The document explores the concept of gradients and slopes concerning curves, outlining methods to find the gradient at specific points for various functions (y = x^2, y = x^3, y = x^4). It addresses the relationship between functions and their gradients, discusses increasing and decreasing functions, and identifies stationary points where the gradient equals zero. Additionally, it covers the identification of points of inflexion, ensuring a comprehensive understanding of gradient functions and their applications.

1. 1
Gradients and Slopes
Questions to think about during this lesson:
We know how to find the gradient of lines but how do we find the gradient of a curve?
Is there a relationship between a function and the gradient of the function at any particular point?
Why do we want to study gradient functions?
On your GDC or graphing software graph the following functions and complete the table by finding the
gradient at each of the given points. Sketch the gradient function on the axes below.
Part A
𝑦 = 𝑥2
x -2 -1 0 1 2 3 4
Gradient
Is there a relationship between a function and its gradient at any particular point?
Part B
𝑦 = 𝑥3
x -3 -2 -1 0 1 2 3
Gradient
Gradient Function

2. 2
Is there a relationship between a function and the gradient at any particular point?
Part C
𝑦 = 𝑥4
x -3 -2 -1 0 1 2 3
Gradient
Is there a relationship between a function and the gradient at any particular point?
Part D
What is the pattern?
Function Gradient function
𝑦 = 𝑥2
𝑦 = 𝑥3
𝑦 = 𝑥4
𝑦 = 𝑥 𝑛
Can you explain your rule in words?
Gradient Function
Gradient Function

3. 3
The Gradient Function
Find the Gradient of y = x2
at x = 1 by completing the table below:
Point A Point Bn
(Remember that point B is on the
curve y = x2
)
Calculation of
Gradient ABn
Gradient ABn
A( 1, 1) B
( 3, 9 )
=
A( 1, 1) B1 ( 2.4, ) =
A( 1, 1) B2 ( 2, ) =
A( 1, 1) B3 ( 1.4, ) =
A( 1, 1) B4 ( 1.1, ) =
A( 1, 1) B5 ( 1.01, ) =
A( 1, 1) B6 ( 1.001, ) =
The Gradient of y = x2
at the point x = 1 is =

4. 4
The Gradient Function
Let us look at the curve y = x2 again.
Consider a fixed point A and a second point B which is very close to A.
B is h more in the x direction (by the way h is very small).
What are the coordinates of B if A is (x, x2)?
Point A Point B
A( x, x2 ) B ( x + h, )
Calculation of Gradient AB
Since h is very small Gradient AB =
The Gradient for the curve y = x2 at any point on the curve is given by the function:
Can you use first principles to find the gradient function for y = x3?
This method is called using
first principles to find the
gradient function
B( )

5. 5
Increasing and Decreasing Functions
An increasing function has a positive gradient.
A decreasing function has a negative gradient.
Look at the gradient of each point of this curve and complete this table. Use you GDC if you need
to.
X -1 -2 0 1 2 3 4 5 6
Gradient
What do you notice about the sign of the gradient for x<0? Is the gradient positive or negative? Is
this increasing or decreasing?
What do you notice about the sign of the gradient for 0<x<4? Is the gradient positive or negative?
Is this increasing or decreasing?
What do you notice about the sign of the gradient for x >4? Is the gradient positive or negative?
Is this increasing or decreasing?
Complete this table:
Increasing Decreasing Increasing
0 < 𝑥 < 4
𝑑𝑦
𝑑𝑥
< 0

6. 6
Stationary Points
A stationary point is when a curve has a gradient of zero or alternatively when a curve stops
increasing and starts decreasing or vice versa.
Sketch:
𝒇( 𝒙) = 𝒙 𝟐
Where is the stationary point for this curve?
What is the sign of the gradient at x = 1?
What is the sign of the gradient at x = -1
Sketch:
𝒇( 𝒙) = −𝒙 𝟐
Where is the stationary point for this curve?
What is the sign of the gradient at x = 1?
What is the sign of the gradient at x = -1
Sketch
𝒇( 𝒙) = (𝒙 + 𝟐)(𝒙 − 𝟑)(𝒙 + 𝟏)
Find out where all the possible stationary
points are.
What is a point of inflexion?
Describe how to find horizontal and non-
horizontal point of inflexion. Include a diagram
and an example to illustrate your explanation.