Graph

Grade 12 Math - Plotting and Interpreting Linear Graphs

Lesson Objectives

Plot linear graphs using given equations
Interpret graphs to find slope, intercepts, and relationships
Use graphs to solve real-world problems and systems of equations

Lesson Introduction

Linear graphs represent equations of the form $y = mx + c$ , where $m$ is the slope (gradient) and $c$ is the y-intercept. Plotting these graphs involves selecting values of $x$ , calculating corresponding $y$ values, and graphing the points. Interpretation includes identifying gradients, intercepts, and solving problems graphically.

Core Lesson Content

Gradient and Intercept

Gradient (m) = rise/run = $(y_2 - y_1) / (x_2 - x_1)$
Y-intercept (c) is the value of $y$ when $x = 0$

Plotting Steps

Choose 3-5 values of $x$
Substitute into the equation to get corresponding $y$
Plot points on a Cartesian plane and draw a straight line

Interpreting Linear Graphs

Identify slope: positive, negative, zero
Find intercepts
Use to solve equations graphically

Worked Examples

Example 1: Plot the graph of

y = 2x + 1

for

x = -2, -1, 0, 1, 2

.
Solution: Table of values:

(-2, -3), (-1, -1), (0, 1), (1, 3), (2, 5)

Example 2: Find the gradient of the line passing through (1, 2) and (3, 6).
Solution:

m = (6 - 2)/(3 - 1) = 4/2 = 2

Example 3: Interpret the graph of

y = -3x + 2

: what does the negative slope mean?
Solution: The line slopes downward from left to right, indicating a decrease in

y

x

increases.

Example 4: Determine the y-intercept of

y = 5x - 7

.
Solution: Y-intercept =

-7

Example 5: A line passes through (0, 4) and has a slope of 3. Write the equation.
Solution:

y = 3x + 4

Example 6: Plot

y = -x

for

x = -2, -1, 0, 1, 2

.
Solution: Table:

(-2, 2), (-1, 1), (0, 0), (1, -1), (2, -2)

Example 7: What is the x-intercept of

y = 2x - 4

?
Solution: Set

y = 0 \Rightarrow 0 = 2x - 4 \Rightarrow x = 2

Example 8: Two lines intersect at point (2, 3). Line A is

y = x + 1

, Line B is

y = 3x - 3

. Verify intersection.
Solution: Substitute

x = 2

into both equations: A →

2 + 1 = 3

, B →

6 - 3 = 3

Example 9: A straight line has equation

y = 0.5x + 6

. What is its slope and intercept?
Solution: Slope = 0.5, Intercept = 6

Example 10: Draw

y = -2x + 4

and describe its behavior.
Solution: Line falls steeply with slope -2, crosses y-axis at 4

Exercises

[NECO] Plot the graph of $y = 3x - 2$ for $x = -2$ to $2$ [Past Question]
Find the slope of the line through (4, 3) and (6, 9)
[WAEC] What is the y-intercept of $y = -4x + 1$ ? [Past Question]
Plot $y = 0.5x - 3$ and find its x-intercept
Write the equation of a line with slope 2 that passes through (0, -1)
Determine the gradient of the line from (2, -2) to (5, 4)
[WAEC] Plot and interpret $y = -x + 5$ [Past Question]
Solve graphically: $y = x + 2$ and $y = 2x - 1$ ; find point of intersection
[NECO] Find the value of $x$ when $y = 0$ in the equation $y = -3x + 6$ [Past Question]
Plot $y = 4x$ and describe the steepness compared to $y = x$

Conclusion/Recap

Understanding and interpreting linear graphs is a key skill in algebra and real-world problem solving. Students should be confident in identifying gradients, intercepts, and using graphs to analyze and solve equations or model data visually.