Unit: Data Handling & Analysis
Chapter: Linear Graphs
Reference: – What is a Linear Graph, Linear Equations in Two Variables, Plotting Points on a Coordinate Plane, Graphing Linear Equations Using Tables, Slope of a Line, Intercepts (x-intercept and y-intercept), Graphing Using Slope-Intercept Form (y = mx + b), Graphing Using Intercepts, Horizontal and Vertical Lines, Real-World Applications of Linear Graphs, Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is a Linear Graph
- How to Plot Points on a Coordinate Plane
- How to Graph a Linear Equation Using a Table, Intercepts, or Slope-Intercept Form
- How to Find and Interpret Slope and Intercepts
Introduction to Linear Graphs
Definition
A linear graph is the graph of a linear equation. It is a straight line on the coordinate plane. Linear equations have the form y = mx + b (slope-intercept form) or Ax + By = C (standard form). Every point on the line is a solution to the equation.
When we study linear graphs, we essentially ask:
"How does this line look on the coordinate plane? Where does it cross the axes? How steep is it?"
Once we understand these features, we can graph any linear equation quickly and interpret what it means.
Importance of Linear Graphs
- Shows the relationship between two variables visually
- Helps predict values between known data points
- Used in science to show constant rates (speed, growth, cost)
- Foundation for understanding more complex functions
- Essential for data analysis and trend lines
Example
The equation y = 2x + 1 graphs as a straight line. Every point on the line, like (0,1), (1,3), and (2,5), makes the equation true. The line has slope 2 and crosses the y-axis at 1.
Subtopics
1. The Coordinate Plane Review
The coordinate plane has two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). They meet at the origin (0, 0).
Points are written as ordered pairs (x, y), where x tells how far to move right (positive) or left (negative), and y tells how far to move up (positive) or down (negative).
Four Quadrants:
Quadrant I: x positive, y positive (top right)
Quadrant II: x negative, y positive (top left)
Quadrant III: x negative, y negative (bottom left)
Quadrant IV: x positive, y negative (bottom right)
2. Graphing Linear Equations Using a Table
The most basic method: choose x-values, compute y-values, plot points, and connect them with a straight line.
Example – Graph y = x + 2
Choose x = -1, 0, 1, 2
When x = -1, y = -1 + 2 = 1 → point (-1, 1)
When x = 0, y = 0 + 2 = 2 → point (0, 2)
When x = 1, y = 1 + 2 = 3 → point (1, 3)
When x = 2, y = 2 + 2 = 4 → point (2, 4)
Plot these points and draw a straight line through them.
3. Slope of a Line
Slope (m) measures the steepness of a line. It is the ratio of vertical change (rise) to horizontal change (run).
Formula: m = (y₂ – y₁) / (x₂ – x₁) = rise / run
What Slope Tells You:
Positive slope (m > 0): line rises from left to right
Negative slope (m < 0): line falls from left to right
Zero slope (m = 0): horizontal line
Undefined slope: vertical line
Example – Find slope between (1, 2) and (4, 8)
m = (8 – 2) / (4 – 1) = 6 / 3 = 2
4. Intercepts
y-intercept (b): The point where the line crosses the y-axis. At this point, x = 0. In y = mx + b, b is the y-intercept.
x-intercept: The point where the line crosses the x-axis. At this point, y = 0.
Example – Find intercepts of 2x + y = 6
y-intercept: set x = 0 → y = 6 → (0, 6)
x-intercept: set y = 0 → 2x = 6 → x = 3 → (3, 0)
5. Graphing Using Slope-Intercept Form (y = mx + b)
This is the fastest method.
Step 1: Identify m (slope) and b (y-intercept)
Step 2: Plot the y-intercept (0, b)
Step 3: Use slope = rise/run to find another point. From the y-intercept, go up/down (rise) and right (run).
Step 4: Draw the line through the two points.
Example – Graph y = (1/2)x – 3
m = 1/2 (rise 1, run 2), b = -3
Plot (0, -3)
From (0, -3): go up 1, right 2 → reach (2, -2)
Draw line through (0, -3) and (2, -2)
Example – Graph y = -2x + 4
m = -2 = -2/1 (rise -2, run 1), b = 4
Plot (0, 4)
From (0, 4): go down 2, right 1 → reach (1, 2)
Draw line through (0, 4) and (1, 2)
6. Graphing Using Intercepts
Useful for equations in standard form (Ax + By = C).
Step 1: Find x-intercept (set y = 0, solve for x)
Step 2: Find y-intercept (set x = 0, solve for y)
Step 3: Plot both intercepts and draw the line through them.
Example – Graph 3x + 2y = 6
x-intercept: set y=0 → 3x = 6 → x = 2 → (2, 0)
y-intercept: set x=0 → 2y = 6 → y = 3 → (0, 3)
Plot (2, 0) and (0, 3) and draw the line
7. Horizontal and Vertical Lines
Horizontal Line: y = c (c is constant)
Slope = 0
Graph is flat, crossing y-axis at (0, c)
Example: y = 4 is a horizontal line through all points where y = 4
Vertical Line: x = c (c is constant)
Slope is undefined
Graph is straight up and down, crossing x-axis at (c, 0)
Example: x = -2 is a vertical line through all points where x = -2
Important: A vertical line is graphed on the coordinate plane, but it is NOT a function (fails the vertical line test).
Solved Examples
Example 1 – Using a Table: Graph y = 3x – 1 using a table of values.
Solution: Choose x = -1, 0, 1, 2
x = -1 → y = 3(-1) – 1 = -4 → (-1, -4)
x = 0 → y = -1 → (0, -1)
x = 1 → y = 2 → (1, 2)
x = 2 → y = 5 → (2, 5)
Plot and connect with a straight line.
Answer: Graph is a line through these points.
Example 2 – Using Slope-Intercept Form: Graph y = -3x + 2.
Solution: m = -3, b = 2
Plot (0, 2)
From (0, 2): go down 3, right 1 → reach (1, -1)
Draw line through (0, 2) and (1, -1)
Answer: Line with slope -3 crossing y-axis at 2.
Example 3 – Finding Slope: Find the slope of the line through (3, 5) and (7, 11).
Solution: m = (11 – 5) / (7 – 3) = 6 / 4 = 3/2
Answer: 3/2
Example 4 – Using Intercepts: Graph 4x – y = 8 using intercepts.
Solution: y-intercept: set x = 0 → -y = 8 → y = -8 → (0, -8)
x-intercept: set y = 0 → 4x = 8 → x = 2 → (2, 0)
Plot (0, -8) and (2, 0) and draw the line.
Answer: Line through (0, -8) and (2, 0).
Common Mistakes to Avoid
Mistake 1 – Reversing rise and run in slope
Slope = rise/run (vertical change over horizontal change), not run/rise.
Correct understanding: Rise = change in y, Run = change in x.
Mistake 2 – Forgetting negative slope direction
Negative slope means go DOWN as you move right, not up.
Correct understanding: Negative rise with positive run gives negative slope.
Mistake 3 – Plotting the y-intercept incorrectly
y-intercept (0, b) is on the y-axis. Do not plot (b, 0) unless b = 0.
Correct understanding: y-intercept always has x = 0.
Mistake 4 – Drawing curves instead of straight lines
Linear graphs are straight lines. A curve means the equation is not linear.
Correct understanding: Use a ruler to ensure the line is straight.
Mistake 5 – Confusing x-intercept with y-intercept
x-intercept: set y = 0; y-intercept: set x = 0.
Correct understanding: Remember: x-intercept has y = 0; y-intercept has x = 0.
Mistake 6 – Thinking horizontal lines have no slope
Horizontal lines have slope = 0, not "no slope."
Correct understanding: "No slope" means undefined (vertical). Horizontal slope is zero.
Quick Reference Summary
Linear Equation: y = mx + b (or Ax + By = C)
Slope (m): m = (y₂ – y₁)/(x₂ – x₁) = rise/run
y-intercept (b): where line crosses y-axis (x = 0)
x-intercept: where line crosses x-axis (y = 0)
Graphing Methods:
Table of values → choose x, find y, plot points
Slope-intercept form → plot y-intercept, use slope for second point
Intercepts → plot x-intercept and y-intercept, draw line
Horizontal Line: y = c, slope = 0
Vertical Line: x = c, slope undefined (not a function)