MathematicsChapter 8: Geometry of Straight Lines
Slope Intercept Form
When a line is not vertical it can be written y=mx+c.
Learning objectives
- Explain slope intercept form in clear mathematical language
- Apply the ideas in meaning of m and c
- Solve problems involving writing the equation
Definition
When a line is not vertical it can be written y=mx+c. Here m is the slope (gradient) and c is the y-intercept, the value of y where the line cuts the y-axis at (0, c). This is the slope-intercept form. If m=0 it collapses to the horizontal line y=c. From a graph, read c where the line hits the y-axis, then read m as rise/run from that point, or as tan theta.
Mathematical form
y=mx+c. Here m is the slope (gradient) and
m=0 it collapses to the horizontal line y
y=mx+c
x=-c/m when m
Where / symbols
- y=mx+c
- When a line is not vertical it can be written y=mx+c.
- slope
- Here m is the slope (gradient) and c is the y-intercept, the value of y where the line cuts the y-axis at (0, c).
- y-intercept
- Here m is the slope (gradient) and c is the y-intercept, the value of y where the line cuts the y-axis at (0, c).
- slope-intercept form
- This is the slope-intercept form.
Relationship
- y=mx+c
Overview
Read m as tilt and c as the landing on the y-axis:
MEANING OF m AND c
In y=mx+c the coefficient of x is m = rise/run = tan theta. The constant c is the y-intercept: plug x=0 and y=c, so the point (0, c) is on the line. Positive m climbs; negative m falls; m=0 is horizontal. Two lines y=2x+1 and y=2x-4 are parallel (same m) with different intercepts.
Examples y=2x+1: m=2, c=1 · y=-x+3: m=-1, c=3 · y=4: m=0, c=4
Writing the Equation
If you know slope m and a point (x1, y1), then c = y1 - m x1, so y = m x + (y1 - m x1). If you know two points, first compute m = y2-y1x2-x1, then find c from one of the points. You cannot force a vertical line into y=mx+c. Rearrange ax+by+c=0 (next page) into this form by solving for y, when b!=0.
Examples m=1/2 through (0,3): y=(1/2)x+3 · Points (0,2) and (2,6): m=2, y=2x+2 · Cannot write x=5 as y=mx+c
What it means
When a line is not vertical it can be written y=mx+c.
Main ideas
1. MEANING OF m AND c: In y=mx+c the coefficient of x is m = rise/run = tan theta. 2. WRITING THE EQUATION: If you know slope m and a point (x1, y1), then c = y1 - m x1, so y = m x + (y1 - m x1).
Why it matters
Graphing y=mx+c: plot (0, c), then from that point count run 1 and rise m (or run 2 and rise 2m if m is a fraction). Join with a straight line. The x-intercept is where y=0: solve 0=mx+c, so x=-c/m when m!=0.
How to answer in an exam
y=mx+c is slope-intercept: m is slope, c is the y-intercept, not the x-intercept. Vertical lines are the exception. If the equation is y=mx+c with c=0, the line goes through the origin.
Explanation
Graphing y=mx+c: plot (0, c), then from that point count run 1 and rise m (or run 2 and rise 2m if m is a fraction). Join with a straight line. The x-intercept is where y=0: solve 0=mx+c, so x=-c/m when m!=0.
In simple words
y=mx+c is slope-intercept: m is slope, c is the y-intercept, not the x-intercept. Vertical lines are the exception. If the equation is y=mx+c with c=0, the line goes through the origin.
Key points
- y=mx+c
- slope
- c
- y-intercept
- m = rise/run
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Slope Intercept Form
Sticky Notes · Quick Revision
Core Idea
- When a line is not vertical it can be written y=mx+c.
Key Terms
- y=mx+c: When a line is not vertical it can be written y=mx+c.
- slope: Here m is the slope (gradient) and c is the y-intercept, the value of y where the line cuts the y-axis at (0, c).
Main Pattern
- 1. MEANING OF m AND c: In y=mx+c the coefficient of x is m = rise/run = tan theta.
- 2. WRITING THE EQUATION: If you know slope m and a point (x1, y1), then c = y1 - m x1, so y = m x + (y1 - m x1).
Worked Reminder
- y=2x+1: m=2, c=1
- y=-x+3: m=-1, c=3
- y=4: m=0, c=4
Common Mistake
- y=mx+c is slope-intercept: m is slope, c is the y-intercept, not the x-intercept. Vertical lines are the exception. If the equation is y=mx+c with c=0, the line goes through the origin.
Exam Tip
- State the rule for slope intercept form, show each algebraic step, and check the final answer against the question.
60-second revision
- When a line is not vertical it can be written y=mx+c.
- In y=mx+c the coefficient of x is m = rise/run = tan theta.
- If you know slope m and a point (x1, y1), then c = y1 - m x1, so y = m x + (y1 - m x1).
- Graphing y=mx+c: plot (0, c), then from that point count run 1 and rise m (or run 2 and rise 2m if m is a fraction).
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