Calculating Gradients
Gradient is the steepness or slope of a line. We've already learnt how to calculate gradient but let's just refresh ourselves on a couple of key points to start.
$m=\frac{y_2-y_1}{x_2-x_1}$m=y2−y1x2−x1
Other handy points to remember:
- The gradient formula can be thought of generally as: $\text{Gradient }=\frac{\text{rise }}{\text{run }}$Gradient =rise run
- Gradient of Vertical Line is undefined
- Gradient of Horizontal Line $\text{= 0}$= 0
Even though we are used to finding the gradient of a line from two given pairs of coordinates, we may also need to reverse the process. Sometimes, we may be asked to find a pair of coordinates when we're given the gradient and a point.
Check out the examples to see the different ways we can work with the gradient formula.
Examples
Question 1
Consider the following ramp:
a) What is the gradient of this skateboard ramp if it measures $0.9$0.9 metres high and $1$1 metre across?
Think: What is the rise and run of this ramp?
Do:
| $\text{Gradient }$Gradient | $=$= | $\frac{\text{Rise }}{\text{Run }}$Rise Run |
| $=$= | $\frac{0.9}{1}$0.91 | |
| $=$= | $0.9$0.9 |
b) It can only be used as a 'beginner’s ramp' if for every $1$1 metre horizontal run, it has a rise of at most $0.4$0.4 metres. Can it be used as a 'beginner’s ramp'?
Think: What is the maximum gradient of a beginners' ramp? Is the ramp steeper than this?
Do: A beginner's ramp needs to have a gradient of $\frac{0.4}{1}$0.41 or $0.4$0.4. Since the ramp has a steeper gradient of $0.9$0.9, it cannot be used as a beginners' ramp.
Question 2
A line passes through the points $\left(11,c\right)$(11,c) and $\left(-20,16\right)$(−20,16) and has a gradient of $-\frac{4}{7}$−47.
Find the value of $c$c.