Graphing Linear Functions: Complete Theoretical and Applied Study Guide

Foundations: Coordinate Planes, Evaluating Expressions, and Graphing Utilities

• Coordinate Plane Structure and Point Plotting:

  • The coordinate plane is formed by the intersection of two perpendicular number lines: the horizontal $x$-axis and the vertical $y$-axis, intersecting at the origin $(0, 0)$.

  • The axes divide the plane into four distinct regions called quadrants:

  • Quadrant I: Located in the upper right, where both coordinates are positive ($x > 0, y > 0$).

  • Quadrant II: Located in the upper left, where the $x$-coordinate is negative and the $y$-coordinate is positive ($x < 0, y > 0$).

  • Quadrant III: Located in the lower left, where both coordinates are negative ($x < 0, y < 0$).

  • Quadrant IV: Located in the lower right, where the $x$-coordinate is positive and the $y$-coordinate is negative ($x > 0, y < 0$).

  • Points lying on the axes do not belong to any quadrant:

  • Points on the $x$-axis possess a $y$-coordinate of zero: $(x, 0)$.

  • Points on the $y$-axis possess an $x$-coordinate of zero: $(0, y)$.

  • Point Plotting Procedure:

  • To plot an ordered pair $(x, y)$, begin at the origin $(0, 0)$.

  • Move horizontally along the $x$-axis according to the sign and magnitude of $x$ (right if positive, left if negative).

  • Move vertically parallel to the $y$-axis according to the sign and magnitude of $y$ (up if positive, down if negative).

  • Example: To plot $A(-3, 4)$, begin at $(0, 0)$, move $3$ units left, and move $4$ units up. The point terminates in Quadrant II.

  • Abstract Coordinate Signs: For any positive real numbers $a$ and $b$:

    • $(a, b)$ is in Quadrant I.

    • $(-a, b)$ is in Quadrant II.

    • $(-a, -b)$ is in Quadrant III.

    • $(a, -b)$ is in Quadrant IV.

• Algebraic Evaluation of Expressions:

  • Evaluating an algebraic expression involves substituting a specific numerical value for each variable instance and computing the result using standard operational precedence (parentheses, exponents, multiplication/division, addition/subtraction).

  • Worked Example 1: Evaluate $4x - 5$ when $x = 3$:

  • Substitute: $4(3) - 5$

  • Multiply: $12 - 5$

  • Subtract: $7$

  • Worked Example 2: Evaluate $-2x + 9$ when $x = -8$:

  • Substitute: $-2(-8) + 9$

  • Multiply: $16 + 9$

  • Add: $25$

• Graphing Calculator Viewing Windows:

  • Screen Aspect Ratio:

  • A standard handheld graphing calculator display screen possesses a physical height-to-width ratio of $2 \text{ to } 3$.

  • Standard Viewing Window:

  • Defined by the boundary settings $X\min = -10$, $X\max = 10$, $Xscl = 1$, $Y\min = -10$, $Y\max = 10$, and $Yscl = 1$.

  • Because the display is wider than it is tall, setting identical numerical intervals ($20$ units total on each axis) compresses the tick marks along the $y$-axis relative to the $x$-axis.

  • Graphs displayed in standard viewing windows fail to present lines in their true geometric perspective; slopes appear flatter or steeper than their actual geometric angle.

  • Square Viewing Window:

  • Formulated such that the distance representing one unit on the horizontal $x$-axis is identical in physical length to one unit on the vertical $y$-axis.

  • Achieved by setting horizontal and vertical ranges proportional to the screen's $3:2$ aspect ratio, such as $-9 \le x \le 9$ with $-6 \le y \le 6$, or $-18 \le x \le 18$ with $-12 \le y \le 12$.

  • A square window ensures perpendicular lines appear truly perpendicular and the geometric angle of inclined lines matches the mathematical slope.

Understanding Relations and Functions

• Core Definitions:

  • Relation: A mathematical mapping or pairing between a set of input values and a set of output values. Expressed as ordered pairs $(x, y)$, the $x$-coordinates represent inputs and the $y$-coordinates represent outputs.

  • Function: A specialized type of relation in which each input value is paired with exactly one output value.

  • Domain: The set of all possible input values ($x$-values) of a relation or function.

  • Range: The set of all possible output values ($y$-values) resulting from the evaluated domain.

  • Independent Variable: The variable that represents the input values of a function. It can assume any value within the defined domain.

  • Dependent Variable: The variable that represents the output values of a function. Its value depends entirely on the value chosen for the independent variable. The phrase "$y$ is a function of $x$" indicates that $y$ varies depending upon the value assigned to $x$.

• Representations of Functions:

  • Functions can be communicated in multiple equivalent formats:

  • Words: Stating the operational rule verbally (e.g., "the output is $3$ more than the input").

  • Equations: Defining the relationship algebraically (e.g., $y = x + 3$).

  • Input-Output Tables: Tabular arrangement matching each domain value to its corresponding range value.

  • Mapping Diagrams: Visual diagrams utilizing input and output ovals with directional arrows connecting inputs to outputs.

  • Graphs: Visual plots of all coordinate pairs $(x, y)$ in the Cartesian plane.

  • Sets of Ordered Pairs: Discrete collection notation, such as $\{(-1, 2), (0, 3), (1, 4), (2, 5)\}$.

• Criteria for Functionality:

  • Analyzing Sets and Tables:

  • A relation is a function if every input $x$ maps to a single output $y$.

  • It is entirely permissible for multiple distinct inputs to share the same output value. For example, the set $\{(-2, 2), (-1, 2), (0, 2), (1, 0), (2, 0)\}$ represents a function because every input has only one output, even though several inputs produce $2$.

  • A relation fails to be a function if a single input value maps to two or more distinct output values. For instance, $\{(4, 0), (8, 7), (6, 4), (4, 3), (5, 2)\}$ is not a function because the input $4$ maps to both $0$ and $3$.

  • Analyzing Equations:

  • An equation represents a function of $x$ if each input value substituted into the equation produces a unique output value.

  • For $y = 5x$, substituting any real number for $x$ yields a single value for $y$, confirming it is a function.

  • For $x = y^2$, substituting $x = 1$ yields two valid solutions: $y = 1$ and $y = -1$. Because a single input generates two distinct outputs, $x = y^2$ does not define $y$ as a function of $x$.

  • The Vertical Line Test:

  • A graphical method for determining functionality:

  • A graph represents a function if and only if no vertical line can be drawn that passes through more than one point on the graph.

  • If any vertical line intersects the curve or scatter plot at two or more points, an input value corresponds to multiple output values, violating the definition of a function.

• Domain and Range from Graphs:

  • Discrete Graphs: Represented by isolated points. The domain and range are listed as discrete sets of values.

  • For points $\{(-3, -2), (-1, 0), (1, 2), (3, 4)\}$:

    • Domain: $\{-3, -1, 1, 3\}$

    • Range: $\{-2, 0, 2, 4\}$

  • Continuous Graphs: Represented by unbroken lines or curves. The domain and range are written as continuous inequalities.

  • For a continuous line segment bounded from $x = -2$ to $x = 3$ horizontally, and extending from $y = -1$ to $y = 2$ vertically:

    • Domain: $-2 \le x \le 3$

    • Range: $-1 \le y \le 2$

• Applied Mathematical Modeling of Variables:

  • Juice Consumption Model: The function $y = -3x + 12$ models the amount $y$ (in fluid ounces) of juice remaining in a bottle after taking $x$ gulps.

  • Independent Variable: Number of gulps, $x$.

  • Dependent Variable: Amount of juice remaining, $y$.

  • Given a discrete domain of $\{0, 1, 2, 3, 4\}$, substitute to compute the range:

    • $x = 0 \implies y = -3(0) + 12 = 12$

    • $x = 1 \implies y = -3(1) + 12 = 9$

    • $x = 2 \implies y = -3(2) + 12 = 6$

    • $x = 3 \implies y = -3(3) + 12 = 3$

    • $x = 4 \implies y = -3(4) + 12 = 0$

  • Range: $\{0, 3, 6, 9, 12\}$.

  • Oven Preheating Model: The equation $t = 19m + 65$ models the internal temperature $t$ (in degrees Fahrenheit) of an oven after preheating for $m$ minutes.

  • Independent Variable: Preheating time in minutes, $m$.

  • Dependent Variable: Temperature in degrees Fahrenheit, $t$.

  • For a target temperature of $350\,^\circ\text{F}$:

    • Set $350 = 19m + 65 \implies 19m = 285 \implies m = 15\,\text{minutes}$.

    • Domain: $0 \le m \le 15$

    • Range: $65 \le t \le 350$

Linear Functions: Identification, Representation, and Discrete vs. Continuous Domains

• Core Mathematical Characteristics:

  • Linear Equation in Two Variables: An equation involving variables $x$ and $y$ that can be written in the form $y = mx + b$, where $m$ and $b$ are constants.

  • Linear Function: A function whose graph is a nonvertical straight line. A linear function is distinguished by having a strictly constant rate of change:     $\frac{\Delta y}{\Delta x} = \text{constant}$

  • Nonlinear Function: A function whose rate of change varies across intervals; its graphical representation is not a line.

• Methods for Classifying Linear and Nonlinear Functions:

  • From Graphs:

  • Inspect the geometric path: If the graph forms an unbroken or aligned straight nonvertical line, it is linear.

  • Any curve, parabolic bend, or piecewise deviation represents a nonlinear function.

  • From Tables:

  • Calculate the change in $x$ ($\Delta x$) and the change in $y$ ($\Delta y$) across successive pairs.

  • If equal changes in $x$ yield constant, equal changes in $y$, the rate of change is constant, confirming linearity.

  • Linear Example:

    • $x$ values: $3, 6, 9, 12$ ($\Delta x = +3$)

    • $y$ values: $36, 30, 24, 18$ ($\Delta y = -6$)

    • Rate of change: $\frac{-6}{3} = -2$ (constant, linear).

  • Nonlinear Example:

    • $x$ values: $1, 3, 5, 7$ ($\Delta x = +2$)

    • $y$ values: $2, 9, 20, 35$ ($\Delta y = +7, +11, +15$)

    • Rate of change is non-constant ($\frac{7}{2} \ne \frac{11}{2} \ne \frac{15}{2}$); therefore, it is nonlinear.

  • From Equations:

  • A function is linear if it can be manipulated via valid algebraic operations into the form $y = mx + b$.

  • Variables cannot be raised to powers other than $1$ (no $x^2$, $x^3$, etc.).

  • Variables cannot appear inside radicals (no $\sqrt{x}$).

  • Variables cannot be situated in denominators (no $\frac{1}{x}$).

  • Independent and dependent variables cannot be multiplied together (no $xy$).

  • Linear Form Validations:

    • $y = 3.8 \implies y = 0x + 3.8$ (linear constant function).

    • $y = 6(x - 1) \implies y = 6x - 6$ (linear).

    • $2 + \frac{1}{6}y = 3x + 4 \implies \frac{1}{6}y = 3x + 2 \implies y = 18x + 12$ (linear).

  • Nonlinear Form Confirmations:

    • $y = \sqrt{x}$ (variable within a radical).

    • $y = 3^x$ (exponential form; variable in exponent).

    • $y = \frac{2}{x}$ (variable in denominator).

    • $x^2 - y = 0 \implies y = x^2$ (quadratic form; degree $2$).

    • $2x + 3y = 9xy$ (cross-product term $xy$).

• Discrete versus Continuous Domains:

  • Discrete Domain:

  • A set of input values consisting only of certain isolated numbers within an interval (e.g., integers, whole numbers, countable objects).

  • The graph consists exclusively of distinct, unconnected points.

  • Application — Museum Admission:

    • A linear function $y = 15.95x$ models the total cost $y$ (in dollars) of purchasing $x$ museum tickets, with a maximum purchase limit of $3$ tickets.

    • Because fractional tickets cannot be purchased, the input $x$ must be a whole number.

    • Domain: $\{0, 1, 2, 3\}$ (discrete).

    • Coordinate solutions: $(0, 0), (1, 15.95), (2, 31.90), (3, 47.85)$.

    • Range: $\{0, 15.95, 31.90, 47.85\}$.

• Continuous Domain:

  • A set of input values consisting of all real numbers within a given interval (measurable quantities such as time, distance, volume, weight).

  • The graph forms a solid, unbroken line or ray connecting the solutions.

  • Application — Nutritional Intake:

  • A cereal bar contains $130\,\text{calories}$. Total calories $c$ consumed is modeled by $c = 130b$, where $b$ is the number of bars eaten.

  • Because fractional parts of a cereal bar can be consumed continuously, $b$ can be any real number greater than or equal to zero.

  • Domain: $b \ge 0$ (continuous).

  • Range: $c \ge 0$.

• Application — Bathtub Draining:

  • A $20\,\text{gallon}$ bathtub drains at a constant rate of $2.5\,\text{gallons per minute}$.

  • Linear Equation: $g = 20 - 2.5m$, where $g$ is remaining gallons and $m$ is elapsed minutes.

  • Drain time to empty ($g = 0$): $0 = 20 - 2.5m \implies 2.5m = 20 \implies m = 8\,\text{minutes}$.

  • Domain: $0 \le m \le 8$ (continuous, since time passes continuously).

  • Range: $0 \le g \le 20$.

Function Notation: Evaluation, Interpretation, and Analysis

• Mathematical Syntax and Meaning:

  • Function Notation: Denoted as $f(x)$, replacing the dependent variable $y$.

  • Read aloud as "the value of $f$ at $x$" or "$f$ of $x$".

  • Parentheses indicate input assignment, not multiplication ($f(x) \ne f \times x$).

  • Letters such as $g$, $h$, $j$, and $k$ are commonly used as function identifiers.

  • The coordinate points on the graph of a function $f$ are ordered pairs of the form $(x, f(x))$.

• Evaluating Functions:

  • To find the value of a function for a specific input, substitute that input value for every occurrence of $x$ within the functional rule.

  • Worked Example: Given $f(x) = -4x + 7$:

  • Evaluate at $x = 2$:       $f(2) = -4(2) + 7 = -8 + 7 = -1$

  • Evaluate at $x = -2$:       $f(-2) = -4(-2) + 7 = 8 + 7 = 15$

• Real-World Contextual Interpretation:

  • If $f(t)$ represents the outdoor temperature in degrees Fahrenheit ($^\circ\text{F}$) at $t$ hours after $6\,\text{a.m.}$:

  • $f(0) = 58$: The initial temperature at $6\,\text{a.m.}$ ($0$ hours elapsed) was exactly $58\,^\circ\text{F}$.

  • $f(6) = n$: At noon ($6$ hours after $6\,\text{a.m.}$), the outdoor temperature was $n\,^\circ\text{F}$.

  • $f(3) < f(9)$: The temperature at $9\,\text{a.m.}$ ($3$ hours after $6\,\text{a.m.}$) was strictly lower than the temperature at $3\,\text{p.m.}$ ($9$ hours after $6\,\text{a.m.}$).

• Solving for the Independent Variable:

  • When given a specific functional output value $f(x) = c$, solve the resulting single-variable equation to determine the corresponding input $x$.

  • Worked Example: For the linear function $h(x) = \frac{2}{3}x - 5$, determine $x$ when $h(x) = -7$:     $-7 = \frac{2}{3}x - 5$     $-7 + 5 = \frac{2}{3}x$     $-2 = \frac{2}{3}x$     $\frac{3}{2} \cdot (-2) = x \implies x = -3$

• Comparative Modeling in Real Life:

  • Helicopter Flight Duration Comparison:

  • Flight 1: Graph indicates an initial distance of $300\,\text{miles}$ from destination at $x = 0$, reaching its destination ($0\,\text{miles}$ remaining) at $x = 3\,\text{hours}$.

  • Flight 2: The helicopter flies $50\,\text{miles}$ farther (initial distance $350\,\text{miles}$) and increases speed by $25\,\text{mph}$ (speed = $100 + 25 = 125\,\text{mph}$). Modeled by the function:       $f(x) = 350 - 125x$

  • Calculate the arrival time for Flight 2 by determining the zero of the function ($f(x) = 0$):       $0 = 350 - 125x$       $125x = 350$       $x = \frac{350}{125} = 2.8\,\text{hours}$

  • Comparison: Flight 2 requires $2.8\,\text{hours}$, which is shorter than Flight 1's duration of $3.0\,\text{hours}$.

Graphing Linear Equations in Standard Form and Finding Zeros

• Standard Form Foundations:

  • The standard form of a linear equation in two variables is expressed as:     $Ax + By = C$     where $A$, $B$, and $C$ are real numbers, and $A$ and $B$ are not both equal to zero.

• Equations of Horizontal and Vertical Lines:

  • Horizontal Lines ($A = 0$):

  • The equation reduces to $By = C \implies y = \frac{C}{B}$, universally represented as $y = b$.

  • Every point on the line shares the identical $y$-coordinate $b$, regardless of $x$.

  • The line passes through $(0, b)$ and is completely parallel to the horizontal $x$-axis.

  • The slope of every horizontal line is strictly $0$.

  • Represents a constant function.

  • Vertical Lines ($B = 0$):

  • The equation reduces to $Ax = C \implies x = \frac{C}{A}$, universally represented as $x = a$.

  • Every point on the line shares the identical $x$-coordinate $a$, regardless of $y$.

  • The line passes through $(a, 0)$ and is completely parallel to the vertical $y$-axis.

  • The slope of every vertical line is undefined (due to division by zero: $\Delta x = 0$).

  • Vertical lines fail the Vertical Line Test; thus, they do not represent functions.

• Intercepts and the Intercept Graphing Method:

  • $x$-intercept: The $x$-coordinate of a point where the graph intersects the $x$-axis. Occurs where $y = 0$, producing the point $(a, 0)$.

  • Calculation: Substitute $y = 0$ into $Ax + By = C$ and solve for $x$:       $Ax + B(0) = C \implies Ax = C \implies x = \frac{C}{A}$

  • $y$-intercept: The $y$-coordinate of a point where the graph intersects the $y$-axis. Occurs where $x = 0$, producing the point $(0, b)$.

  • Calculation: Substitute $x = 0$ into $Ax + By = C$ and solve for $y$:       $A(0) + By = C \implies By = C \implies y = \frac{C}{B}$

  • Graphing via Intercepts:

  • Calculate the $x$-intercept $(a, 0)$ and the $y$-intercept $(0, b)$.

  • Plot both points on the coordinate grid.

  • Draw a straight line passing through the two plotted points.

  • Example: Graph $3x + 4y = 12$:

    • Let $y = 0 \implies 3x = 12 \implies x = 4$. Plot $(4, 0)$.

    • Let $x = 0 \implies 4y = 12 \implies y = 3$. Plot $(0, 3)$.

    • Connect $(4, 0)$ and $(0, 3)$ with a line.

• Zeros of Functions:

  • Zero of a Function: An input value $x$ for which the functional value equals zero ($f(x) = 0$ or $y = 0$).

  • Geometrically, the zero of a function is precisely the $x$-intercept of the graph of that function.

  • Worked Example: Find the zero of $f(x) = 2x + 8$:     $0 = 2x + 8 \implies 2x = -8 \implies x = -4$     The zero of the function is $-4$.

  • Applied Meaning of Zeros — Business Break-Even:

  • An artist rents an exhibition booth for \n300. Profit is modeled by $f(x) = 50x - 300$, where $x$ represents the number of paintings sold.

  • Find the zero: $0 = 50x - 300 \implies 50x = 300 \implies x = 6$.

  • Meaning: The artist must sell exactly $6\,\text{paintings}$ to recover the cost of the booth rental (yielding zero net profit, or the break-even point).

• Real-World Modeling with Standard Form:

  • Banquet Seating Allocation:

  • An awards banquet requires seating for $180\,\text{people}$. Small tables seat $6\,\text{people}$, and large tables seat $10\,\text{people}$.

  • Model: $6x + 10y = 180$, where $x$ is small tables and $y$ is large tables.

  • Determine Intercepts:

    • Let $y = 0 \implies 6x = 180 \implies x = 30$. Intercept: $(30, 0)$. Interpretation: Renting $30\,\text{small tables}$ seats all attendees with $0\,\text{large tables}$.

    • Let $x = 0 \implies 10y = 180 \implies y = 18$. Intercept: $(0, 18)$. Interpretation: Renting $18\,\text{large tables}$ seats all attendees with $0\,\text{small tables}$.

  • Contextual Integer Solutions (Discrete pairs lying on line):

    • $(0, 18)$: $0\,\text{small}$, $18\,\text{large}$

    • $(10, 12)$: $6(10) + 10(12) = 60 + 120 = 180$

    • $(20, 6)$: $6(20) + 10(6) = 120 + 60 = 180$

    • $(30, 0)$: $30\,\text{small}$, $0\,\text{large}$

Graphing Linear Equations in Slope-Intercept Form

• The Concept of Slope:

  • Slope ($m$): The measure of the steepness and direction of a nonvertical line. Defined mathematically as the constant ratio of vertical change (rise) to horizontal change (run):     $m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$     where $(x_1, y_1)$ and $(x_2, y_2)$ are any two distinct points on the line, and $x_2 \ne x_1$.

  • Classification of Slopes:

  • Positive Slope ($m > 0$): The line rises from left to right.

  • Negative Slope ($m < 0$): The line falls from left to right.

  • Zero Slope ($m = 0$): The line is horizontal ($y_2 - y_1 = 0$).

  • Undefined Slope: The line is vertical ($x_2 - x_1 = 0$, resulting in division by zero).

• Calculating Slope from Points and Tables:

  • Two Points: Given $(-3, -2)$ and $(3, 2)$:     $m = \frac{2 - (-2)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3}$

  • From Tabular Data:

  • Select any two coordinate pairs $(x_1, y_1)$ and $(x_2, y_2)$ from the table.

  • If the table displays $x = 4, y = 20$ and $x = 7, y = 14$:       $m = \frac{14 - 20}{7 - 4} = \frac{-6}{3} = -2$

  • If all $y$-values are identical while $x$ varies, $m = 0$.

  • If all $x$-values are identical while $y$ varies, the slope is undefined.

• Slope-Intercept Form:

  • A linear equation written in the format:     $y = mx + b$     where $m$ represents the slope of the line, and $b$ represents the $y$-intercept, corresponding to the Cartesian point $(0, b)$.

  • Converting to Slope-Intercept Form:

  • Solve the linear equation explicitly for $y$.

  • Example: Convert $-5x - y = -2$:       $-y = 5x - 2$       $y = -5x + 2$       Slope $m = -5$, $y$-intercept $b = 2$ (point $(0, 2)$).

• Graphing Procedure Using Slope-Intercept Form:

  • Step 1: Write the equation in slope-intercept form ($y = mx + b$).

  • Step 2: Identify the slope $m$ and the $y$-intercept $b$.

  • Step 3: Plot the $y$-intercept at the coordinates $(0, b)$.

  • Step 4: Interpret the slope $m$ as a fraction $\frac{\text{rise}}{\text{run}}$. Starting from $(0, b)$, move vertically by the rise and horizontally by the run to locate and plot a second point. Alternatively, moving in reverse ($\frac{-\text{rise}}{-\text{run}}$) locates points in the opposite direction.

  • Step 5: Draw a straight line extending through both points with arrows at the ends.

  • Worked Example: Graph $2x + y = 2$:

  • Solve for $y$: $y = -2x + 2$.

  • Identify: $m = -2 = \frac{-2}{1}$, $b = 2$.

  • Plot $(0, 2)$.

  • Move down $2$ units and right $1$ unit to plot $(1, 0)$.

  • The line crosses the $x$-axis at $(1, 0)$, so the $x$-intercept is $1$.

• Physical Modeling of Slope as a Rate of Change:

  • Submersible Ascent Model:

  • An oceanographic exploration submersible ascends from the sea floor according to the elevation function:       $h(t) = 650t - 13{,}000$       where $h$ is elevation (in feet relative to sea level) and $t$ is time elapsed (in minutes).

  • Slope Interpretation: The slope is $650\,\text{ft/min}$. This indicates that the submersible ascends at a constant rate of $650\,\text{feet per minute}$.

  • $h$-intercept Interpretation: The $h$-intercept is $-13{,}000$, corresponding to $(0, -13{,}000)$. This represents the starting depth of $13{,}000\,\text{feet below sea level}$ at $t = 0\,\text{minutes}$.

  • $t$-intercept Interpretation: Setting $h(t) = 0$:       $0 = 650t - 13{,}000 \implies 650t = 13{,}000 \implies t = 20\,\text{minutes}$       The submersible requires exactly $20\,\text{minutes}$ to reach the surface ($0\,\text{feet}$ elevation).

  • Domain and Range Constraints:

    • Domain: $0 \le t \le 20$

    • Range: $-13{,}000 \le h \le 0$

Modeling Direct Variation

• Core Mathematical Principles:

  • Direct Variation: A functional relationship between two quantities $x$ and $y$ characterized by the equation:     $y = ax \quad \text{where } a \ne 0$

• Constant of Variation ($a$): The non-zero constant coefficient representing the constant ratio between $y$ and $x$:   $\frac{y}{x} = a \quad (x \ne 0)$

  • Connection to Slope-Intercept Form:

  • A direct variation equation $y = ax$ is equivalent to slope-intercept form $y = mx + b$ where the slope is $m = a$ and the $y$-intercept is explicitly $b = 0$.

  • Consequently, the graph of every direct variation equation is a straight line that must pass through the origin $(0, 0)$.

• Identification and Verification Techniques:

  • Testing Equations:

  • Isolate $y$. If the resulting equation possesses a non-zero $y$-intercept ($b \ne 0$), it is not direct variation.

• Example A: $2x - 3y = 0 \implies -3y = -2x \implies y = \frac{2}{3}x$. This matches $y = ax$; hence, it represents direct variation with constant of variation $a = \frac{2}{3}$.

• Example B: $-x + y = 4 \implies y = x + 4$. Because $b = 4 \ne 0$, this does not represent direct variation.

• Testing Data Tables:

  • Compute the ratio $\frac{y}{x}$ for every non-zero coordinate pair $(x, y)$.

  • If every ratio evaluates to the identical constant $a$, the quantities exhibit direct variation.

  • Example — Digital Music Downloads:

  • $3\,\text{songs}$ cost \n2.97: $\frac{2.97}{3} = 0.99$

• $5\,\text{songs}$ cost \n4.95: $\frac{4.95}{5} = 0.99$

• $7\,\text{songs}$ cost \n6.93: $\frac{6.93}{7} = 0.99$

  • Constant ratio $a = 0.99\,\text{dollars/song}$. Direct variation model: $C = 0.99s$.

• Problem-Solving Applications:

  • Saltwater Aquarium Calibration:

  • The number of tablespoons $s$ of sea salt varies directly with the number of gallons $w$ of water in an aquarium tank ($s = aw$).

  • A $20\,\text{gallon}$ tank requires $100\,\text{tablespoons}$ of salt.

  • Find constant of variation: $100 = a(20) \implies a = 5\,\text{tablespoons per gallon}$.

  • Model: $s = 5w$.

  • For a $30\,\text{gallon}$ tank:       $s = 5(30) = 150\,\text{tablespoons}$

• Planetary Gravitational Weight:

  • An object weighing $100\,\text{pounds}$ on Earth ($E$) weighs $6\,\text{pounds}$ on Pluto ($P$). Assuming weight on Pluto varies directly with weight on Earth:     $P = aE \implies 6 = a(100) \implies a = 0.06$     Direct variation model: $P = 0.06E$.

  • To find the Earth weight of a boulder weighing $45\,\text{pounds}$ on Pluto:     $45 = 0.06E \implies E = \frac{45}{0.06} = 750\,\text{pounds}$

Transformations of Graphs of Linear Functions

• Foundational Transformation Theory:

  • Family of Functions: A group of functions sharing identical structural characteristics.

  • Parent Function: The most fundamental, basic function within a family. For all nonconstant linear functions, the parent function is:     $f(x) = x$     The graph of $f(x) = x$ passes through $(0, 0)$ with a slope of $1$.

  • Transformation: An operation that alters the position, size, orientation, or shape of a parent graph in the Cartesian plane.

• Types of Functional Transformations:

  • Translations (Rigid Shifts):

  • A transformation that shifts every point on a graph a fixed distance horizontally or vertically without altering its shape, size, or slope.

  • Horizontal Translation: Given by $y = f(x - h)$.

    • Subtracting $h$ from the inputs before function evaluation shifts the graph horizontally.

    • If $h > 0$, the graph shifts right by $h$ units: $y = f(x - h)$.

    • If $h < 0$, the graph shifts left by $|h|$ units: $y = f(x + |h|)$.

  • Vertical Translation: Given by $y = f(x) + k$.

    • Adding $k$ to function outputs shifts the graph vertically.

    • If $k > 0$, the graph shifts up by $k$ units.

    • If $k < 0$, the graph shifts down by $|k|$ units.

  • Reflections (Flipping Orientation):

  • A transformation that mirrors a graph over a designated line of reflection.

  • Reflection in the $x$-axis: Given by $y = -f(x)$.

    • All output values ($y$-values) are multiplied by $-1$, reversing their signs.

    • Each point $(x, y)$ maps to $(x, -y)$.

  • Reflection in the $y$-axis: Given by $y = f(-x)$.

    • All input values ($x$-values) are multiplied by $-1$, reversing their signs.

    • Each point $(x, y)$ maps to $(-x, y)$.

  • Stretches and Shrinks (Non-Rigid Distortions):

  • Horizontal Stretches and Shrinks: Given by $y = f(ax)$, where $a > 0$ and $a \ne 1$.

  • Every input $x$ is scaled by a factor of $\frac{1}{a}$.

  • If $a > 1$: The graph is a horizontal shrink toward the $y$-axis by a factor of $\frac{1}{a}$.

  • If $0 < a < 1$: The graph is a horizontal stretch away from the $y$-axis by a factor of $\frac{1}{a}$.

  • The $y$-intercept remains invariant (fixed).

• Vertical Stretches and Shrinks: Given by $y = a \cdot f(x)$, where $a > 0$ and $a \ne 1$.

  • Every output $y$ is scaled directly by a factor of $a$.

  • If $a > 1$: The graph is a vertical stretch away from the $x$-axis by a factor of $a$.

  • If $0 < a < 1$: The graph is a vertical shrink toward the $x$-axis by a factor of $a$.

  • The $x$-intercept remains invariant (fixed).

• Summary Rules for Transformations on $f(x)$:

  • Horizontal Translation: $y = f(x - h)$ (shifts right if $h > 0$, left if $h < 0$)

  • Vertical Translation: $y = f(x) + k$ (shifts up if $k > 0$, down if $k < 0$)

  • Reflection in $x$-axis: $y = -f(x)$ (output values negated)

  • Reflection in $y$-axis: $y = f(-x)$ (input values negated)

  • Horizontal Stretch: $y = f(ax)$ with $0 < a < 1$ (factor: $\frac{1}{a}$)

  • Horizontal Shrink: $y = f(ax)$ with $a > 1$ (factor: $\frac{1}{a}$)

  • Vertical Stretch: $y = a \cdot f(x)$ with $a > 1$ (factor: $a$)

  • Vertical Shrink: $y = a \cdot f(x)$ with $0 < a < 1$ (factor: $a$)

• Systematic Procedure for Combining Transformations:

  • To transform the parent graph $y = f(x)$ into the composite form $y = a \cdot f(x - h) + k$ or $y = f(ax - h) + k$:

  • Step 1 (Horizontal Shift): Translate the graph horizontally by $h$ units.

  • Step 2 (Scaling): Apply the horizontal or vertical stretch or shrink using factor $a$.

  • Step 3 (Reflection): If the multiplier is negative ($a < 0$), reflect the resulting graph across the appropriate axis ($x$-axis for vertical, $y$-axis for horizontal).

  • Step 4 (Vertical Shift): Translate the resulting graph vertically by $k$ units.

  • Worked Example: Transform $f(x) = x$ to create $g(x) = -2x + 3$:

  • Express in terms of $f$: $g(x) = -2f(x) + 3$.

  • Step 1: No horizontal translation ($h = 0$).

  • Step 2: Vertically stretch the graph by a factor of $2$, producing $h(x) = 2x$.

  • Step 3: Reflect the graph in the $x$-axis, producing $r(x) = -2x$.

  • Step 4: Vertically translate the graph $3\,\text{units up}$, yielding $g(x) = -2x + 3$.

• Real-World Transformational Modeling:

  • Telecommunications Subscription Plans:

  • Standard service cost: $c(m) = 60m$, where $m$ is months of service (\n60/\text{month}, no setup fee).

• Promotional service cost: $r(m) = 30m + 45$, offering a reduced monthly rate of \n30/\text{month} with a mandatory \n45 installation fee.

• Express promotional model $r$ in terms of baseline model $c$:   $r(m) = \frac{1}{2}c(m) + 45$

• Transformational Sequence:

  • Vertical shrink of the graph of $c(m)$ by a factor of $\frac{1}{2}$.

  • Vertical translation upward by $45\,\text{units}$ .