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Drawing Linear Graphs

CSEC Mathematics: Introduction to Graphs

Essential Understanding: Linear graphs are straight lines that represent relationships between two quantities. By understanding how to draw and interpret these graphs, you can visualize mathematical relationships, solve equations graphically, and analyze real-world situations involving proportional and linear relationships.

Slope-Intercept: y = mx + c
Y-Intercept: Where line crosses y-axis
Gradient: Steepness of line

Understanding Linear Functions

A linear function is a function whose graph forms a straight line. All linear functions can be written in the form:

Slope-Intercept Form

\[ y = mx + c \]

Where:

  • m = gradient (slope) of the line
  • c = y-intercept (where the line crosses the y-axis)
  • (x, y) = coordinates of any point on the line

Gradient (m)

The steepness of the line, calculated as rise over run.

m = rise/run

Think: How much y changes for each unit of x

Y-Intercept (c)

The point where the line crosses the y-axis.

At this point, x = 0

Think: The starting value when x = 0

m

Types of Gradient

Positive: Line rises (m > 0)

Negative: Line falls (m < 0)

Zero: Horizontal line (m = 0)

Interactive Graph Drawer

Draw Your Own Linear Graphs

1
2
y = 1x + 2
(0, 2)
(-2, 0)
Positive

Methods for Drawing Linear Graphs

There are several methods to draw a linear graph accurately. The most common methods are:

Method 1: Using Two Points

The simplest method is to find two points that lie on the line and draw a straight line through them.

1
Find the y-intercept: Set x = 0 and solve for y. This gives you point (0, c).
2
Find another point: Choose a convenient value for x (such as 1, 2, or 5) and calculate the corresponding y-value.
3
Plot both points: Mark the two points on your coordinate grid.
4
Draw the line: Draw a straight line through both points, extending it across the grid.

Method 2: Using Intercepts

Using the x-intercept and y-intercept is particularly useful when the intercepts are easy to find.

Finding Intercepts

Y-Intercept:
Set \(x = 0\)
Point: \((0, c)\)
X-Intercept:
Set \(y = 0\)
Point: \(\left(-\frac{c}{m}, 0\right)\)
1

Worked Example: Drawing Using Intercepts

Problem: Draw the graph of \(y = 2x + 3\)

1
Find y-intercept: Set x = 0
\[ y = 2(0) + 3 = 3 \]
Point A: (0, 3)
2
Find x-intercept: Set y = 0
\[ 0 = 2x + 3 \]
\[ 2x = -3 \]
\[ x = -1.5 \]
Point B: (-1.5, 0)
3
Plot and connect: Plot points (0, 3) and (-1.5, 0), then draw a straight line through them.

Special Cases of Linear Graphs

Horizontal Lines

Equation: y = c

Gradient: m = 0

Example: y = 3

Y-value is constant, x can be anything

Vertical Lines

Equation: x = k

Gradient: Undefined

Example: x = 5

X-value is constant, y can be anything

Lines Through Origin

Equation: y = mx

Y-intercept: c = 0

Example: y = -x

Line passes through (0, 0)

Direct Proportion

Equation: y = kx

Relationship: y ∝ x

Example: y = 2x

When x doubles, y doubles

Remembering Linear Graph Forms

  • Horizontal: "Y stays the same" → y = constant
  • Vertical: "X stays the same" → x = constant
  • Through origin: "No extra" → y = mx (no +c)
  • General: "Y equals mx plus c" → y = mx + c

Practice: Equation Builder

Click on different values of m and c to see how they affect the graph:

Explore Different Equations

y = 1x + 0
Rising line through origin

Solving Problems with Linear Graphs

Finding Coordinates on a Line

Once you have drawn a linear graph, you can use it to find missing coordinates or solve equations.

2

Worked Example: Finding Missing Values

Problem: The cost of renting a car is given by C = 50 + 10d, where C is the total cost in dollars and d is the number of days.

(a) Draw the graph of this relationship

(b) Find the cost for 5 days

(c) If the cost is $150, how many days was the car rented?

a
Draw the graph:
Y-intercept: When d = 0, C = 50 → Point (0, 50)
Another point: When d = 5, C = 50 + 10(5) = 100 → Point (5, 100)
Draw a line through these points.
b
Find cost for 5 days:
From the graph, when d = 5, C = $100
(Or calculate: C = 50 + 10(5) = 100)
c
Find days for $150:
From the graph, when C = 150, d = 10 days
(Or calculate: 150 = 50 + 10d, 10d = 100, d = 10)

Comparing Two Linear Relationships

Linear graphs are excellent for comparing two different situations or relationships.

3

Worked Example: Comparing Two Companies

Problem: Company A charges $5 per hour plus a fixed fee of $20. Company B charges $8 per hour with no fixed fee.

(a) Write equations for both companies

(b) Draw both graphs on the same axes

(c) After how many hours do they cost the same?

a
Write equations:
Company A: \( C_A = 5h + 20 \) (where h = hours)
Company B: \( C_B = 8h \)
b
Draw graphs:
Company A: y-intercept at (0, 20), gradient 5
Company B: y-intercept at (0, 0), gradient 8
Both lines will intersect.
c
Find when costs are equal:
Set equations equal: \( 5h + 20 = 8h \)
\( 20 = 3h \)
\( h = \frac{20}{3} = 6\frac{2}{3} \) hours
From graph: Intersection point at approximately (6.67, 53.33)

Past Paper Style Questions

CSEC-Style Question 1

(a) Complete the table of values for the equation y = 3x - 2:

x -1 0 1 2 3
y ? ? ? ? ?

(b) Using a scale of 2 cm to 1 unit on both axes, draw the graph of y = 3x - 2 for x = -1 to x = 3.

(c) State the y-intercept of the line.

(d) Find the gradient of the line.

Solutions:

(a) y values: -5, -2, 1, 4, 7

(b) Plot the points (-1, -5), (0, -2), (1, 1), (2, 4), (3, 7) and draw a straight line through them.

(c) Y-intercept = -2 (point where x = 0)

(d) Gradient = 3 (coefficient of x)

CSEC-Style Question 2

(a) Two points P(2, 5) and Q(6, 17) lie on a straight line. Find the equation of the line in the form y = mx + c.

(b) Find the y-intercept of this line.

(c) Find the value of y when x = 0.

(d) If y = 29, find the value of x.

Solutions:

(a) Gradient m = (17 - 5)/(6 - 2) = 12/4 = 3
Using point P(2, 5): 5 = 3(2) + c, 5 = 6 + c, c = -1
Equation: y = 3x - 1

(b) Y-intercept = -1

(c) When x = 0, y = -1

(d) 29 = 3x - 1, 3x = 30, x = 10

CSEC-Style Question 3

The temperature of a liquid cooling from boiling point is given by T = 90 - 5t, where T is temperature in °C and t is time in minutes.

(a) State the initial temperature when t = 0.

(b) After how many minutes will the temperature reach 40°C?

(c) Draw the graph of T against t for t = 0 to t = 15 minutes.

(d) What does the gradient represent in this context?

Solutions:

(a) Initial temperature = 90°C (when t = 0)

(b) 40 = 90 - 5t, 5t = 50, t = 10 minutes

(c) Points: (0, 90), (5, 65), (10, 40), (15, 15). Draw line through these points.

(d) Gradient = -5 represents the rate of cooling (5°C per minute)

CSEC Practice Arena

Test Your Understanding

1
What is the y-intercept of the line y = 4x - 3?
4
x = 4
(0, -3)
(3, 0)
Solution: The y-intercept is found by setting x = 0: y = 4(0) - 3 = -3, so the point is (0, -3).
2
Which of the following equations represents a horizontal line?
y = 2x
y = 5
x = 5
y = x + 1
Solution: y = 5 is a horizontal line because the y-value is constant (gradient = 0).
3
Calculate the gradient of the line passing through (1, 3) and (4, 9).
1
2
2
3
Solution: m = (9 - 3)/(4 - 1) = 6/3 = 2
4
What is the x-intercept of the line y = 2x + 4?
(0, 4)
2
(-2, 0)
(4, 0)
Solution: Set y = 0: 0 = 2x + 4, 2x = -4, x = -2, so the x-intercept is (-2, 0).
5
Which line has the steepest gradient?
y = 2x
y = -3x
y = 5x
y = x
Solution: y = 5x has the steepest gradient because |5| > |2|, |-3|, and |1|. The absolute value determines steepness.
Target

CSEC Examination Tips

  • Label your axes: Always label x and y axes with the correct quantities and units
  • Use appropriate scale: Choose a scale that allows your graph to fill most of the grid
  • Plot accurately: Use the grid lines to plot points precisely
  • Draw straight lines: Use a ruler to draw the straight line through your points
  • Find intercepts first: The intercepts are often the easiest points to calculate
  • Check your points: Verify that plotted points satisfy the equation
  • Extended line: Extend your line across the entire grid as required
  • Read graphs carefully: When solving problems, read values from the graph accurately

Summary: Key Points for Drawing Linear Graphs

Equation Forms

  • General: y = mx + c
  • Horizontal: y = constant
  • Vertical: x = constant
  • Through origin: y = mx

Finding Points

  • Y-intercept: Set x = 0
  • X-intercept: Set y = 0
  • Other points: Choose x values and calculate y

Gradient Types

  • Positive: Rises left to right
  • Negative: Falls left to right
  • Zero: Horizontal
  • Undefined: Vertical

Drawing Steps

  • Calculate two or more points
  • Plot points accurately
  • Draw straight line through points
  • Label the line with its equation
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