Gradient and <em>y</em>-intercept Method

Gradient and \(y\)-intercept method

We can draw a straight line graph of the form \(y=mx+c\) using the gradient (\(m\)) and the \(y\)-intercept (\(c\)).

We calculate the \(y\)-intercept by letting \(x = 0\). This gives us one point \((0;c)\) for drawing the graph and we use the gradient to calculate the second point.

The gradient of a line is the measure of steepness. Steepness is determined by the ratio of vertical change to horizontal change:

\[m = \cfrac{\text{change in }y}{\text{change in }x} = \cfrac{\text{vertical change}}{\text{horizontal change}}\]

For example, \(y = \cfrac{3}{2}x - 1\), therefore \(m > 0\) and the graph slopes upwards.

\[m = \cfrac{\text{change in }y}{\text{change in }x} = \cfrac{3\uparrow}{2\to }=\cfrac{-3\downarrow}{-2arrow}\]

Example

Question

Sketch the graph of \(p(x) = \cfrac{1}{2}x - 3\) using the gradient-intercept method.

Use the intercept

\(c=-3\), which gives the point \((0;-3)\).

Use the gradient

\[m=\cfrac{\text{change in }y}{\text{change in }x}=\cfrac{1\uparrow}{2\to }=\cfrac{-1\downarrow}{-2arrow}\]

Start at \((0;-3)\). Move \(\text{1}\) unit up and \(\text{2}\) units to the right. This gives the second point \((2;-2)\).

Or start at \((0;-3)\), move \(\text{1}\) unit down and \(\text{2}\) units to the left. This gives the second point \((-2;-4)\).

Plot the points and draw the graph

Always write the function in the form \(y = mx + c\) and take note of \(m\). After plotting the graph, make sure that the graph increases if \(m > 0\) and that the graph decreases if \(m < 0\).