College Algebra - Week 5
Graphing Lines
A linear function is the simplest function that changes at a constant rate. This week you master the slope concept, the three forms of a line's equation, and the mechanics of graphing. You also study parallel and perpendicular lines, which are prerequisites for geometry, calculus, and physics. Every graph you draw reinforces the connection between equations and visual structure.
Learning Objectives
- Calculate slope from two points and interpret slope as rate of change. - Graph a line given its equation in any form (slope-intercept, point-slope, standard). - Write the equation of a line given two points, or a point and the slope. - Convert between slope-intercept form, point-slope form, and standard form. - Identify and write equations of horizontal and vertical lines. - Determine whether two lines are parallel, perpendicular, or neither. - Find the x-intercept and y-intercept of a line and use them to graph.
1. Slope: The Rate of Change
Slope measures how steeply a line rises or falls. Given two points (x1, y1) and (x2, y2): m = (y2 - y1)/(x2 - x1) This is the rise over the run. Example: Find the slope of the line through (2, 3) and (6, 11). m = (11 - 3)/(6 - 2) = 8/4 = 2 Interpretation: for every 1 unit increase in x, y increases by 2. Positive slope → line rises left to right. Negative slope → line falls left to right. Zero slope → horizontal line. Undefined slope → vertical line.
2. Slope-Intercept Form
y = mx + b where m is slope and b is the y-intercept. The y-intercept is where the line crosses the y-axis (where x = 0). To graph using slope-intercept form: 1. Plot the y-intercept (0, b). 2. Use the slope: rise/run from that point. 3. Draw the line through both points. Example: Graph y = 2x - 3. Y-intercept: (0, -3). Slope: 2 = 2/1. From (0,-3), go up 2, right 1 to (1,-1). Draw the line.
3. Point-Slope Form
y - y1 = m(x - x1) where (x1, y1) is a known point and m is the slope. This form is best when you know the slope and one point, but NOT the y-intercept. Example: Write the equation of the line with slope 3 passing through (2, -1). y - (-1) = 3(x - 2) y + 1 = 3x - 6 y = 3x - 7 (slope-intercept form) You can leave the answer in point-slope form or convert to slope-intercept form as needed.
4. Writing the Equation Through Two Points
Step 1: Find the slope using m = (y2 - y1)/(x2 - x1). Step 2: Plug the slope and ONE point into point-slope form. Step 3: Simplify. Example: Find the equation of the line through (-1, 4) and (3, -8). Slope: m = (-8 - 4)/(3 - (-1)) = -12/4 = -3 Point-slope (using (-1,4)): y - 4 = -3(x - (-1)) y - 4 = -3(x + 1) y - 4 = -3x - 3 y = -3x + 1
5. Standard Form
Standard form: Ax + By = C where A, B, C are integers and A >= 0. To convert y = 3x - 5 to standard form: Subtract 3x from both sides: -3x + y = -5. Multiply by -1: 3x - y = 5. To find intercepts from standard form: - x-intercept: set y = 0 and solve. - y-intercept: set x = 0 and solve. Example: 4x - 2y = 8. X-intercept: 4x = 8 → x = 2. Point (2, 0). Y-intercept: -2y = 8 → y = -4. Point (0, -4).
6. Horizontal and Vertical Lines
Horizontal lines have slope 0. Equation: y = k (a constant). All points on the line have the same y-coordinate. Example: y = -4 is a horizontal line passing through all points where y = -4. Vertical lines have undefined slope. Equation: x = k. All points have the same x-coordinate. Example: x = 3 is a vertical line passing through all points where x = 3. Vertical lines are NOT functions (fail the vertical line test). Horizontal lines ARE functions (constant functions).
7. Parallel Lines
Parallel lines never intersect. They have the SAME slope but DIFFERENT y-intercepts. Examples of parallel lines: y = 3x + 1 and y = 3x - 5 (both have slope 3). Example: Find the equation of a line parallel to y = 2x - 4 passing through (1, 7). Same slope: m = 2. Point-slope: y - 7 = 2(x - 1) y = 2x + 5
8. Perpendicular Lines
Perpendicular lines meet at a 90° angle. Their slopes are negative reciprocals. If m1 and m2 are slopes of perpendicular lines: m1 * m2 = -1. Examples: Slope 2 → perpendicular slope is -1/2. Slope -3/4 → perpendicular slope is 4/3. Slope 0 (horizontal) → perpendicular is a vertical line (undefined slope). Example: Find the line perpendicular to y = (3/4)x + 2 through (-3, 5). Perpendicular slope: -4/3. y - 5 = (-4/3)(x + 3) y = (-4/3)x + 1
9. Graphing Using Intercepts
Two intercepts give two points — enough to graph a line. Example: Graph 3x + 2y = 12. X-intercept (y = 0): 3x = 12 → x = 4. Point (4, 0). Y-intercept (x = 0): 2y = 12 → y = 6. Point (0, 6). Plot both points, draw the line. This method is fastest when a line is in standard form.
10. Slope as Rate of Change in Applications
Slope has a real meaning outside of geometry. Example: A phone plan charges $0.05 per text plus a $20/month base fee. Cost = 0.05t + 20 where t = number of texts. Slope = 0.05 = the rate at which cost increases per text. Y-intercept = 20 = the fixed monthly cost. Example: Temperature in Celsius can be modeled as a linear function of altitude. As altitude increases by 1000 m, temperature drops by about 6.5°C. Slope = -6.5°C per 1000 m (rate of change).
11. Summary of Line Forms
Slope-intercept: y = mx + b — best for graphing and identifying slope/y-intercept. Point-slope: y - y1 = m(x - x1) — best for writing equations. Standard: Ax + By = C — required for some contexts, best for finding intercepts. Convert between forms using algebra. All three describe the same line — just written differently.
12. Common Mistakes
- Computing slope as (x2 - x1)/(y2 - y1) instead of rise over run. - Mistaking the slope for the y-intercept in y = mx + b. - Forgetting that parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes. - Writing the equation of a vertical line as y = k instead of x = k. - Sign errors when converting from point-slope to slope-intercept form.