Quadratic function

Additional Mathematics

Form 4

QUADRATIC FUNCTIONS

Learning objectives :

Understand the concept of quadratic functions and their graphs.

Understand and use the condition for quadratic equations to have 2 roots, 1

root and no root.

Recognise quadratic functions

Recognise shapes and locations of graph of quadratic functions

Determine the root of quadratic equation from the value

Shapes of graphs of quadratic functions

Relating the position of quadratic function graphs with types of roots for f(x) = 0

Learning outcomes :

acb 42

Part A

What is the equation of quadratic?

f(x) = Ax2 + Bx + C

A ≠ 0

Do you know how to plot a quadratic function graph?

What does it mean by A,B and C?

f(x) = Ax2 + Bx + C

We can plot a quadratic function graph based on given tabulated values of the quadratic

function. The parabolic curve obtained in the plot is a symmetrical curve with respect to a middle

line. This middle line is called the axis of symmetry. The axis of symmetry intersects the parabola at the maximum or minimum

point.

Plotting the graph of a quadratic function

The x intercept is the value of x where the curve intersects the x axis. Similarly, the y intercept is the value of y where the curve

intersects the y axis. The maximum or minimum point is also called

the turning point

SHAPES OF GRAPHS OF QUADRATIC FUNCTIONS

The shapes of graphs of a quadratic function f(x)= a2+ bx+c depend on the value

of a. 

Please choose 5 positive numbers from the list below :

1,2,3,4,5,6,7,8,9,10

POSITIVE NUMBERS : …. , …. , …. , …. , ….

Please choose 5 negative numbers from the list below :

-1,-2,-3,-4,-5,-6,-7,-8,-9,-10

NEGATIVE NUMBERS : …. , …. , …. , …. , ….

By using the positive numbers and negative numbers that you choose,substitute it to A in quadratic function by using GeoGebra.(Let B = C = 0)

f(x) = Ax2

What do you get from the graph?

When the value of A become bigger, the graph is narrow

When the value of A become smaller, the graph is wide

When the value of A is positive, the graph is looks smile

When the value of A is negative, the graph is looks sad.

ADDITIONAL INFORMATIONS

Let say A = 1, what can you determine when you substitute the positive and negative numbers

you have choosed before into B?

( Let C = 0 )

f(x) = Ax2 + Bx

What do you get from the graph?

When the value of B is positive, the graph is going to

the left.

When the value of B is negative, the graph is going to the right.

Let say A = 1 and B = 0, what can you determine when you substitute the positive and negative

numbers

you have choosed before into C?

f(x) = Ax2 + C

What do you get from the graph?

When the value of C is positive, the graph is going upwards.

When the value of C is negative, the graph is going downwards.

Part B

RECOGNISE THE GENERAL FORM OF

QUADRATIC FUNCTIONS

RELATING THE POSITION OF QUADRATIC

FUNCTION GRAPHS WITH TYPES OF

ROOTS FOR F(X) = 0

RELATING THE POSITION OF QUADRATIC FUNCTION GRAPHS WITH TYPES OF ROOTS FOR F(X) = 0

b2-4ac>0 (two distinct roots); here, the graph intersects the x axis at two different points, which are the roots of the equationf(x)=0 . b2-4ac=0 (two equal roots ); here, the graph touches the x axis at one point only, which is the only root of the equation f(x)=0. 

b2-4ac<0 (no roots); here, the graph f(x) does not intersect the x axis, thus there is no root for the equation f(x)=0 

We can relate the positions of quadratic function graphs f(x) when f(x)=0 to the types of roots as

follows:

Roots of the quadratic function graphs are the value of    where the graph cuts the axis.There are 3 ways to find roots of a quadratic functions, which are:-factorization-completing the square-using the quadratic formulaThe types of roots depends on the value of

~discriminant of a quadratic functions equation

x x

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Draw the quadratic functions given below by using Geogebra software, and fill in the blank

cbxaxxf 2)(

Function

1 2 1

-1 2 1

1 3 4

-1 3 -4

1 4 4

-1 4 -2

a b c b2

ac4 acb 42

4

4

9

9

16

16

4 0

-4

16

16

16

8

8

-7

-7

0

8

)(xg

)(xh

)(xj

)(xk

)(xm

)(xn

CONCLUSION

Two distinct roots. Here, the graph intersects at the x axis at two different points, where the roots of the equation f(x)=0

Two equal roots. Here, the graph touches at the x axis at one point only, where the only root of the equation f(x)=0

No real roots. Here, the graph does not touch the x axis, thus, there is no root for the equation f(x)=0

042 acb

042 acb

042 acb

EXERCISE

GivenDetermine the shape of the graph.

736)( 2 xxxf

Smile shape

No real root

What type of root of the function?