Linear Graphs
A linear graph is a graph that forms a straight line. It represents a linear relationship between two variables — when one variable changes, the other changes at a constant rate.
Linear graphs are among the most important types of graphs in mathematics. They appear in distance-time relationships, cost calculations, temperature conversions, and many other situations.
The general equation of a linear graph is y = mx + c, where m is the slope (gradient) and c is the y-intercept. When c = 0, the line passes through the origin and represents direct proportion.
What is Linear Graphs?
Definition: A linear graph is a graph of an equation of the form y = mx + c, which always produces a straight line.
y = mx + c
Where:
- m = slope (gradient) — how steep the line is
- c = y-intercept — where the line crosses the y-axis
- x = independent variable (horizontal axis)
- y = dependent variable (vertical axis)
Special cases:
- If c = 0: y = mx → line passes through origin (direct proportion)
- If m = 0: y = c → horizontal line
- If m > 0: line goes upward from left to right
- If m < 0: line goes downward from left to right
Methods
Steps to draw a linear graph:
- Write the equation in the form y = mx + c.
- Choose at least 3 values of x.
- Calculate the corresponding values of y.
- Plot the points (x, y) on graph paper.
- Join the points — they should form a straight line.
Steps to read a linear graph:
- Locate the required value on one axis.
- Draw a line (horizontal or vertical) to meet the graph.
- From that point, draw a line to the other axis.
- Read the value.
Finding the equation from a graph:
- Find the y-intercept (where the line crosses the y-axis) — this is c.
- Pick two points on the line and calculate slope: m = (y₂ − y₁)/(x₂ − x₁).
- Write y = mx + c.
Solved Examples
Example 1: Example 1: Drawing y = 2x + 1
Problem: Draw the graph of y = 2x + 1 for x = −1, 0, 1, 2, 3.
Solution:
Table of values:
- x = −1: y = 2(−1) + 1 = −1
- x = 0: y = 2(0) + 1 = 1
- x = 1: y = 2(1) + 1 = 3
- x = 2: y = 2(2) + 1 = 5
- x = 3: y = 2(3) + 1 = 7
Points: (−1, −1), (0, 1), (1, 3), (2, 5), (3, 7).
Join these points to get a straight line with slope 2 and y-intercept 1.
Answer: The graph is a straight line crossing the y-axis at (0, 1).
Example 2: Example 2: Graph through the origin
Problem: Draw the graph of y = 3x.
Solution:
- x = 0: y = 0
- x = 1: y = 3
- x = 2: y = 6
- x = 3: y = 9
Points: (0, 0), (1, 3), (2, 6), (3, 9). This line passes through the origin.
Answer: The graph is a straight line through the origin (direct proportion with k = 3).
Example 3: Example 3: Distance-time graph
Problem: A person walks at 5 km/h. Draw the distance-time graph for 0 to 4 hours.
Solution:
d = 5t:
- t = 0: d = 0
- t = 1: d = 5
- t = 2: d = 10
- t = 3: d = 15
- t = 4: d = 20
The graph is a straight line through the origin. The slope = 5 (speed).
Answer: A straight line from (0, 0) to (4, 20).
Example 4: Example 4: Temperature conversion graph
Problem: Draw the conversion graph for C to F using F = (9/5)C + 32.
Solution:
- C = 0: F = 32
- C = 10: F = 50
- C = 20: F = 68
- C = 30: F = 86
- C = 100: F = 212
The graph is a straight line with slope 9/5 and y-intercept 32.
It does NOT pass through the origin (because c = 32 ≠ 0).
Answer: A rising straight line crossing the F-axis at 32.
Example 5: Example 5: Reading from a graph
Problem: From the graph of y = 2x + 3, find y when x = 4.
Solution:
- On the graph, locate x = 4 on the horizontal axis.
- Draw a vertical line up to the graph line.
- From that point, draw a horizontal line to the y-axis.
- Read: y = 2(4) + 3 = 11
Answer: When x = 4, y = 11.
Example 6: Example 6: Finding the equation from a graph
Problem: A straight-line graph passes through (0, 2) and (3, 8). Find the equation.
Solution:
- y-intercept: c = 2 (the line crosses y-axis at 2)
- Slope: m = (8 − 2)/(3 − 0) = 6/3 = 2
- Equation: y = 2x + 2
Answer: The equation is y = 2x + 2.
Example 7: Example 7: Negative slope
Problem: Draw the graph of y = −x + 4 for x = 0, 1, 2, 3, 4.
Solution:
- x = 0: y = 4
- x = 1: y = 3
- x = 2: y = 2
- x = 3: y = 1
- x = 4: y = 0
The line goes downward from left to right (negative slope = −1).
Answer: A descending straight line from (0, 4) to (4, 0).
Example 8: Example 8: Fixed charge + per unit cost
Problem: An auto fare is Rs 20 (fixed) plus Rs 8 per km. Draw the graph for 0 to 5 km.
Solution:
Fare = 8d + 20:
- 0 km: Rs 20
- 1 km: Rs 28
- 2 km: Rs 36
- 3 km: Rs 44
- 4 km: Rs 52
- 5 km: Rs 60
The graph is a straight line with slope 8 and y-intercept 20.
Answer: The graph starts at (0, 20) — not at the origin — because of the fixed charge.
Example 9: Example 9: Horizontal line
Problem: Draw the graph of y = 5.
Solution:
- For any value of x, y = 5.
- Points: (−2, 5), (0, 5), (3, 5), (7, 5).
All points have the same y-coordinate. The graph is a horizontal line at y = 5.
Answer: A horizontal straight line at height 5 (slope = 0).
Example 10: Example 10: Identifying linear from data
Problem: Data: x = 1, 2, 3, 4 and y = 7, 10, 13, 16. Is this linear? Find the equation.
Solution:
- Check differences in y: 10−7=3, 13−10=3, 16−13=3 → constant difference of 3.
- Since the difference is constant, the relationship is linear.
- Slope m = 3. When x = 1, y = 7: 7 = 3(1) + c → c = 4.
- Equation: y = 3x + 4
Answer: Yes, it is linear. Equation: y = 3x + 4.
Real-World Applications
Real-world applications of linear graphs:
- Distance-time graphs: Constant speed produces a straight line; slope = speed.
- Conversion graphs: Currency, temperature, units — all linear conversions can be shown as straight lines.
- Bills and charges: Fixed charge + variable rate gives a linear graph (e.g., taxi fare, electricity bill).
- Science experiments: Plotting experimental data to check if two quantities are linearly related.
- Business: Revenue = price × quantity (straight line through origin).
Key Points to Remember
- A linear graph is always a straight line.
- Equation: y = mx + c where m = slope and c = y-intercept.
- If c = 0, the line passes through the origin (direct proportion).
- Positive slope: line rises left to right. Negative slope: line falls.
- Only 2 points are needed to draw a line, but use 3 for checking.
- Slope = (change in y)/(change in x) = (y₂ − y₁)/(x₂ − x₁).
- If the y-differences are constant for equal x-steps, the data is linear.
- Parallel lines have equal slopes.
Practice Problems
- Draw the graph of y = x + 3 for x = −2, −1, 0, 1, 2, 3.
- Draw the graph of y = −2x + 6 and find where it crosses the x-axis.
- A taxi charges Rs 15 fixed + Rs 10 per km. Draw the fare graph for 0 to 6 km.
- From the data x: 2, 4, 6, 8 and y: 5, 9, 13, 17, find the equation and draw the graph.
- Draw the graphs of y = 2x and y = 2x + 4. What do you notice?
- A water tank is being filled at 5 litres per minute. It already has 20 litres. Draw the graph for 0 to 10 minutes.
Frequently Asked Questions
Q1. What is a linear graph?
A linear graph is a graph that forms a straight line. It represents the equation y = mx + c.
Q2. How many points do you need to draw a linear graph?
You need at least 2 points. However, plotting 3 points is recommended for verification — if the third point doesn't lie on the line, there's an error.
Q3. What does the slope of a linear graph tell us?
The slope tells how steeply the line rises or falls. A slope of 3 means y increases by 3 for every 1 unit increase in x.
Q4. What does the y-intercept mean?
The y-intercept is the value of y when x = 0. It is where the line crosses the y-axis.
Q5. Is every straight line a direct proportion graph?
No. Only straight lines through the origin (c = 0) represent direct proportion. A line like y = 2x + 5 is linear but not direct proportion.
Q6. What does a horizontal line on a graph mean?
A horizontal line means y is constant regardless of x. The slope is zero. Example: y = 4.
Q7. How do you find the equation from a linear graph?
Read the y-intercept (c) from where the line crosses the y-axis. Pick two points and calculate slope m = (y₂−y₁)/(x₂−x₁). Then y = mx + c.
Q8. Can a linear graph have a negative slope?
Yes. A negative slope means the line goes downward from left to right. Example: y = −3x + 10.
