3 Quadratic Function

3.1 32 3.3 3.4

Quadratic Functions and their graphs. Maximum and minimum values of QF. Sketch graphs of Quadratic functions. Quadratic Inequalities.

Analisa soalan tahun-tahun lepas

3

Additional Mathematics (3472) Chapter Form 4 Quadratic Functions

2003 N M 1 3

N: number of question M: total marks of question

General form of a quadratic function

2004 N M 2 6

Paper 1 2005 2006 N M N M 1 3 2 5

2007 N M 2 6

2008 N M 2 6

3 Quadratic Functions

Additional M athematics

f ( x) = ax 2 + bx + c

Form 4

a, b and c are constants and a ≠ 0.

The different between Quadratic Equations and Quadratic Functions Quadratic Equations 2

Quadratic Functions

f ( x ) = ax 2 + bx + c

ax + bx + c = 0

Exercise 1 Determine whether each of the following functions is a quadratic function. (a)

f ( x) = 5 − 4 x + x 2

(d) f ( x ) = ( x − 1)( x + 3)

f ( x ) = x 2 (6 − x )

(b)

(e) f ( x ) =

1 x

2

(c)

f ( x) = x ( x 2 + 4)

(f)

f ( x) = 1 − 3 x + x 2

+5

Exercise 2: Shape of the graph of a Quadratic Function Make a table of values and plot the graph of f ( x ) = − x 2 + x + 6 for the values of x in the range

x -1 0 1 2 3 4 5 y 8 3 0 -1 0 3 8 The above table shows the values of x and f(x) for f ( x ) = x 2 − 4 x + 3 . Plot the graph of f(x) for the values of x in the range −1 ≤ x ≤ 5 .

−3 ≤ x ≤ 4. x y f(x)

f(x)

x

x

Note

f ( x) = ax 2 + bx + c

if a >0, graph minimum shaped obtained. If a < 0, graph maximum shaped obtained.

2

3 Quadratic Functions

Additional M athematics

3

Form 4

3 Quadratic Functions

Additional M athematics

Form 4

Exercise 3 1. Determine the types of roots of the equation f(x) = 0. (√ the right answer)

Two real and distinct roots Two real and equal roots No real roots

Two real and distinct roots Two real and equal roots No real roots

f(x)

(c)

(d)

f(x)

x x Two real and distinct roots Two real and equal roots No real roots

Two real and distinct roots Two real and equal roots No real roots

2. The graph of the quadratic function f ( x) = x 2 − 3kx + 5k − 1 touches the x-axis at only one point. Find the possible values of k. a= b= c=

(keyword: f(x) touches the x-axis at only one point, -> b 2 − 4ac = 0 ) 3. Find the range of values of p if the graph of the quadratic function f ( x) = 5 − p + 6 x − 3 x 2 does not intersect the x-axis. General form for quadratic function, f(x)=ax²+bx+c f(x) = a= b= c= (keyword: f(x )does not touches the x-axis, -> b 2 − 4ac < 0 ) 4

3 Quadratic Functions

Additional M athematics

Form 4

4. Find the range of values of k if the graph of the quadratic functions f ( x) = x 2 − 2kx + k + 6 intersect the xaxis at two different points.

a= b= c=

(keyword: f(x )does not touches the x-axis, -> b 2 − 4ac > 0 ) 5. The quadratic equation x(x+1)=px - 4 has two distinct roots. Find the range of values of p. [3m] .

6. A quadratic equation x 2 + px + 9 = 2 x has two equal roots. Find the possible values of p. [3m]

7. The quadratic equation hx 2 + kx + 3 = 0 , where h and k are constants, has two equal roots. Express h in terms of k.

8.

5

3 Quadratic Functions

Additional M athematics

Form 4

3.2 The maximum &the minimum values of a QF.

Exercise 4 1. The figures below show the shapes of the graph of quadratic function f(x) = a(x+p)² +q. Determine the Values of a, p and q for each of the graphs.

p= q=

p= q=

at(0,5)

at(0,-2)

2

6

3 Quadratic Functions

Additional M athematics

Form 4

2

3. The quadratic function f ( x ) = p( x + q ) 2 + r , where p, q and r are constants, has a minimum value of - 4. The equation of the axis of symmetry is x = 3. State (a) the range of values of p, (b) the value of q, (c) the value of r. [3m]

7

3 Quadratic Functions

Additional M athematics

Form 4

3.3 Sketching the Graph of a Quadratic Function.

We use b²-4ac to test the graph whether intersects the x-axis or not. If b²-4ac>0. it intersects the x-axis by 2 points If b²-4ac=0, it intersect the x-axis by 1 point. If b²-4ac<0, it doesn’t touch x-axis.

Example Sketch the graph of each of the following QF. State the axis of symmetry in each of the graphs.

f ( x) = x 2 + x − 6

Step 1

a=

, the graph has a

shape.

Step 4 Sketch the graph

Find whether the graph intercept the x-axis. b 2 − 4ac = 1-4(1)(-6) = 25 > 0 Step 2

By completing the square b b f ( x) = x 2 + x + ( ) 2 − ( ) 2 + c 2 2 = x2 + x + ( )2 − ( )2 + ( 2 2 1 = (x + )2 2 1 2 25 = (x + ) − 2 4

Step 3

(

)

) +(

)

a>0, the ______________ point = (

)

When x = 0, f(x) = When f(x) = 0, x2 + x − 6 = 0 x=

(0,

) Axis of symmetry,

(

8

, 0), (

,0)

_________________

3 Quadratic Functions

Additional M athematics

Form 4

3.4 Quadratic Inequality Sketching the graph can be used to determine the range of values of x which satisfies a f(x).

Therefore, the range of values of x is 2<x<5.

Therefore, the range of values of x is x<-4 or x>6.

9

3 Quadratic Functions

Additional M athematics

Form 4

Example 1 Find the range of values of x for which x 2 − 2 x − 15 > 0

Example 2 Find the range of values of x for which

Solution a = 1>0, graph shape U from calculator, EQN, find the value x.

Solution a = -1 < 0, graph shape from calculator, EQN, find the value x.

-3

∩

-3

5

x 2 − 2 x − 15 > 0 , y >0 shape the graph.

-3

− x2 − x + 6 > 0

− x2 − x + 6 > 0 ,

y > 0 shape the graph.

-3

5

For x 2 − 2 x − 15 > 0 x < −3 or x > 5

2

(y>0)

For − x 2 − x + 6 > 0 -3 < x < 2

Exercise 5 1. Find the range of values of x for which (2x + 3)(x + 2) > 10.

2. (x – 1)(5 – x) > 3.

10

2

3 Quadratic Functions

Additional M athematics

3.

4.

11

Form 4

3 Quadratic Functions

Additional M athematics

12

Form 4