Functions and Graphs (1)

3

3.1 Introduction to Functions

3.2 Quadratic Functions

3.3 Graphs of Other Functions

Chapter Summary

Case Study

Functions and Graphs (1)

P. 2

Therefore, if the height of the building is 80 m, while the speed of the lift is 2 m/s, then the time it takes to reach the top floor from the ground floor is (80 2) s = 40 s.

In junior forms, we learnt that the relationship between time, speed and

distance can be expressed as: Time DistanceSpeed

The above relationship can be treated as a function and we will discuss the concept of a function in details in Section 3.1 of this chapter.

Case Study How long does it take for us to reach the top floor of the building by taking this lift?It depends on the speed of

the lift and the heightof the building.

P. 3

A. Basic Idea of a FunctionA. Basic Idea of a Function

The concept of a function is one of the basic concepts in mathematics.

Suppose the price of petrol is $12 per litre.

The following table shows amount paid and the corresponding volume of petrol bought.

Volume of petrol bought (L) 5 10 15 20 25

Amount paid ($) 60 120 180 240 300

If we use $y to represent the amount paid and x L to represent the volume of petrol bought, then we have:

y 12xThe amount paid $12 multiplied by the volume of petrol boughtis

3.1 Introduction to Functions3.1 Introduction to Functions

P. 4

A. Basic Idea of a FunctionA. Basic Idea of a Function

The value of y (the amount paid) depends on a given value of x (the volume of petrol bought).

We call y the dependent variable and x the independent variable.

Each value of x gives one and exactly one value of y.

A variable y is said to be a function of a variable x if there is a relation between x and y such that every value of x gives exactly one value of y.

In this case, we say that y is a function of x.

y 12xThe amount paid $12 multiplied by the volume of petrol boughtis


P. 5

B. Different Ways to Represent FunctionsB. Different Ways to Represent Functions

Volume of petrol bought (L) 0 5 10 15 20 25

Amount paid ($) 0 60 120 180 240 300

Tabular representation

Algebraic representation

y 12x

Graphical representation

Plot the corresponding values of x and y according to the above table.

The relation between x and y can be expressed in different ways:


P. 6

B. Different Ways to Represent FunctionsB. Different Ways to Represent Functions

Consider a square with side x cm and perimeter y cm.

x (cm) 1 2 3 4 5

y (cm) 4 8 12 16 20

Tabular representation

Algebraic representation

y 4x

Graphical representation

Plot the corresponding values of x and y according to the above table.

We can use the graph to find the value of y for other value of x, for example:

When x 2.5, y 10.


P. 7

A function can be expressed by an equation, such as y 3x + 2.It is common to use the notation f(x) to denote a function of x, such as f(x) 3x + 2.

The symbol ‘f(x)’ is read as ‘f of x’,

Notes: y 3x + 2 and f(x) 3x + 2 represent the same function.

f(x) does not mean f is multiplied by x. f(x) represents the value of the function at x.

For example, f(10) is the value of the function f(x) at x 10, that is,

f(10) 3(10) + 2 32

Consider the representations f(x) 3x + 2 and y 3x + 2. The meaning of f(10) is the same as that of the phrase ‘the value of y when x 10’.

The letter ‘f ’ in the notation f(x) may be replaced by other letters to represent different functions of x, such as g(x), h(x), F(x) and G(x).

C. Notation for a FunctionC. Notation for a Function


P. 8

Example 3.1TConsider f(x) x3 – 4x2 + 5.(a) Find f(–2), f(0) and f(2).(b) Is f(–2) + f(2) f(0)?

Solution:(a) f(–2) (–2)3 – 4(–2)2 + 5 –19

f(0) (0)3 – 4(0)2 + 5 5

f(2) (2)3 – 4(2)2 + 5 –3

(b) f(–2) + f(2) –19 + (–3) –22

∴ f(–2) + f(2) f(0)

In general,f(a + b) f(a) + f(b).

and f(0) 5



P. 9

Example 3.2TConsider f(x) k – 3x2 and f(3) –7.(a) Find the value of k.(b) Find the values of x such that f(x) 8.Solution:

(a) f(3) –7

k 20(b) From (a), f(x) 20 – 3x2.

∴ 20 – 3x2 8

Substitute x 3 into f(x), then find the value of k by solving the equation f(3) 7.

k – 3(3)2 –7 k – 27 –7

∵ f(x) 8

3x2

12 x2

4 x 2



P. 10

Example 3.3TConsider f(x) –4x2 + 7x – 2.(a) Express f(–a) and f(–a + 1) in terms of a.(b) If f(–a) f(–a + 1), find the value of a.

Solution:(a) f(x) –4x2 + 7x – 2 f(–a) –4(–a)2 + 7(–a) – 2

(b) ∵ f(–a) f(–a + 1)∴ –4a2 – 7a – 2 –4a2 + a + 1

–4a2 – 7a – 2

f(–a + 1) –4(–a + 1)2 + 7(–a + 1) – 2 –4(a2 – 2a + 1) – 7a + 7 – 2 –4a2 + 8a – 4 – 7a + 7 – 2 –4a2 + a + 1

–8a 3 a 8

3



P. 11

D. Domain and Co-domain of a FunctionD. Domain and Co-domain of a Function

When describing a function y f(x), we may want to know:

(i) Domain

(ii) Co-domain

(iii) Range

all possible values that the independent variable x can take

Examples:

Consider f(x) x2 + 1. x can take any real numbers.∴ The domain is the set of all real numbers.

x1

Consider g(x) . x cannot be 0.

∴ The domain is the set of all real numbers except 0.

The domain of the function

y is the set of all

real numbers except 1.

1

1

x


P. 12

D. Domain and Co-domain of a FunctionD. Domain and Co-domain of a Function

(i) Domain: the set of all real numbers

(iii) Range: the set of real numbers larger than or equal to 1

(ii) Co-domain: the set of all real numbers For the function y x2 + 1, even though x can be negative, y must be positive.

When describing a function y f(x), we may want to know:

(i) Domain

(ii) Co-domain

(iii) Range

all possible values that the independent variable x can take

all possible values that the dependent variable y can take

all output values of dependent variable y of the function, that is, the corresponding values of independent variable x

e.g. Consider f(x) y x2 + 1.


P. 13

E. Variables in FunctionsE. Variables in Functions

We have been using x and y as the independent and the dependent variables of a function.

In fact, we can have functions of other variables:

Consider f(l) l2. f(l) is a function of l.

Consider h() sin . h() is a function of .

Consider g(t) . g(t) is a function of t.1

22 t

t

We can also use different variables for the same function.For example, f(l) l2, f(x) x2 and f(t) t2 represent the same function. We call the variables l, x and t the dummy variables.


P. 14

A. Graphs of Quadratic FunctionsA. Graphs of Quadratic Functions

3.2 Quadratic Functions3.2 Quadratic Functions

A function in the form y ax2 + bx + c, where a, b and c are constants and a 0 is called a quadratic function.

Plot the graph of y –x2 – 2x + 6 for –4 x 2:

x –4 –3 –2 –1 0 1 2

y –2 3 6 7 6 3 –2

The graph is a curve which is called a parabola.

y x2 2x 6

P. 15

A. Graphs of Quadratic FunctionsA. Graphs of Quadratic Functions

Example 3.4T

Solution:

Consider y x2 + 2x – 1.(a) Complete the following table.

(b) Plot the graph of y x2 + 2x – 1 for –4 x 2.

x –4 –3 –2 –1 0 1 2

y

x –4 –3 –2 –1 0 1 2

y 7 2 –1 –2 –1 2 7

(a) y x2 + 2x – 1

(b) Refer to the figure on the right.

y x2 2x 1


P. 16

B. Properties of Quadratic FunctionsB. Properties of Quadratic Functions

The following figure shows two parabolas y ax2 + bx + c where a > 0 and a < 0 respectively.

Each parabola has a vertex.

Each parabola is symmetrical about the axis of symmetry. The parabolas cut the y-axis at (0, c) and c is called the y-intercept.


P. 17

1. Direction of opening:(a) If a > 0, then it opens upwards.(b) If a < 0, then it opens downwards.

Properties of the graph of a quadratic function y ax2 + bx + c:

2. Since the graph cuts the y-axis at (0, c), c is the y-intercept of the graph.

3. (a)If a > 0, then the vertex is the lowest point of the parabola.(b) If a < 0, then the vertex is the highest point of the parabola.

4. The axis of symmetry passes through the vertex.



P. 18

Apart from determining the direction of opening of the graph, the values of a (in y ax2 + bx + c) also affect the width of the opening.

When a > 0,the opening is narrower for a larger value of a.

When a < 0,the opening is wider for a larger value of a.(i.e., the more negative the value of a, the narrower the opening.)



P. 19

Example 3.5TConsider the graphs of the following three functions:

(I) y (x – 3)2 – 1(II) y 2(x – 3)2 – 1(III) y –(x – 3)2 + 1

(a) Compare the shapes of the graphs of (I) and (III).(b) Describe the difference in the shapes of the graphs between (I) and (II).

Solution:(a) Consider function (III):

y –(x – 3)2 + 1 –[(x – 3)2 – 1]

∴(I) and (III) are symmetrical about the x-axis.

(b) The coefficients of the x2 terms in (I) and (II) are positive.

∴The opening of the graph of (I) is wider than that of (II).The coefficient of the x2 term in (II) is greater than that in (I).



P. 20

C. Solving Quadratic Equations byC. Solving Quadratic Equations by the Graphical Methodthe Graphical Method


Consider the graph of y ax2 + bx + c:

Suppose the graph cuts the x-axis at two points (p, 0) and (q, 0).

The two values p and q are called thex-intercepts of the graph of y ax2 + bx + c.

Since the y-coordinates of these two points are 0,

ax2 + bx + c 0.

∴ The x-intercepts of the graph of y ax2 + bx + c satisfy the equation ax2 + bx + c 0.

∴ The x-intercepts of the graph of y ax2 + bx + c are the roots of the quadratic equation ax2 + bx + c 0.

P. 21

Example 3.6TThe figure shows the graph of y 5x2 + 8x – 4.Solve 5x2 + 8x – 4 0 graphically.Solution:

The graph cuts the x-axis at the points (–2.0, 0) and (0.4, 0). (cor. to the nearest 0.05)

Hence the roots of 5x2 + 8x – 4 0 are–2.0 and 0.4.



P. 22

Example 3.7TThe figure shows the graph of y 4x2 + 12x + 9.Solve 4x2 + 12x + 9 0 graphically.Solution:

The graph touches the x-axis at the point (–1.5, 0). (cor. to the nearest 0.05)

Hence the root of 4x2 + 12x + 9 0 is –1.5.



P. 23

Example 3.8TThe figure shows the graph of y –x2 + x – 1.Solve –x2 + x – 1 0 graphically.Solution:

The graph does not intersect with the x-axis.

Hence the equation –x2 + x – 1 0 has no real roots.



P. 24

Example 3.9TConsider the graph of y x2 + 3x + 1.(a) Solve x2 + 3x + 1 0 graphically.(b) Solve the above equation by the quadratic

formula and compare the result with (a).

Solution:(a) The graph cuts the x-axis at the points (–2.6, 0) and (–0.4, 0).

Hence the roots of x2 + 3x + 1 0 are –2.6 and –0.4. (cor. to 1 d. p.)(b) By the quadratic formula, we have

)1(2

)1)(1(433 2 x

2

53

According to the scale of the graph, we can only find the approximate roots of the equation correct to 1 decimal place in (a) while the roots in (b) are exact values.



P. 25

D. Quadratic Graphs and Nature of RootsD. Quadratic Graphs and Nature of Roots

In Chapter 2, we learnt that the discriminant of a quadratic equation can help us to determine the nature of the roots.

We can also use b2 – 4ac to find the number of x-intercepts of the graph of y ax2 + bx + c.

If b2 – 4ac > 0,then the graph of y ax2 + bx + c cuts the x-axis at two distinct points.

If b2 – 4ac 0,then the graph of y ax2 + bx + c touches the x-axis at only one point.

If b2 – 4ac < 0,then the graph of y ax2 + bx + c does not cut the x-axis.


P. 26


b2 – 4acNature of rootsof the equation

ax2 + bx + c 0

Number of x-intercepts of the graph of

y ax2 + bx + c

> 0 Two unequal real roots 2

0One double real root

(Two equal real roots)1

< 0 No real roots 0

The following table summarizes the nature of roots of quadratic equations and the corresponding number of x-intercepts of their graphs.


P. 27

Example 3.10TThe graph of y (2 – k)x2 – 16x – 16 touches the x-axis at only one point.(a) Find the value of k.(b) Hence solve (2 – k)x2 – 16x – 16 0.Solution:

(a) Since the graph touches the x-axis, 0.

(b) The equation is –4x2 – 16x – 16 0.

∴ x2 + 4x + 4 0

162 – 4(2 – k)(–16) 0 256 + 128 – 64k 0

64k 384 k

6

(x + 2)2

0 x

–2



P. 28

A. Linear FunctionsA. Linear Functions

3.3 Graphs of Other Functions3.3 Graphs of Other Functions

A function in the form y mx + c, where m and c are constants and m 0 is called a linear function.

If m 0, then the function becomes y c which is called a constant function.

Notes:

c is the y-intercept of the graph.If c 0, then the graph is a straight line passing through the origin.

m 0 m 0

P. 29

B. Other FunctionsB. Other Functions


Apart from linear and quadratic functions, Example 3.11T and Example 3.12T show some other functions.

P. 30

The function y ax3 + bx2 + cx + d, where a, b, c and d are constants and a 0, is called a cubic function.

Example 3.11TConsider the function y x3 – 2x2 – x + 2.(a) Plot the graph of y x3 – 2x2 – x + 2

for –2 x 3.(b) Find the x-intercepts and the y-intercept

of the graph. (Give the answers correct to 1 decimal place if necessary.)

Solution:

x –2 –1.5 –1 –0.5 0

y –12 –4.4 0 1.9 2

x 0.5 1 1.5 2 2.5 3

y 1.1 0 –0.6 0 2.6 8

(a)

(b) The x-intercepts are –1.0, 1.0 and 2.0.The y-intercept is 2.0.



P. 31

Example 3.12TThe figure shows the graphs of y 3(2 – x), y x2 + 9 and y – 2.

Compare the graphs of the functions with respect to the following:(a) domain(b) existence of maximum or minimum value(c) number of x-intercepts

x6

Solution:(a) For y 3(2 – x), the domain is the set

of all real numbers.

For y – 2, the domain is the set of all

real numbers except 0.x6

For y x2 + 9, the domain is the set of all real numbers.



P. 32

Example 3.12TThe figure shows the graphs of y 3(2 – x), y x2 + 9 and y – 2.

Compare the graphs of the functions with respect to the following:(a) domain(b) existence of maximum or minimum value(c) number of x-intercepts

x6

Solution:(b) For y 3(2 – x), there is no maximum

or minimum value.

For y – 2, there is no maximum or minimum value.x6

For y x2 + 9, there is a minimum value.

(c) For y 3(2 – x), there is one x-intercept.

For y – 2, there is one x-intercept.x6

For y x2 + 9, there is no x-intercept.



P. 33

3.1 Introduction to Functions

Chapter Summary

1. A variable y is said to be a function of x if there is a relation between x and y such that every value of x corresponds to exactly one value of y.

2. We use f(x) to denote a function of x, where x is the independent variable.

3. The domain is the set of values that the independent variable can take.The co-domain is the set of possible values that the dependent variable can take.The range is the set of values that the dependent variable takes corresponds to the independent variable.

P. 34


Chapter Summary

Properties of the graph of y ax2 + bx + c:

1. (a) If a > 0, then the graph opens upwards.(b) If a < 0, then the graph opens downwards.

2. The graph cuts the y-axis at (0, c), that is, the y-intercept c.

3. The vertex is the turning point of the graph.

(a) If a > 0, the vertex is the minimum point of the graph.(b) If a < 0, the vertex is the maximum point of the graph.

4. The axis of symmetry is a line that the graph is symmetrical about it. It passes through the vertex of the graph.

P. 35


Chapter Summary

For two quadratic functions:

(I) y a1x2 + b1x + c1

(II) y a2x2 + b2x + c2

If 0 < a1 < a2 or 0 > a1 > a2, then graph (II) opens narrower than the graph (I).

The roots of the quadratic equation ax2 + bx + c 0 (a 0) can be obtained by finding the x-intercept(s) of the graph.

b2 – 4acNumber of real roots

of the equationax2 + bx + c 0

Number of x-intercepts of the graph of

y ax2 + bx + c > 0 2 2 0 1 1 < 0 0 0

P. 36

3.3 Graphs of Other Functions

Chapter Summary

1. The graph of a linear function y mx + c is a straight line.

2. Without drawing the graph of a function y f(x), they-intercept can be found by putting x 0 into the function, that is, the y-intercept of y f(x) is f(0).

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