3.4 Linear Graphs, Slopes & Coordinate Planes

3.4 Linear Graphs, Slopes & Coordinate Planes

Quick Summary: Coordinate geometry bridges algebraic equations and visual graphs. Linear graphs on the Cartesian coordinate plane depict relationships with a constant rate of change, known as the slope ($m = \frac{y_2 - y_1}{x_2 - x_1}$). Lines are represented using slope-intercept form ($y = mx + b$). Parallel lines share identical slopes ($m_1 = m_2$), while perpendicular lines feature negative reciprocal slopes ($m_1 \cdot m_2 = -1$). In electrical circuit analysis, plotting Voltage ($V$) against Current ($I$) produces a linear slope representing circuit Resistance ($R = \frac{\Delta V}{\Delta I}$), directly demonstrating Ohm's Law.

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The Cartesian Coordinate System

The Cartesian coordinate plane is formed by two perpendicular number lines that intersect at a central point called the origin $(0, 0)$:

• Horizontal Axis ($x$-axis): Positive values extend to the right; negative values extend to the left.
• Vertical Axis ($y$-axis): Positive values extend upward; negative values extend downward.

Any point on the plane is identified by an ordered pair $(x, y)$, where $x$ represents the horizontal position (abscissa) and $y$ represents the vertical position (ordinate). The plane is divided into four quadrants:

• Quadrant I: $(+, +)$ [Upper Right]
• Quadrant II: $(-, +)$ [Upper Left]
• Quadrant III: $(-, -)$ [Lower Left]
• Quadrant IV: $(+, -)$ [Lower Right]

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The Concept of Slope ($m$)

The slope of a non-vertical line measures its steepness and direction. Mathematically, slope is defined as the ratio of vertical change (rise) to horizontal change (run) between any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on the line.

The Slope Formula

$$m = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

Where $x_2 \neq x_1$.

Four Classifications of Slope

1. Positive Slope ($m > 0$): The line rises from left to right ($\nearrow$).
2. Negative Slope ($m < 0$): The line falls from left to right ($\searrow$).
3. Zero Slope ($m = 0$): The line is completely horizontal ($\rightarrow$). The $y$-coordinates are identical ($y_2 = y_1 \implies \text{Rise} = 0$).
4. Undefined Slope: The line is completely vertical ($\uparrow$). The $x$-coordinates are identical ($x_2 = x_1 \implies \text{Run} = 0$, causing division by zero).

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Step-by-Step Example: Calculating Slope

Problem: Calculate the slope of the line passing through points $(3, 8)$ and $(7, 0)$.

1. Designate coordinates: $$(x_1, y_1) = (3, 8), \quad (x_2, y_2) = (7, 0)$$
2. Substitute into the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 8}{7 - 3}$$
3. Simplify numerator and denominator: $$m = \frac{-8}{4}$$
4. Divide: $$m = -2$$

The slope of the line is $-2$ (indicating that for every $1$ unit increase in $x$, $y$ decreases by $2$ units).

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Forms of Linear Equations

Linear relationships can be expressed algebraically in several standard forms:

1. Slope-Intercept Form ($y = mx + b$)

The most versatile form for graphing and analysis on the EIAT is slope-intercept form:

$$y = mx + b$$

Where:

• $m$ = Slope of the line.
• $b$ = $y$-intercept (the point $(0, b)$ where the line crosses the vertical $y$-axis).

2. Point-Slope Form ($y - y_1 = m(x - x_1)$)

Useful when you know the slope $m$ and one point $(x_1, y_1)$ on the line:

$$y - y_1 = m(x - x_1)$$

3. Standard Form ($Ax + By = C$)

Where $A$, $B$, and $C$ are integers (with $A \ge 0$). The slope of a line in standard form is $m = -\frac{A}{B}$.

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Parallel & Perpendicular Line Dynamics

Geometric relationships between two lines on a coordinate plane depend strictly on their relative slopes.

Parallel Lines ($m_1 = m_2$)

Parallel lines run side-by-side in the same direction and never intersect.

• Rule: Two non-vertical lines are parallel if and only if they have identical slopes and distinct $y$-intercepts: $$m_1 = m_2 \quad \text{and} \quad b_1 \neq b_2$$

Example: $y = 3x + 4$ and $y = 3x - 7$ are parallel because both have slope $m = 3$.

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Perpendicular Lines ($m_1 \cdot m_2 = -1$)

Perpendicular lines intersect at a precise $90^\circ$ right angle.

• Rule: Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other: $$m_2 = -\frac{1}{m_1} \quad \text{or} \quad m_1 \cdot m_2 = -1$$

To find the negative reciprocal of a fraction $\frac{a}{b}$:

1. Flip the fraction upside down: $\frac{b}{a}$.
2. Change the sign: $-\frac{b}{a}$.

Step-by-Step Example: Finding a Perpendicular Line

Problem: Which equation represents a line perpendicular to $y = -\frac{2}{5}x + 4$ and passing through the point $(0, -3)$?

1. Identify the original slope ($m_1$): $$m_1 = -\frac{2}{5}$$
2. Calculate the perpendicular slope ($m_2$): Take the negative reciprocal of $-\frac{2}{5}$: $$m_2 = -\left( -\frac{5}{2} \right) = +\frac{5}{2}$$
3. Identify the $y$-intercept ($b$): The line passes through $(0, -3)$, which is the vertical $y$-intercept, so $b = -3$.
4. Construct the equation in slope-intercept form: $$y = m_2 x + b \implies y = \frac{5}{2}x - 3$$

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Electrical Engineering Application: V-I Graphs & Ohm's Law

In electrical engineering, linear coordinate graphs provide a visual representation of Ohm's Law:

$$V = I \cdot R$$

When circuit test data is plotted on a coordinate plane with Voltage ($V$, in Volts) on the vertical $y$-axis and Current ($I$, in Amperes) on the horizontal $x$-axis:

$$\text{Slope } m = \frac{\Delta y}{\Delta x} = \frac{\Delta V}{\Delta I}$$

According to Ohm's Law, $\frac{V}{I} = R$. Therefore, the slope of a $V$-$I$ line directly equals the electrical Resistance ($R$) of the circuit component in Ohms ($\Omega$).

Trade Scenario: Determining Resistance from $V$-$I$ Graph Data

An electrician conducts testing on a heating element and plots the resulting circuit voltage versus current:

• Point 1: $(2\text{ A}, 24\text{ V})$
• Point 2: $(5\text{ A}, 60\text{ V})$

Step 1: Calculate the Slope of the $V$-$I$ Line

$$m = \frac{V_2 - V_1}{I_2 - I_1} = \frac{60\text{ V} - 24\text{ V}}{5\text{ A} - 2\text{ A}} = \frac{36\text{ V}}{3\text{ A}} = 12\text{ Volts per Ampere}$$

Step 2: Physical Interpretation

Since $1\text{ Volt per Ampere} = 1\ \Omega$, the slope represents Electrical Resistance, and its value is $12\ \Omega$.

• Ohmic Components (Linear Slopes): Resistors and heating elements maintain a constant straight-line slope, indicating constant resistance regardless of voltage level.
• Non-Ohmic Components (Curved Slopes): Devices such as diodes and incandescent lamp filaments display curved $V$-$I$ lines, indicating variable resistance under thermal load.

Mastery of linear graphing, slope calculations, and coordinate geometry enables electrical apprentices to analyze circuit diagnostics quickly and excel on the EIAT.