# 3.9.1 Finding Gradients of Non-Linear Graphs

#### What is a non-linear graph?

• A linear graph is a straight-line graph
• These are easily identified as their equations can always be written in the form
y = mx + c, where m is the gradient and c is the y-axis intercept

• All other graphs are non-linear – ie. curves
• The equations of non-linear graphs take various forms
• Here are a few you could plot quickly using graphing software
• y = x2 – 4x + 3 (a quadratic graph – called a parabola)
• y = sin x (a trigonometric graph)
• y = x3 + 2x2 – 4 (a cubic graph)
• y = 1 / x (a reciprocal graph)

• Another way of thinking about gradient is how y changes as x changes
• On a graph this means how steep the graph is at a certain point on it
• ie. how is y changing at a particular value of x
• For a linear graph the gradient is constant – the value of x is irrelevant
• For a non-linear graph, the gradient is dependent on the x-coordinate

#### How do I find the gradient of a non-linear graph?

• Using a copy of the graph it will only be possible to find an estimate of a gradient
• Differentiation allows gradients to be found exactly for certain graphs
• First, a tangent to the curve must be drawn
• A tangent to a curve is a straight line that touches it at one point only
• The gradient of a curve, at point (x , y) is equal to the gradient of the tangent at point (x , y)

• STEP 1 Draw a tangent to the curve at the required x-coordinate
• STEP 2 Turn the tangent into a rightangled triangle
• STEP 3 Measure/Read off (some estimating usually involved here) the rise and the run
• STEP 4 The gradient is given by rise ÷ run
(Alternatively this is “Change in y” ÷ “Change in x”)

#### Exam Tip

A sharp pencil helps – but not too sharp – pencil markings made with very sharp pencils are difficult for examiners to see once papers have been scanned into a computer.

Remember your answer is an estimate so can vary a fair amount from someone else’s attempt.