3 Quadratic Function

3.1 Quadratic Functions and their graphs.32 Maximum and minimum values of QF.3.3 Sketch graphs of Quadratic functions.3.4 Quadratic Inequalities.

Analisa soalan tahun-tahun lepas

Additional Mathematics (3472) Paper 1Chapter 2003 2004 2005 2006 2007 2008Form 4 N M N M N M N M N M N M

3 Quadratic Functions 1 3 2 6 1 3 2 5 2 6 2 6

N: number of question M: total marks of question

General form of a quadratic function

3 Quadratic Functions Additional M athematics Form 4

a, b and c are constants and a ≠ 0.

The different between Quadratic Equations and Quadratic Functions

Quadratic Equations Quadratic Functions

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

(d) (e) (f)

Exercise 2: Shape of the graph of a Quadratic Function

x -1 0 1 2 3 4 5y 8 3 0 -1 0 3 8

The above table shows the values of x and f(x) for . Plot the graph of f(x) for the values of x in the range .

f(x)

x

Make a table of values and plot the graph of for the values of x in the range

.

xy

f(x)

x

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

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3 Quadratic Functions Additional M athematics Form 4

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3 Quadratic Functions Additional M athematics Form 4

Exercise 31. Determine the types of roots of the equation f(x) = 0. (√ the right answer)

Two real and distinct roots Two real and distinct roots

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

(c) (d)

Two real and distinct roots Two real and distinct roots Two real and equal roots Two real and equal roots No real roots No real roots 2. The graph of the quadratic function 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, -> ) 3. Find the range of values of p if the graph of the quadratic function 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, -> )4. Find the range of values of k if the graph of the quadratic functions intersect the x-

4

f(x)

x

x

f(x)

3 Quadratic Functions Additional M athematics Form 4

axis at two different points.

a = b = c =

(keyword: f(x )does not touches the x-axis, -> )

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 has two equal roots. Find the possible values of p. [3m]

7. The quadratic equation , where h and k are constants, has two equal roots. Express h in terms of k.

8.

3.2 The maximum &the minimum values of a QF.

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3 Quadratic Functions Additional M athematics Form 4

Exercise 41. 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.

6

2

p=q=

at(0,5)

p=q=

at(0,-2)

3 Quadratic Functions Additional M athematics Form 4

3. The quadratic function , 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]

3.3Sketching the Graph of a Quadratic Function.

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2

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 pointsIf b²-4ac=0, it intersect the x-axis by 1 point.If b²-4ac<0, it doesn’t touch x-axis.

3 Quadratic Functions Additional M athematics Form 4

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

Step 1 a = , the graph has a shape. Step 4 Sketch the graph

Find whether the graph intercept the x-axis.= 1-4(1)(-6) = 25 > 0

Step 2 By completing the square

- ( ) + ( )

a>0, the ______________ point = ( )

Step 3 When x = 0, f(x) = (0, )When f(x) = 0, x = ( , 0), ( ,0)

Axis of symmetry,

_________________3.4 Quadratic Inequality

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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.

Example 1 Example 2

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Find the range of values of x for which

Solutiona = 1>0, graph shape Ufrom calculator, EQN, find the value x.

-3 5

, y >0 shape the graph.

-3 5

For (y>0) or

Find the range of values of x for which

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

-3 2

, y > 0 shape the graph.

-3 2

For -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.

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3.

4.

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