Additional Mathematics Form 4 QUADRATIC FUNCTIONS
Dec 09, 2014
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
acb 42
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?