Introduction to Curve Sketching

Expert reviewed • 22 November 2024 • 4 minute read

HSC Maths Advanced Syllabus

use the first derivative to investigate the shape of the graph of a function
- deduce from the sign of the first derivative whether a function is increasing, decreasing or stationary at a given point or in a given interval

Note:

Video coming soon!

Curve Sketching Notes

Before beginning to use calculus to assist with the sketching of graphs, we need to first understand the nature of a curve.

Any point on a graph can be defined as increasing, decreasing or stationary. To determine the nature of the graph at a certain point, we must find the derivative of the graph (at that point).

For example, Let $f(x)$ be a function which passes through point $p$ .

If $f'(p) >0$ , the function $f(x)$ is increasing, when $x=p$
If $f'(p) <0$ , the function $f(x)$ is decreasing, when $x=p$
If $f'(p) =0$ , the function $f(x)$ is stationary, when $x=p$

Practice Question 1

Determine if the function $y=\frac{3}{x+5}$ has any stationary points.

We know that a function is stationary when it’s derivative at a point is equal to 0. Thus, to determine if this function has any stationary points, we must derive it and see at what values of $x$ occur, when $f'(x)=0$ :

y=3(x+5)^{-1}\\\frac{dy}{dx}=-1\times1\times 3(x+5)^{-2}\\=-\frac{3}{(x+5)^2}

Now that we have found the derivative, we must determine if any points are equal to 0.

0=-\frac{3}{(x+5)^2}\\

Looking at this equation, we can see that no value of $x$ is equal to 0, and as such this graph will have no stationary points.

Up Next

Stationary and Turning Points

Return to Module 2: Curve Sketching Using the Derivative