Gradient and Intercept Revision Notes for HSC SSCE Mathematics Standard

Gradient and Intercept

What is gradient?

The gradient tells us how steep a line is. You might also hear it called the slope of the line. We represent gradient with the letter m.

The gradient shows us how much a line rises vertically for every unit it moves horizontally. A larger gradient value means a steeper line.

To calculate the gradient, we use this formula:

$m = \frac{\text{Vertical rise}}{\text{Horizontal run}}$

The vertical rise is how far the line goes up or down, and the horizontal run is how far the line moves across.

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Think of gradient like walking up a hill - the steeper the hill, the larger the gradient value. A flat path has a gradient of zero, while a steep mountain path has a large gradient.

Positive and negative gradients

Gradients can be positive or negative:

Positive gradients: Lines that slope upward as they move to the right. These represent increasing relationships.
Negative gradients: Lines that slope downward as they move to the right. These represent decreasing relationships.

chatImportant

Remember the direction:

Positive gradient = line goes up to the right (increasing)
Negative gradient = line goes down to the right (decreasing)

The sign of the gradient tells you the direction of the line!

Worked example: Finding the gradient of a line

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Worked Example: Finding the Gradient of a Line

Let's find the gradient of a line passing through the points $(1, 1)$ and $(3, 4)$ .

Step 1: Draw a coordinate plane with the $x$ -axis horizontal and the $y$ -axis vertical.

Step 2: Plot the two points $(1, 1)$ and $(3, 4)$ .

Step 3: Draw a straight line connecting these points.

Step 4: Create a right-angled triangle by drawing a vertical line and a horizontal line between the two points.

Step 5: Notice that this line slopes upward to the right, so it has a positive gradient.

Step 6: Calculate the vertical rise: $4 - 1 = 3$

Step 7: Calculate the horizontal run: $3 - 1 = 2$

Step 8: Substitute these values into the formula:

$m = \frac{\text{Vertical rise}}{\text{Horizontal run}} = \frac{3}{2} = +\frac{3}{2}$

Therefore, the gradient is $+\frac{3}{2}$ .

What is intercept?

An intercept is the point where a line crosses an axis.

The y-intercept is where the line crosses the vertical axis (the $y$ -axis). We denote this with the letter b.
The x-intercept is where the line crosses the horizontal axis (the $x$ -axis). We denote this with the letter a.

The y-intercept is particularly useful because it tells us the starting value of $y$ when $x = 0$ .

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The y-intercept is often easier to identify than the x-intercept because it's simply the constant term in the equation $y = mx + b$ . This makes it a convenient starting point when sketching graphs.

Worked example: Finding gradient and vertical intercept

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Worked Example: Finding Gradient and Vertical Intercept

Let's find the gradient and vertical intercept for the line $y = -2x + 1$ .

Step 1: Create a table of values for $x$ and $y$ .

Step 2: Choose convenient values for $x$ (let's use $x = -1, 0, 1$ ) and calculate the corresponding $y$ values using the equation $y = -2x + 1$ :

$x$	$-1$	$0$	$1$
$y$	$3$	$1$	$-1$

Step 3: Draw a coordinate plane and plot the points $(-1, 3)$ , $(0, 1)$ , and $(1, -1)$ .

Step 4: Draw a line through these points.

Step 5: Construct a right-angled triangle using vertical and horizontal lines.

Step 6: This line slopes downward to the right, so it has a negative gradient.

Step 7: Calculate the vertical rise: $3 - 1 = 2$

Step 8: Calculate the horizontal run: $-1 - 0 = -1$

Step 9: Substitute these values into the gradient formula:

$m = \frac{\text{Vertical rise}}{\text{Horizontal run}} = \frac{2}{-1} = -2$

Step 10: The line crosses the vertical axis at the point $(0, 1)$ .

Therefore, the gradient is $-2$ and the vertical intercept is $1$ .

The gradient-intercept formula

When we write a straight line's equation in the form $y = mx + b$ , we call this the gradient-intercept formula. This format is extremely useful because we can immediately identify:

m is the gradient (the coefficient of $x$ )
b is the y-intercept (the constant term)

The gradient-intercept formula is:

$y = mx + b$

where:

$m$ = slope or gradient of the line (vertical rise divided by horizontal run)
$b$ = y-intercept where the line crosses the y-axis

In this formula, $x$ is the independent variable and $y$ is the dependent variable. This means that the value of $y$ depends on the value of $x$ .

chatImportant

The Power of $y = mx + b$

This is one of the most important formulas in mathematics! Once you can recognize this form, you can instantly identify:

How steep the line is (the m value)
Where it crosses the y-axis (the b value)

If an equation isn't in this form, your first step should be to rearrange it to $y = mx + b$ .

Worked example: Identifying gradient and y-intercept from equations

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Worked Example: Identifying Gradient and y-Intercept from Equations

Let's identify the gradient and y-intercept for each of these equations:

a) $y = -2 + 5x$

First, rearrange the equation to gradient-intercept form:

$y = 5x - 2$

The gradient is the coefficient of $x$ : $m = 5$

The y-intercept is the constant term: $b = -2$

b) $y = 8 - x$

Rearrange to gradient-intercept form:

$y = -1x + 8$

The gradient is: $m = -1$

The y-intercept is: $b = 8$

c) $y = 6x$

Write in gradient-intercept form:

$y = 6x + 0$

The gradient is: $m = 6$

The y-intercept is: $b = 0$

d) $y - 3x = 4$

Rearrange to gradient-intercept form:

$y = 3x + 4$

The gradient is: $m = 3$

The y-intercept is: $b = 4$

Sketching graphs using gradient-intercept form

To sketch a straight-line graph, we need at least two points. When an equation is in gradient-intercept form, we already know one point: the y-intercept. We can find a second point using the gradient.

Here's the process:

Identify the gradient ( $m$ ) and y-intercept ( $b$ )
Mark the y-intercept on the y-axis at the point $(0, b)$
Use the gradient to find a second point (rise over run from the y-intercept)
Draw a straight line through the two points

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Tip for Success: Always start by marking the y-intercept - it's your anchor point. Then, think of the gradient as instructions: "From the y-intercept, rise this many units, run this many units."

Worked example: Sketching with positive gradient

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Worked Example: Sketching a Graph with Positive Gradient

Let's sketch the graph of $y = 3x + 1$ .

Step 1: The equation is already in gradient-intercept form: $y = 3x + 1$

Step 2: The gradient is $m = 3$ (the coefficient of $x$ )

Step 3: The y-intercept is $b = 1$ (the constant term)

Step 4: Mark the y-intercept at the point $(0, 1)$ on the y-axis.

Step 5: The gradient of $3$ can be written as $\frac{3}{1}$ , which means a vertical rise of 3 and a horizontal run of 1.

Step 6: Starting from the y-intercept $(0, 1)$ , move $1$ unit horizontally to the right, then move $3$ units vertically upward.

Step 7: This gives us the point $(1, 4)$ .

Step 8: Draw a straight line through the points $(0, 1)$ and $(1, 4)$ .

Worked example: Sketching with negative gradient

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Worked Example: Sketching a Graph with Negative Gradient

Let's sketch the graph of $y = -2x + 1$ .

Step 1: The equation is already in gradient-intercept form: $y = -2x + 1$

Step 2: The gradient is $m = -2$ (the coefficient of $x$ )

Step 3: The y-intercept is $b = 1$ (the constant term)

Step 4: Mark the y-intercept at the point $(0, 1)$ on the y-axis.

Step 5: The gradient of $-2$ can be written as $\frac{-2}{1}$ or $\frac{2}{-1}$ . This means a vertical rise of $2$ and a horizontal run of $1$ to the left (or alternatively, down $2$ and right $1$ ).

Step 6: Starting from the y-intercept $(0, 1)$ , move $1$ unit horizontally to the left, then move $2$ units vertically upward.

Step 7: This gives us the point $(-1, 3)$ .

Step 8: Draw a straight line through the points $(0, 1)$ and $(-1, 3)$ .

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Key Points to Remember:

Gradient measures the steepness of a line and is calculated as $m = \frac{\text{Vertical rise}}{\text{Horizontal run}}$
Positive gradients slope upward to the right; negative gradients slope downward to the right
Intercepts are where a line crosses an axis: the y-intercept ( $b$ ) is on the vertical axis, the x-intercept ( $a$ ) is on the horizontal axis
The gradient-intercept formula is $y = mx + b$ , where $m$ is the gradient and $b$ is the y-intercept
To sketch a line from its equation, mark the y-intercept first, then use the gradient (rise over run) to find a second point

Gradient and Intercept (HSC SSCE Mathematics Standard): Revision Notes