Graphing Lines and Finding Slope in College Algebra

Section 2.4: Slope and Graphing Using Slope-Intercept

Introduction

This section explores the methods for graphing linear equations and determining the slope of lines. Understanding these concepts is fundamental in College Algebra, as they form the basis for analyzing and interpreting linear relationships.

Graphing Lines

Graphing Lines Using X and Y Intercepts

One common method for graphing a line is to find its intercepts with the axes.

• X-intercept: The point where the line crosses the x-axis. To find it, set y = 0 in the equation and solve for x.

• Y-intercept: The point where the line crosses the y-axis. To find it, set x = 0 in the equation and solve for y.

Example 1: Find the x and y intercepts of the graph of $y = -5x + 6$ and graph this line.

• Set $y = 0$: $0 = -5x + 6 \implies x = \frac{6}{5}$ (x-intercept: $(\frac{6}{5}, 0)$)

• Set $x = 0$: $y = -5(0) + 6 = 6$ (y-intercept: $(0, 6)$)

Example 2: Find the x and y intercepts of $2x - 6y = 12$ and graph this line.

• Set $y = 0$: $2x = 12 \implies x = 6$ (x-intercept: $(6, 0)$)

• Set $x = 0$: $-6y = 12 \implies y = -2$ (y-intercept: $(0, -2)$)

Example 3: Complete the ordered pairs so they are solutions of $3y = 2x - 6$ and graph this line.

• When $x = 0$: $3y = -6 \implies y = -2$ (point: $(0, -2)$)

• When $y = 0$: $0 = 2x - 6 \implies x = 3$ (point: $(3, 0)$)

Example 4: For $-5x + 2y = 10$, find $(0, ?)$ and $(?, 0)$.

• When $x = 0$: $2y = 10 \implies y = 5$ (point: $(0, 5)$)

• When $y = 0$: $-5x = 10 \implies x = -2$ (point: $(-2, 0)$)

Graphing Lines Using Point and Slope

The slope of a line measures its steepness and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points. The slope is often denoted by m.

• Point-Slope Form: The equation of a line with slope m passing through point $(x_1, y_1)$ is $y - y_1 = m(x - x_1)$.

Example 5: Graph the line passing through $(2, 0)$ with slope $-\frac{4}{5}$.

• Equation: $y - 0 = -\frac{4}{5}(x - 2)$

Example 6: Graph the line passing through $(-3, 1)$ with slope $\frac{7}{2}$.

• Equation: $y - 1 = \frac{7}{2}(x + 3)$

Graphing "Special" Lines

Some lines have equations involving only one variable:

• Vertical Line: $x = a$ is a vertical line crossing the x-axis at $x = a$.

• Horizontal Line: $y = b$ is a horizontal line crossing the y-axis at $y = b$.

Example 7: Graph $x = -6$ (vertical line through $x = -6$).

Example 8: Graph $y - 3 = 0$ or $y = 3$ (horizontal line through $y = 3$).

Finding the Slopes of Lines

Finding the Slope When Given Two Points

The slope m of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example 9: Through $(4, 2)$ and $(7, 3)$:

  • $m = \frac{3 - 2}{7 - 4} = \frac{1}{3}$

• Example 10: Through $(-2, 3)$ and $(-5, 7)$:

  • $m = \frac{7 - 3}{-5 + 2} = \frac{4}{-3} = -\frac{4}{3}$

• Example 11: Through $(2, -3)$ and $(-4, -3)$:

  • $m = \frac{-3 - (-3)}{ -4 - 2 } = \frac{0}{-6} = 0$ (horizontal line)

• Example 12: Through $(2, -1)$ and $(2, -5)$:

  • $m = \frac{ -5 - (-1) }{ 2 - 2 } = \frac{ -4 }{ 0 }$ (undefined; vertical line)

Finding the Slope When Given the Equation of the Line

The slope-intercept form of a line is $y = mx + b$, where m is the slope and b is the y-intercept.

• Example 13: $3x - 2y = 12$

  • Solve for $y$:

  • $-2y = -3x + 12$

  • $y = \frac{3}{2}x - 6$

  • Slope $m = \frac{3}{2}$

• Example 14: $5y = -7x$

  • $y = -\frac{7}{5}x$

  • Slope $m = -\frac{7}{5}$

Finding the Slope of "Special" Lines

• Vertical Line: $x = a$ has an undefined slope.

• Horizontal Line: $y = b$ has a slope of zero.

Example 15: The slope of a horizontal line through $(5, 7)$ is $0$.

Example 16: The slope of a vertical line through $(-9, -1)$ is undefined.

Summary Table: Types of Lines and Their Slopes

Equation Form	Type of Line	Slope	Graph Description
$y = mx + b$	Oblique (non-vertical, non-horizontal)	$m$	Slanted line
$x = a$	Vertical	Undefined	Vertical line through $x = a$
$y = b$	Horizontal	0	Horizontal line through $y = b$

Additional info: The concepts of slope and intercepts are foundational for understanding more advanced topics such as systems of equations, analytic geometry, and calculus. Mastery of these basics is essential for success in College Algebra and beyond.