For any smoothly varying function $f(x)$f(x) we can find the gradient at any point on the curve. If we know the equation of $f(x)$f(x), we can do this by differentiating to find $f'(x)$f′(x), the gradient function. Alternatively, we can draw a tangent at any point on the curve and estimate a selection of individual gradient values. These values become the $y$y-coordinates on the gradient function. Once we are satisfied that we have enough values, we can join these together to create a sketch, despite the fact we don't know the equation of $f(x)$f(x).
It might seem like a difficult task ahead - but we have learnt so much about the gradient function in this chapter!
As a first step, you should mark positive, negative and zero values of gradient on the given function. Imagine that you're drawing tangents all along the graph. Gradient is zero at the stationary points on the curve as this is where the gradient of the tangent at that point is equal to zero.
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When the gradient function of a curve is sketched, the stationary points are represented as $x$x-intercepts. They will link the values on the curve that are above the $x$x-axis - the positive gradient values - with the negative gradient values which will lie below.
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The steeper the gradient of the tangent, the further away from the $x$x-axis that the gradient function will be at this point.
Even without an equation for the function, you can draw on your understanding of polynomials. When a polynomial of degree $n$n is graphed on a coordinate plane, the gradient function of the polynomial is one degree less than the original polynomial.
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So if you the shape of the function appears to be a cubic, your derivative sketch should be a quadratic. And of course, any sections of the curve which appear to be linear will have a constant gradient, so your associated derivative sketch should be a horizontal line.
Worked example
Example 1
The diagram below shows a curve $y=f(x)$y=f(x). Create a sketch of $y=f'(x)$y=f′(x).
On the diagram above we have labelled areas of positive, negative and zero gradient. This allows us to sketch the gradient, or derivative function shown below.
The positive gradient regions correspond to the $x$x values where the gradient function is above the $x$x axis. The negative gradient regions correspond to the $x$x values where the gradient function is below the $x$x axis. Stationary points on the original curve become $x$x intercepts on the gradient function sketch.
As $x$x increases, we can see the original function graph smooths off to a section of the curve that looks like a straight line. The gradient of this piece of the graph is approaching zero. This is indicated in the gradient function sketch below, where the curve approaches the $x$x axis.
Families of curves with the same derivative function
Certain curves will have the same derivative function because their equations have the same derivative. Consider the functions $y=x^2$y=x2, $y=x^2+4$y=x2+4, and $y=x^2-8$y=x2−8 shown below:
These functions are vertical translations of each other. The functions will all have the same derivative of $y'=2x$y′=2x because the constant term always has a derivative of zero. Similarly the curves of all of these functions will have the same derivative function which is the line $y=2x$y=2x as shown below:
So functions that only differ by a constant term, and curves that are vertically translated, will have the same derivative function.
Practice questions
Question 1
Consider the function $y=4x-3$y=4x−3.
Find the gradient function of $y=4x-3$y=4x−3.
Hence graph the gradient function.
Loading Graph...
Question 2
Consider the function $y=-\left(x+7\right)^2+5$y=−(x+7)2+5 graphed below.
State the $x$x-coordinate of the $x$x-intercept of the gradient function.
For $x<-7$x<−7, are the values of the gradient function above or below the $x$x-axis?
Above
ABelow
BFor $x>-7$x>−7, are the values of the gradient function above or below the $x$x-axis?
Above
ABelow
B
Question 3
Consider the functions $f\left(x\right)=x^5$f(x)=x5 and $g\left(x\right)=x^4$g(x)=x4.
Which of the following shows the graph of $f\left(x\right)$f(x) and its derivative?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich of the following shows the graph of $g\left(x\right)$g(x) and its derivative?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich of the following statements are true? Select all that apply.
The graph of the derivative can be found by translating and/or stretching the original function.
ANear the origin, the derivative has a greater value than the function.
BA function and its derivative have the same sign for all values of $x$x.
CIf the degree of a function is even, then the degree of its derivative is odd and vice versa.
D
Question 4
Draw the gradient function of the following graph.