7$, move right $4$ units to $x = 4$. Plot point $(4, -7)$.

  • Interpretation B ($\text{rise} = 3$, $\text{run} = -4$): Move up $3$ units to $y = -1$, move left $4$ units to $x = -4$. Plot point $(-4, -1)$.

  • Step 3: Draw a straight line connecting points $(-4, -1)$, $(0, -4)$, and $(4, -7)$. Plotting three or four points increases alignment accuracy over using the minimum requirement of two points.

Special Cases: Horizontal and Vertical Lines

• Horizontal Lines (Constant Functions):

  • Slope $m = 0$ ($\text{rise} = 0$, $\text{run} = 1$).

  • Division validity: $\frac{0}{1} = 0$ (defined).

  • Equation form: $y = b$ or $f(x) = b$ (equivalent to $y = 0x + b$).

  • Example: $f(x) = 3$ or $y = 3$

  • Function status: Horizontal lines are linear functions, specifically termed constant functions because the output $y$ remains constantly equal to a single value.

• Vertical Lines:

  • Equation form: $x = c$, where $c$ is a constant value.

  • Example: $x = 3$ or $x = -3$

  • Graphical definition: The collection of all coordinate points whose $x$-value equals $-3$

  • Function Status: Vertical lines are NOT functions.

  • Reason: They fail the Vertical Line Test. A single vertical line drawn over the graph intersects infinitely many points, meaning one input $x$ produces infinitely many outputs $y$

  • Slope Status: The slope of a vertical line is undefined (or informally described as infinite) because $\text{run} = 0$, leading to division by zero ($\frac{1}{0}$).

Monotonicity: Increasing, Decreasing, and Constant Functions

• Relationship between Slope and Function Behavior:

  • Increasing Function: Occurs when slope is positive ($m > 0$). The graph rises from left to right.

  • Decreasing Function: Occurs when slope is negative ($m < 0$). The graph falls from left to right.

  • Constant Function: Occurs when slope is zero ($m = 0$). The graph is horizontal and is neither increasing nor decreasing.

Point-Slope Form and Form Conversions

• Point-Slope Form Formula:

  • $y - y_1 = m(x - x_1)$

  • $m$ is the slope.

  • $(x_1, y_1)$ is any known point on the line.

  • Variables $x$ and $y$ remain unsubstituted.

• Example 1: Finding Point-Slope Form and Converting to Slope-Intercept Form

  • Given points: $(4, -5)$ and $(-9, 3)$.

  • Step 1: Calculate slope $m$:

  • $m = \frac{3 - (-5)}{-9 - 4} = \frac{3 + 5}{-13} = \frac{8}{-13} = -\frac{8}{13}$

  • Step 2: Write equation in Point-Slope Form using point $(4, -5)$:

  • $y - (-5) = -\frac{8}{13}(x - 4)$

  • Simplified Point-Slope Form: $y + 5 = -\frac{8}{13}(x - 4)$

  • Step 3: Convert to Slope-Intercept Form ($y = mx + b$):

  • Distribute the slope: $y + 5 = -\frac{8}{13}x + \tau-\frac{8}{13} \times -4\tau$

  • $y + 5 = -\frac{8}{13}x + \frac{32}{13}$

  • Isolate $y$ by subtracting $5$: $y = -\frac{8}{13}x + \frac{32}{13} - 5$

  • Convert $5$ into a fraction with denominator $13$: $5 = \frac{65}{13}$

  • Perform fraction subtraction: $\frac{32}{13} - \frac{65}{13} = \frac{32 - 65}{13} = -\frac{33}{13}$

  • Final Slope-Intercept Form: $y = -\frac{8}{13}x - \frac{33}{13}$

• Example 2: Converting Function Notation to Ordered Pairs

  • Given inputs: $f(3) = -2$ and $f(8) = 1$

  • Translate function notation $f(x) = y$ into ordered pairs:

  • $f(3) = -2 \tau (3, -2)$

  • $f(8) = 1 \tau (8, 1)$

  • Calculate slope: $m = \frac{1 - (-2)}{8 - 3} = \frac{1 + 2}{5} = \frac{3}{5}$

  • Substitute into Point-Slope Form using point $(3, -2)$:

  • $y - (-2) = \frac{3}{5}(x - 3)$

  • $y + 2 = \frac{3}{5}(x - 3)$

Real-World Applications: Cost Functions

• Business Cost Model:

  • Problem Scenario: Ben starts a business. He incurs fixed monthly operational costs (e.g., rent, utilities) regardless of production, plus variable production costs per item.

  • Given numerical data:

  • Fixed overhead cost: \n1250

  • Variable cost per item produced: \n37.50

  • Formulating Total Cost Function $C(x)$ where $x$ is the number of items produced:

  • General Linear Model: $C(x) = mx + b$

  • $y$-intercept $b$: Represents initial cost when zero items are manufactured ($x = 0$). Thus, $b = 1250$

  • Slope $m$: Represents the rate of cost increase per unit manufactured. Thus, $m = 37.50$

  • Resulting Cost Function:

  • $C(x) = 37.50x + 1250$

  • Reason for Linearity: The input $x$ is multiplied by a constant rate ($37.50$) and added to a constant base ($1250$), without any powers or transformations applied to $x$

Questions and Discussion

• Topic: Alternative Names for Slope

  • Question: Is slope not also called a gradient?

  • Answer: Yes, gradient is an equivalent term used to describe rate of change or the steepness of a linear function.

• Topic: Common Algebraic Errors in Slope Formula

  • Question: What error occurred during the setup of the slope calculation between $(-5, 3)$ and $(2, -1)$?

  • Answer: The $x$ and $y$ coordinates were mixed up. The value $-5$ is an $x$-coordinate and must be placed in the denominator, not the numerator. Furthermore, $-1 - 3$ evaluates to $-4$, not $-2$

• Topic: Graphical Shortcut vs. Formula

  • Question: Can slope be determined on a graph simply by counting up and over (rise over run) without writing out the complete formula?

  • Answer: Yes. Counting units visually on a graph executes the exact same mathematical process as the formula. However, care must be taken to ensure downward direction is assigned a negative sign.