The First Derivative and Shaping of a Function

In this lesson, we will look at how to start to draw graphs of our functions by using the derivative. The derivative of a function, gives us the value of the slope of the tangent of that function. This value also tells us the “direction” that the graph is going in at that point. For example, if we have a function f(x), and its derivative f’(x), and there is a point along that graph, say x = 1, if we input x = 1 into f’(x) and it gives us f’(1) = 2, then we know the gradient of the slope of the tangent at x = 1 is 2. Generally, we can also look at the positive or negative value of the gradient of the tangent. A positive value means the line is going up and a negative value means the line is going down. If it is zero, it means the line is flat.

Definition - Increasing and Decreasing Functions for Shapes

To highlight what may seem like common sense, we have the definition for increasing and decreasing functions:

Increasing Function: A function is increasing if as you move “from left to right” along the x-axis, the value of f(x) (y-axis) increases between two certain points. More simply, it is increasing if the line is “going up”.

Decreasing Function: A function is decreasing if as you move “from left to right” along the x-axis, the value of f(x) (y-axis) decreases between two points. More simply, it is decreasing if the line is “going down”.

We will get into more mathematical explanations later in this lesson.

Definition - Monotonic

Monotonic Function: A function is monotonic if it is entirely increasing or decreasing. The function is either one or the other for all of the function. If the function is monotonically increasing then there are no points where it is decreasing. If the function is monotonically decreasing then there are no points where it is decreasing.

We verbally described in the first paragraph of this lesson how derivatives describe the direction of the graph of a function. Now we will describe them mathematically.

If we have a function of x, that is f(x), and we have x values on the x-axis and f(x) values on the y-axis, and we are looking at the function between the intervals (a, b) then,

f(x) is increasing on the interval when f’(x) > 0

f(x) is decreasing on the interval when f’(x) < 0

f(x) is flat or constant on the interval when f’(x) = 0

The visual below may help make sense of it,

This is the graph of a function, f(x)

a graphed function positive, negative, and zero tangent gradients highlighted.

The lines in purple, light blue, and pink represent the slope of the tangent at that point.

At points a1, a2, and a3 we can see the slope of the tangent is positively increasing.

At points b1 and b2 we can see the slope of the tangent is negatively decreasing.

At points c1 and c2 we can see the slope of the tangent is zero/ constant/ flat.

The points marked along the x-axis have a corresponding line for the tangent at that point.

We can calculate the value of the slope of this tangent line by inputting the x-value into the derivative of the function.

For example, for a1, we would do,

f’(a1) = a positive number

Or for b1, we would do

f’(b1) = a negative number

Or for c1, we would do,

f’(c1) = 0

For the points where the graph is increasing, we have a marked tangent line, the slope of this tangent line will be a positive value.

A Reminder on Tangents and Gradients

We will be working a lot with derivatives and what they mean throughout this Unit (and all of mathematics!). So it is important we have a very firm understanding of what may often be taken for granted. It is easy to forget what some of these phrases mean, as we learn them once, and then never go over them again in great detail. Lets recall what Tangents and Gradients mean. To understand what the gradient of a slope of a tangent is, lets first go over what a gradient is.

The gradient is the steepness or slope of a line, but it can be thought of numerically as how much the value moves in the y-axis over how much the value moves in the x-axis,

Gradient = 
y-axis change
x-axis change
a straight line graphed on an x-y grid showing a positive gradient of 1

As we can see when the y-value goes up by 1, the x-value goes across by 1.

This means we have a gradient of 1/1 = 1


a straight line graphed on an x-y grid showing a negative gradient of -2

Here we have the line where the y-value goes down by -2 and the x-value goes across by 1.

This means the gradient would be -2/1 = -2

The Tangent to a point on a line is the straight line that “just touches” that point along the line. Tangents are useful, because when we look at the gradient of that tangent at that point, it gives us the rate of change at that point. Rather than looking at a graph and drawing its tangent and then calculating its slope, we can more easily and accurately use the derivative of that graph to determine the slope of the tangent at a given point. This is largely what derivatives are all about.

a function graphed with tangent points illustrated

Tangents or Tangent Lines are the straight lines for a specific point along a graph. They are the straight line that “just touches” that point. They give the exact gradient at that exact time for that exact point.

As an example, you can imagine a graphed curved line as a path, and you are following that path on a bicycle. As you are cycling along a bend in the path by turning your handlebars, then you come to a point, and at that exact point you straighten your handlebars and go off on a straight line, going off course off the path. Think of that new straight path as the tangent to the point where you stopped following the path and straightened your handlebars.

When we differentiate a function, it gives us the equation that will tell us the gradient/slope of the tangent to a point x from the original function. To go back to our path example, if the path is a function of x, f(x), then the derivative gives us the equation that tells us the gradient of the line that we take if we were to straighten our handlebars at a point x.

Shaping a Graph Using the Derivative

We will learn how to “shape” a graph by using the derivative. This can be done by taking the derivative of a function and then finding the points where we have possible Local Extremes or Critical Numbers i.e. where f’(x) = 0 i.e. where the gradient is zero i.e. flat.

Also, recall that f’(x) = + means an upward direction of a gradient and f’(x) = - means a downward direction of a gradient.

Example - Is a point positive, negative or zero gradient and sketch the direction

For the following functions, determine the "direction" of each of the tangent gradients at the given x-values. (i.e. upwards, downwards, flat. Hint: Find f '(x), then input the x-value.)

a) f(x) = 2x3 - 9x2 + 12x + 5

Find the "direction" for the points:

  1. x = 1
  2. x = -1
  3. x = 3

Reminder: When asked for the "direction", we are looking for the slope of the gradient at the given point. Is it positive (upwards), negative (downwards), or zero (flat)?

So, first we find f '(x)

f(x) = 2x3 - 9x2 + 12x + 5

f '(x) = (3)2x2 - (2)9x + 12 + 0

f '(x) = 6x2 - 18x + 12

Now we have our f' (x), we can input our x-values and discover the direction of the gradient.

i.) x = 1 (input this into f' (x))

f '(x) = 6x2 - 18x + 12

f '(1) = 6(1)2 - 18(1) + 12

= 6 - 18 + 12

= 0

→ at x = 1, the curve will be "flat".


ii.) x = -1

f '(-1) = 6(-1)2 - 18(-1) + 12

= 6 + 18 + 12

= 36

→ This tells us that the gradient is +36, but all we need to know is that it is positive which tells us it is an "upward" gradient.


iii.) x = 3

f '(3) = 6(3)2 - 18(3) + 12

= 54 - 54 + 12

= 12

→ Again this tells us that the gradient is +12. As it is positive, we know it is an "upward" gradient.

b) f(x) = x3 + 3x2 - 24x + 3

Find the "direction" for the points:

  1. x = 3
  2. x = 2
  3. x = 1

Again, finding the "direction" means finding whether the slope of the gradient at the given point is positive, negative, or zero. To do this we first find the derivative.

f(x) = x3 + 3x2 - 24x + 3

f '(x) = 3x2 + 6x - 24

Now we can we can input our x-values and determine the gradient.

i.) x = 3 (input this into f' (x))

f '(x) = 3x2 + 6x - 24

f '(3) = 3(3)2 + 6(3) - 24

= 27 + 18 - 24

= 21

→ we have a positive gradient which means an "upward" direction.


ii.) x = 2 (input this into f' (x))

f '(2) = 3(2)2 + 6(2) - 24

= 12 + 12 - 24

= 0

→ we have a zero gradient which means an "flat" direction.


iii.) x = 1 (input this into f' (x))

f '(1) = 3(1)2 + 6(1) - 24

= 3 + 6 - 24

= -15

→ we have a negative gradient which means an "downward" direction.


In the example above, we worked on identifying how the gradient tells us the direction of the curve. By using these skills, we can start to roughly graph functions that are otherwise too complicated to plot out.

Gradients on a Graphed Function

See here an example of a graphed function, f(x), with noted positive / negative / zero values for the gradient. We can see how these gradient values correlate to the “direction” of the graphed function.

a graphed function positive, negative, and zero tangent gradients highlighted and correlated with the related derivatives at these points.

Note, at the bottom of the visual, the “Rough drawing with straight lines”, we will soon use this to plot out our graphs.