Comprehensive Guide to Linear Relations, Graph Interpretation, and Slope-Intercept Form

Fundamentals of the Cartesian Plane and Relations

• The Cartesian Plane is a two-dimensional grid system formed by the perpendicular intersection of two numerical axes:

  • The x-axis is the horizontal numerical axis.

  • The y-axis is the vertical numerical axis.

  • The Origin is the intersection point of both axes, defined by the coordinates $(0, 0)$.

• The grid is divided into four distinct Quadrants, numbered counter-clockwise starting from the top-right:

  • Quadrant 1 ($Q_1$): Both coordinates are positive ($x > 0$, $y > 0$).

  • Quadrant 2 ($Q_2$): The horizontal coordinate is negative and the vertical coordinate is positive ($x < 0$, $y > 0$).

  • Quadrant 3 ($Q_3$): Both coordinates are negative ($x < 0$, $y < 0$).

  • Quadrant 4 ($Q_4$): The horizontal coordinate is positive and the vertical coordinate is negative ($x > 0$, $y < 0$).

• Coordinates are written as ordered pairs in the form $(x, y)$, maintaining alphabetical order:

  • To plot a point, start at the origin $(0, 0)$.

  • Move horizontally along the $x$-axis (left for negative values, right for positive values).

  • Move vertically along or parallel to the $y$-axis (down for negative values, up for positive values).

  • Specific coordinate examples plotted on the plane:

  • $(-4, 3)$ is located in Quadrant 2.

  • $(0, 1)$ is located directly on the vertical $y$-axis.

  • $(3, -2)$ is located in Quadrant 4.

• A Table of Values displays sets of ordered pairs $(x, y)$ representing points on a coordinate plane:

  • A given set of ordered pairs includes:

  • $(-6, 7)$

  • $(-4, 4)$

  • $(-2, 1)$

  • $(0, -2)$

  • $(2, -5)$

• A Relation is a mathematical or physical connection between two variables that describes how a change in one affects the other. Relations can be represented via:

  • Equations

  • Tables of values

  • Graphs

  • Contextual verbal descriptions

• Independent Variable ($x$):

  • The variable that changes independently on its own.

  • Represents the cause of change in the relationship.

  • Found on the horizontal axis of a graph and in the input column of a table.

• Dependent Variable ($y$):

  • The variable that depends on the independent variable.

  • Represents the outcome or quantity being measured/calculated.

  • Typically isolated on one side of an algebraic equation and plotted on the vertical axis.

• Scenario: Rising Hot Air Balloon

  • A hot air balloon starts from the ground and ascends at a constant rate of $2\,\text{m/s}$.

  • Variable identification:

  • Independent variable ($x$): Time ($\text{s}$), because time elapses steadily on its own.

  • Dependent variable ($y$): Height ($\text{m}$), because the balloon's elevation depends directly on elapsed time.

  • Algebraic equation:

  • $H = 2s$ or $y = 2x$

  • Table of values and corresponding ordered pairs:

  • At $0\,\text{s}$, height is $0\,\text{m}$ $\rightarrow (0, 0)$

  • At $1\,\text{s}$, height is $2\,\text{m}$ $\rightarrow (1, 2)$

  • At $2\,\text{s}$, height is $4\,\text{m}$ $\rightarrow (2, 4)$

  • At $3\,\text{s}$, height is $6\,\text{m}$ $\rightarrow (3, 6)$

  • At $4\,\text{s}$, height is $8\,\text{m}$ $\rightarrow (4, 8)$

Rate of Change and Initial Value

• The general form of a linear relation is:

  • $y = mx + b$

  • $y$ is the dependent variable (isolated).

  • $m$ is the Rate of Change (ROC) or slope.

  • $x$ is the independent variable.

  • $b$ is the Initial Value (IV) or vertical intercept.

• Initial Value (IV):

  • Represents the starting amount or baseline value of the dependent variable when the independent variable is zero ($x = 0$).

  • On a graph, it corresponds to the $y$-intercept where the line crosses the $y$-axis.

  • Direct Variation: The initial value is zero ($b = 0$), meaning the graph passes directly through the origin $(0, 0)$.

  • Partial Variation: The initial value is non-zero ($b \neq 0$), meaning the graph starts above or below the origin on the $y$-axis.

• Rate of Change (ROC):

  • A rate that compares the change in the dependent variable to the change in the independent variable.

  • Mathematical formulas:

  • $\text{ROC} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

  • Verbal indicators of rate of change include words such as "per" ($1$) and "every".

• Identifying IV and ROC across representations:

  • From an Equation ($y = 2x + 4$):

  • Initial Value (IV): The constant term, $b = 4$.

  • Rate of Change (ROC): The numerical coefficient multiplying $x$, $m = 2 = \frac{2}{1}$.

  • From a Graph:

  • Initial Value (IV): The vertical intercept ($y$-intercept) where $x = 0$. In a line crossing at $(0, -2)$, $\text{IV} = -2$.

  • Rate of Change (ROC): Computed by taking $\frac{\text{rise}}{\text{run}}$ between any two points. For points $(0, -2)$ and $(1, 1)$, the rise is $+3$ and the run is $+1$, giving $\text{ROC} = \frac{3}{1} = 3$.

  • From a Table of Values:

  • Given table entries: $(-3, 4)$, $(-2, 8)$, $(-1, 12)$, $(0, 16)$, $(1, 20)$.

  • Initial Value (IV): The value of $y$ when $x = 0$, which yields $\text{IV} = 16$.

  • Rate of Change (ROC): Evaluated by computing $\frac{\Delta y}{\Delta x}$. Here, $y$ increases by $+4$ each time $x$ increases by $+1$, yielding $\text{ROC} = \frac{4}{1} = 4$.

• Specific Context Examples:

  • Hot Air Balloon:

  • Starts from the ground: $\text{IV} = 0\,\text{m}$

  • Ascends at $2\,\text{m/s}$: $\text{ROC} = 2\,\text{m/s}$

  • Classification: Direct variation.

  • The Palasad Bowling Facility:

  • Charges a flat fee of $\\$2$to rent shoes plus$\$2.50$ for every game bowled.

  • Independent variable: Number of games bowled ($x$).

  • Dependent variable: Total cost ($y$).

  • Initial Value: $\text{IV} = \\$2\n - Rate of Change: \text{ROC} = \frac{\2.50}{1\,\text{game}} = \\2.50/\text{game}\n - Classification: Partial variation.\n- Algebraic Variations:\n - y = 10x - 1:\n - \text{IV} = -1\n - \text{ROC} = 10\n - Type: Partial variation.\n - y = -x + 0:\n - \text{IV} = 0\n - \text{ROC} = -1\n - Type: Direct variation.\n - y = 5 + \frac{1}{2}x:\n - \text{IV} = 5\n - \text{ROC} = \frac{1}{2}\n - Type: Partial variation.\n - Graph passing through (0, 0)$and$(1, -1):\n - \text{IV} = 0\n - \text{ROC} = \frac{-1}{1} = -1\n - Type: Direct variation.\n - Table containing (0, -5), (1, 0), (2, 5), (3, 10), (4, 15):\n - \text{IV} = -5\n - \text{ROC} = \frac{5}{1} = 5\n - Type: Partial variation.\n - Table containing (0, 0), (1, 3), (2, 6), (3, 9), (4, 12), (5, 15):\n - \text{IV} = 0\n - \text{ROC} = \frac{3}{1} = 3\n - Type: Direct variation.\n- Applied Scenario: Math Tutoring Earnings\n - A student earns \$20$ per hour tutoring mathematics.

  • Variables:

  • Independent variable ($x$): Time ($\text{h}$

  • Dependent variable ($y$): Money earned (\

  • Equation:

  • $y = 20x + 0$

  • Table of Values:

  • $0\,\text{h} \rightarrow \\$0\n - 1\,\text{h} \rightarrow \$20$

  • $2\,\text{h} \rightarrow \\$40\n - 3\,\text{h} \rightarrow \$60$

  • Initial Cost: $\text{IV} = \\$0\n - Rate of Change: \text{ROC} = \frac{20}{1} = 20\,\$\text{/h}$

  • Classification: Direct variation.

Distinguishing Linear and Non-Linear Relations

• Linear Relation Definition:

  • A relationship between two variables that produces a straight line when graphed.

  • Maintains a constant, uniform Rate of Change (ROC is always identical; values increase or decrease by the exact same amount every iteration).

• Methods to Determine Linearity:

  • By Equation:

  • In a linear equation, both variables ($x$ and $y$) must have an explicit degree/exponent of $1$ ($y^1 = mx^1 + b$).

  • Equations must always be simplified/expanded first before assessing degree.

  • $y = 2x + 1$ is linear ($x$ and $y$ have exponent $1$).

  • $y = 3x^2$ is non-linear (degree $2$ on $x$).

  • $y = 2x(x + 1)$ simplifies to $y = 2x^2 + 2x$, which is non-linear due to the degree of $2$.

  • By Graph:

  • Linear graphs form a perfectly straight line with identical ROC along any two selected points.

    • Example: On a line passing through $(0, 1)$, $(1, 2)$, $(2, 3)$, and $(5, 6)$, the ROC between $(0, 1)$ and $(1, 2)$ is $\frac{1}{1} = 1$, and the ROC between $(2, 3)$ and $(5, 6)$ is $\frac{3}{3} = 1$.

  • Non-linear graphs curve; their local slope changes across intervals.

    • Example: On an exponential curve passing through $(0, 1)$, $(1, 2)$, $(2, 4)$, and $(3, 8)$, the ROC between $(0, 1)$ and $(1, 2)$ is $\frac{1}{1} = 1$, whereas the ROC between $(2, 4)$ and $(3, 8)$ is $\frac{4}{1} = 4$. Because $1 \neq 4$, the relationship is non-linear.

  • By Table of Values (First Differences):

  • First differences ($\Delta y$) represent the differences between consecutive $y$-values, calculated as $y_{n} - y_{n-1}$, provided $x$ increases by a consistent step size.

  • If first differences are identical, the relation is linear.

  • If first differences vary, the relation is non-linear.

  • Linear Table Example:

    • Values: $(1, -5)$, $(2, -8)$, $(3, -11)$, $(4, -14)$

    • Step size in $x$: $+1$

    • First differences ($\Delta y$):

    • $-8 - (-5) = -3$

    • $-11 - (-8) = -3$

    • $-14 - (-11) = -3$

    • First differences are constant ($-3$); therefore, the relation is linear.

  • Non-Linear Table Example:

    • Values: $(0, 0)$, $(5, 1)$, $(10, 3)$, $(15, 6)$

    • Step size in $x$: $+5$

    • First differences ($\Delta y$):

    • $1 - 0 = +1$

    • $3 - 1 = +2$

    • $6 - 3 = +3$

    • First differences change ($+1, +2, +3$); therefore, the relation is non-linear.

• Rate of Change Computations in Context:

  • Elevator descent: Height decreases by $150\,\text{feet}$ every minute:

  • $\text{ROC} = \frac{-150\,\text{ft}}{1\,\text{min}} = -150\,\text{ft/min}$

  • Automobile travel: Car travels $240\,\text{km}$ every $3\,\text{hours}$:

  • $\text{ROC} = \frac{240\,\text{km}}{3\,\text{h}} = 80\,\text{km/h}$

  • Table of distance: $(1, 80)$, $(2, 160)$, $(3, 240)$, $(4, 320)$, $(5, 400)$

  • $\Delta y = +80\,\text{km}$, $\Delta x = +1\,\text{h}$

  • $\text{ROC} = \frac{80\,\text{km}}{1\,\text{h}} = 80\,\text{km/h}$

  • Equation $y = 4 - \frac{1}{2}x$:

  • Coefficient on $x$ is $-\frac{1}{2}$, so $\text{ROC} = -\frac{1}{2}$.

  • Graph of area over time passing through $(0, 0)$ and $(1, 8)$:

  • $\text{ROC} = \frac{\text{rise}}{\text{run}} = \frac{8\,\text{m}^2}{1\,\text{min}} = 8\,\text{m}^2/\text{min}$

• Block Pattern Growth Example:

  • Figure 1 contains $1$ square block.

  • Figure 2 contains $5$ square blocks (a central block with $1$ block extending in each of the $4$ directions).

  • Figure 3 contains $9$ square blocks (a central block with $2$ blocks extending in each of the $4$ directions).

  • Figure 4 contains $13$ square blocks (a central block with $3$ blocks extending in each of the $4$ directions).

  • Table of values:

  • Figure 1: $1$ block

  • Figure 2: $5$ blocks (change of $+4$)

  • Figure 3: $9$ blocks (change of $+4$)

  • Figure 4: $13$ blocks (change of $+4$)

  • Conclusion: The pattern represents a linear relationship because the first differences ($\Delta y$) are consistently $+4$.

Interpreting and Connecting Graphs to Real-World Events

• Real-world processes can be tracked and interpreted through piecewise or continuous graphs by examining the slope ($\text{ROC}$) of each segment.

• Case Study 1: Patient Temperature Tracking (Nurse Joan)

• Graph features:

  • Independent variable: Time ($\text{h}$) on the horizontal axis.

  • Dependent variable: Temperature ($^∘\text{C}$) on the vertical axis.

  • Axis scale note: The vertical axis skips lower values and begins at $37^∘\text{C}$.

• Segment Analysis:

  • Interval 7 am – 10 am ($3\,\text{hours}$ elapsed):

    • Temperature rises from $37^∘\text{C}$ to $40^∘\text{C}$.

    • $\text{ROC} = \frac{\text{rise}}{\text{run}} = \frac{3^∘\text{C}}{3\,\text{h}} = 1^∘\text{C/h}$

    • Interpretation: The patient's temperature increases steadily by $1^∘\text{C}$ every hour.

  • Interval 10 am – 1 pm ($3\,\text{hours}$ elapsed):

    • Temperature remains flat at $40^∘\text{C}$.

    • $\text{ROC} = \frac{\text{rise}}{\text{run}} = \frac{0^∘\text{C}}{3\,\text{h}} = 0$

    • Interpretation: The patient's fever stabilizes with zero change in body temperature.

  • Interval 1 pm – 3 pm ($2\,\text{hours}$ elapsed):

    • Temperature falls from $40^∘\text{C}$ to $37^∘\text{C}$.

    • $\text{ROC} = \frac{\text{rise}}{\text{run}} = \frac{-3^∘\text{C}}{2\,\text{h}} = -1.5^∘\text{C/h}$

    • Interpretation: The patient's temperature decreases at a rate of $1.5^∘\text{C}$ per hour.

• Hypothesized causes for temperature shifts: Medication administered causing the fever to break, active biological immune response, or ambient room temperature changes.

  • Case Study 2: Taxi Ride Cost Relation

• Graph features:

  • Independent variable: Distance ($\text{km}$) on the horizontal axis.

  • Dependent variable: Cost (\) on the vertical axis.

• Specific questions and calculated values:

  • Cost of a $3\,\text{km}$ ride: Read point $(3, 6)$; the ride costs $\\$6.\n - Distance achievable for \$8$: Read point $(4.5, 8)$; the distance is $4.5\,\text{km}$.

  • Breakdown of associated costs:

    • Base fare / Initial Value ($b$): The vertical intercept is at $(0, 2)$, indicating a flat starting fee of $\\$2.\n - Mileage rate / Rate of Change (m$): Calculated from$(0, 2)$to$(3, 6)$as$\frac{\text{rise}}{\text{run}} = \frac{\4}{3\,\text{km}} \approx \\1.33/\text{km}.\n\n- **Case Study 3: Hot Water Tap Operation**\n\n\n\n - Graph features:\n - Independent variable: Time (\text{min}).\n - Dependent variable: Temperature (^∘\text{C}).\n - Observations:\n - Temperature of the water after 40\,\text{minutes}$: Approximately$29^∘\text{C}.\n - Time at which water is 15^∘\text{C}$: At$55\,\text{minutes}.\n - Peak temperature: The water reaches its maximum heat of 60^∘\text{C}$at approximately$3\,\text{minutes}.\n - Physical explanation for decline: Water temperature drops after the initial surge because the water heater tank depleted its hot water supply.\n\n- **Case Study 4: Samuel's Trip to Mac's Variety Store**\n - Scenario Details:\n - Samuel lives 600\,\text{m} away from the store.\n - Walks partway, then stops to talk to a friend for 3\,\text{minutes}.\n - Runs to the store before it closes.\n - Walks back home after visiting the store.\n\n\n\n- Evaluation of Graph Options:\n - **Graph 1 (Flawed):** Displays an instantaneous vertical rise from 100\,\text{m}$to$400\,\text{m}$at$t = 3\,\text{min}$. A person cannot teleport$300\,\text{m}$in$0 seconds (infinite rate of change). In addition, the graph continues away from home without showing the return trip.\n - **Graph 2 (Correct Graph):**\n - From t = 0$to$t = 1\,\text{min}$, walks$100\,\text{m}.\n - From t = 1$to$t = 4\,\text{min}$, a horizontal line indicates a stationary period of$3\,\text{minutes}$($\text{ROC} = 0).\n - From t = 4$to$t = 6\,\text{min}$, a steep line indicates running rapidly from$100\,\text{m}$to the destination at$600\,\text{m} (steeper slope = faster rate of change).\n - From t = 6$to$t = 11\,\text{min}$, a line with a negative slope returns to$0\,\text{m} (home). The shallower slope represents walking rather than running.\n - **Graph 3 (Flawed):** The horizontal stationary segment at 400\,\text{m}$only lasts from$t = 4$to$t = 5\,\text{min}$($1\,\text{minute}$stop instead of the required$3\,\text{minutes}).\n\n# Equation of a Line in Slope-Intercept Form\n\n- **Slope-Intercept Form:**\n - y = mx + b\n - Reading convention: Graphs are always evaluated from left to right to determine the sign of the slope.\n- **The Four Slope Categories:**\n - **Positive Slope (m > 0):**\n - Line rises from left to right (\frac{+}{+} = +).\n\n\n\n- **Negative Slope (m < 0):**\n - Line falls from left to right (\frac{-}{+} = -).\n\n\n\n- **Zero Slope (m = 0):**\n - Completely horizontal line (\frac{0}{\text{run}} = 0).\n - Equation simplifies to y = 0x + b \implies y = b.\n\n\n\n- **Undefined Slope:**\n - Completely vertical line (\frac{\text{rise}}{0} = \text{undefined}, division by zero is mathematically undefined).\n - The line does not cross the y$-axis (unless it is the$y$-axis itself), so the$y-intercept does not exist.\n - Equation takes the form x = a$, where$a$is the constant$x-coordinate.\n\n\n\n- **Deriving the Equation from a Table of Values:**\n - Procedure:\n 1. Find \Delta x$by calculating differences between consecutive$x-values.\n 2. Find \Delta y$by calculating differences between consecutive$y-values.\n 3. Calculate slope: m = \frac{\Delta y}{\Delta x}.\n 4. Find the initial value b$by identifying the$y$-value where$x = 0.\n 5. Write the final equation: y = mx + b.\n\n\n\n- Table 1:\n - Data points: (-6, 16)$,$(-4, 11)$,$(-2, 6)$,$(0, 1)$,$(2, -4)$,$(4, -9)$,$(6, -14)\n - Increments: \Delta x = +2$,$\Delta y = -5\n - Slope: m = \frac{\Delta y}{\Delta x} = -\frac{5}{2}\n - Vertical intercept: At x = 0$,$y = 1$, so$b = 1\n - Equation:\n - y = -\frac{5}{2}x + 1\n\n\n\n - Table 2:\n - Data points: (-12, -19)$,$(-8, -16)$,$(-4, -13)$,$(0, -10)$,$(4, -7)$,$(8, -4)$,$(12, -1)\n - Increments: \Delta x = +4$,$\Delta y = +3\n - Slope: m = \frac{\Delta y}{\Delta x} = \frac{3}{4}\n - Vertical intercept: At x = 0$,$y = -10$, so$b = -10\n - Equation:\n - y = \frac{3}{4}x - 10\n\n- **Deriving the Equation from a Graph:**\n - Example 1 (Negative Slope):\n\n\n\n - Line passes through (0, 7)$and$(1, 5).\n - y$-intercept:$b = 7\n - Slope: m = \frac{\text{rise}}{\text{run}} = \frac{-2}{1} = -2\n - Equation:\n - y = -2x + 7\n- Example 2 (Fractional Positive Slope):\n\n\n\n - Line passes through (0, 1)$and$(3, 2).\n - y$-intercept:$b = 1\n - Slope: m = \frac{\text{rise}}{\text{run}} = \frac{1}{3}\n - Equation:\n - y = \frac{1}{3}x + 1\n- Example 3 (Large Scale Vertical Values):\n - Line passes through (0, -100)$and$(1, -80).\n - y$-intercept:$b = -100\n - Slope: m = \frac{\text{rise}}{\text{run}} = \frac{20}{1} = 20\n - Equation:\n - y = 20x - 100\n- Example 4 (Fractional Slope with Larger Span):\n - Line passes through (0, 12)$and$(8, 14).\n - y$-intercept:$b = 12\n - Slope: m = \frac{\text{rise}}{\text{run}} = \frac{2}{8} = \frac{1}{4}\n - Equation:\n - y = \frac{1}{4}x + 12\n- Example 5 (Horizontal Line):\n\n\n\n - Line passes horizontally through coordinates (-3, -2)$,$(0, -2)$,$(4, -2)$, and$(5, -2).\n - Slope: m = 0\n - Vertical intercept: b = -2\n - Equation:\n - y = 0x - 2 \implies y = -2\n- Example 6 (Vertical Line):\n\n\n\n- Line passes vertically through coordinates (4, 4)$,$(4, 2)$,$(4, 0)$,$(4, -2)$, and$(4, -5).\n- Slope: Undefined (run is 0, cannot divide by zero).\n- y-intercept: Does not exist.\n- Equation:\n - x = 4$$