When we talk about linear functions, one important part is the gradient, or slope.

Knowing how to show this gradient on a graph can really help us understand functions better in Year 10 math.

The gradient tells us how steep a line is and whether it goes up or down. Let's take a look at how to show the gradient of a linear function on a graph, okay?

What is Gradient?

The gradient of a linear function can be found from its equation. This is usually written like this:

[ y = mx + c ]

Here:

• m is the gradient (slope).
• c is the y-intercept (the point where the line crosses the y-axis).

Graphical Representation

1. Plotting the Function:

To see the gradient, we can first plot the linear function on a graph.

For example, let’s look at the function:

[ y = 2x + 1 ]

We can find a few important points by putting values into x:

• When ( x = 0 ), ( y = 1 ) (that’s the point (0,1)).
• When ( x = 1 ), ( y = 3 ) (that’s the point (1,3)).
• When ( x = -1 ), ( y = -1 ) (that’s the point (-1,-1)).

Now, plot these points on a graph and draw a straight line through them.

2. Understanding the Gradient:

The gradient can also be thought of as "rise over run." This means:

• Rise: the vertical change (how much the line goes up or down).
• Run: the horizontal change (how much the line goes left or right).

3. Calculating the Gradient:

From our points (0,1) to (1,3):

• Rise = ( 3 - 1 = 2 )
• Run = ( 1 - 0 = 1 )

So, the gradient ( m ) is calculated like this:

[ m = \frac{\text{Rise}}{\text{Run}} = \frac{2}{1} = 2 ]

Visualizing the Gradient

You can actually see the gradient on the graph:

• Draw a right triangle where:
  • One side goes up from the point (0,1) to (1,3).
  • The other side goes across from (0,1) to (1,1).

This triangle shows the rise and run clearly, which helps you understand the gradient better.

Summary

In summary, showing the gradient of a linear function on a graph means plotting points, understanding the rise and run, and seeing these changes on the graph.

This makes it easier to understand the slope and how the function works.

So, the next time you graph a linear function, remember to pay attention to how steep the line is and what that means for the relationship between x and y!