4.09 Gradients in context
Introduction
We have already learnt that a rate is a ratio between two measurements with different units.
When we graph these rates, the rate of change can be understood as the gradient, steepness or slope of a line. Further, we look at the equations in gradient-intercept form (that is, y=mx+b, where m is the gradient), the larger the absolute value of m, the steeper the slope of the line.
For example, a line with a slope of of 4 is steeper than a line with a slope of \dfrac23. Similarly, a line with a slope of -2 is steeper than a line with a slope of 1, even though one is positive and one is negative.
Gradient
The rate of change of a line is a measure of how steep it is. In mathematics we also call this the gradient.
The rate of change in a graph can be positive or negative.
The rate of change in a graph can be positive or negative. The lines below have positive rates of change. Notice how as the values on the x-axis increase, the values on the y-axis also increase.
These next graphs have negative rates of change. Unlike graphs with a positive slope, as the values on the x-axis increase, the values on the y-axis decrease.
The rate of change is a single value that describes:
if a line is increasing (has positive gradient)
if a line is decreasing (has negative gradient)
how far up or down the line moves (change in the y-value) with every unit step to the right (for every 1 unit increase in x)
Sometimes it is difficult to measure how far the line goes up or down (how much the y-value changes) in 1 horizontal unit. In this case we calculate the gradient by using a formula: \text{gradient}= \dfrac{\text{rise}}{\text{run}}
Where you take any two points on the line whose coordinates are known or can be easily found, and look for the rise and run between them.
Remember that an increasing line has a positive gradient, while a decreasing line has a negative gradient.
Gradient from coordinates
If you have a pair of coordinates, such as A (3, 6) and B (7, -2), we can find the gradient of the line between these points using the rule: m=\dfrac{y_2-y_1}{x_2-x_1}
Let's substitute our coordinates into this formula: \begin{aligned} m&=\dfrac{y_2-y_1}{x_2-x_1} & \text{Write the formula}\\ &=\dfrac{6-(2)}{3-7} & \text{Substitute the coordinates}\\ &=\dfrac{6+2}{3-7} & \text{Simplify the numerator}\\ &=\dfrac{8}{-4} & \text{Evaluate both parts} \\ &=-2 & \text{Simplify}\end{aligned}
So the rate of change between these coordinates is -2.
Examples
Example 1
After Mae starts running, her heartbeat increases at a constant rate.
Complete the table.
| \text{Number of minutes passed }(x) | 0 | 2 | 4 | 6 | 8 | 10 | 11 |
|---|---|---|---|---|---|---|---|
| \text{Heart rate }(y) | 49 | 55 | 61 | 67 | 73 | 79 |
By how much is her heartbeat increasing each minute?
Form an equation that describes the relationship between the number of minutes passed (x) and Mae’s heartbeat (y).
In the equation, y=3x+49, what does 3 represent?
Example 2
Petrol costs a certain amount per litre. The table shows the cost of various amounts of petrol in dollars:
| \text{Number of litres }(x) | 0 | 10 | 20 | 30 | 40 |
|---|---|---|---|---|---|
| \text{Cost of petrol }(y) | 0 | 16.40 | 32.80 | 49.20 | 65.60 |
Write an equation linking the number of litres of petrol pumped (x) and the cost of the petrol (y).
How much does petrol cost per litre?
How much would 47 litres of petrol cost at this unit price?
In the equation, y=1.64x, what does 1.64 represent?
We can find the gradient of the line between these points using the rule. m=\dfrac{y_2-y_1}{x_2-x_1}
Gradient of horizontal and vertical lines
Horizontal lines have no rise value. The \text{rise}=0. So:
\begin{aligned} \text{gradient}&=\dfrac{\text{rise}}{\text{run}} \\ &= \dfrac{0}{\text{run}} \\ &= 0 \end{aligned}
It doesn't matter what the run is, the gradient will be 0.
Vertical lines have no run value. The \text{run}=0. So:
\begin{aligned} \text{gradient}&=\dfrac{\text{rise}}{\text{run}} \\ &= \dfrac{\text{rise}}{\text{0}} \end{aligned}
It doesn't matter what the rise is, any division by 0 results in the value being undefined.
Examples
Example 3
What kind of slope does the following line have?
Description of gradient: \text{gradient}=\dfrac{\text{rise}}{\text{run}}
The gradient of a horizontal is always 0.
The gradient of a vertical is always undefined.