Finding the Equation of a Straight-Line Graph (VCE SSCE General Mathematics): Revision Notes

Finding the Equation of a Straight-Line Graph

Introduction

When working with linear graphs, you'll often need to determine the equation of a line from its graph. This is a crucial skill that allows you to move from a visual representation to a mathematical formula. There are two main approaches you can use, depending on what information is visible on the graph.

Using the intercept and slope to find the equation

Understanding the intercept-slope form

Every straight line can be written in the form:

$y = a + bx$

where:

• $a$ is the y-intercept (where the line crosses the y-axis when $x = 0$)
• $b$ is the slope or gradient (how steep the line is)

Steps for the intercept-slope method

This method works well when your graph clearly shows where the line crosses the y-axis. Follow these steps:

Step 1: Identify where the line crosses the y-axis. This gives you the value of $a$.

Step 2: Calculate the slope using two clearly marked points on the line. Remember that:

$\text{slope} = \frac{\text{rise}}{\text{run}}$

This gives you the value of $b$.

Step 3: Insert both values into the equation $y = a + bx$ to get your final answer.

Worked example: intercept-slope method

Practice problem

Here's a similar problem for you to try:

Hints:

• First, identify where the line crosses the y-axis
• Choose two convenient points on the line to calculate the slope
• Remember to use the formula: slope = rise ÷ run
• Finally, substitute your values for $a$ and $b$ into $y = a + bx$

Using two points on a graph to find the equation

When to use the two-point method

Sometimes a graph doesn't show the y-intercept, or the scale doesn't include $x = 0$. In these situations, you need to use a different approach called the two-point method. This technique is more involved but works in all circumstances.

Steps for the two-point method

The general equation is still $y = a + bx$, but we need to find both $a$ and $b$ using only two points.

Step 1: Use the coordinates of two points on the line to calculate the slope $(b)$ using:

$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Step 2: Substitute the value of $b$ into the equation $y = a + bx$. Now you have only one unknown value, $a$.

Step 3: Take the coordinates of either of your two points and substitute them into the equation. Then solve for $a$.

Step 4: Write the final equation by substituting both $a$ and $b$ into $y = a + bx$.

Worked example: two-point method

Verification tip

Practice problem

Find the equation of the line passing through $(2, 4)$ and $(3, 10)$.

Hints:

• Begin with $y = a + bx$
• Calculate the slope using the two points
• Substitute the slope value into the equation
• Use either point to solve for $a$

Summary of key methods

Remember!