graphs of functions 2
TRANSCRIPT
![Page 1: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/1.jpg)
![Page 2: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/2.jpg)
![Page 3: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/3.jpg)
![Page 4: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/4.jpg)
CHAPTER 2
GRAPHS OF FUNCTIONS II
2.1 GRAPHS OF FUNCTIONS
• The graph of a function is a set of points on
the Cartesian Plane that satisfy the function
• Information is presented in the form of graphs
• Graph are widely used in science and technology
• Graphs are very useful to researchers, scientists
and economist
![Page 5: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/5.jpg)
The different type of functions and respective power of x
Type of
function
General form Example Highest
power of variable x
Linear
Quadratic
Cubic
Reciprocal
y = ax + c
y = ax2 + bx + c
y = ax3 + bx2 + cx + d
y = a
x
y = 3x
y = -4x + 5
y = 2x2
y = -3x2 + 2x y = 2x2 + 5x + 1
y = 2x3
y = -3x3 + 5x
y = 2x3 - 3x + 6
y = 4
x
y = - 2
x
1
3
-1
2
y = ax3
y = ax3 + bxy = ax3 + bx + c
a ≠ 0
![Page 6: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/6.jpg)
LINEAR FUNCTION
![Page 7: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/7.jpg)
LINEAR FUNCTION
![Page 8: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/8.jpg)
QUADRATIC FUNCTION
![Page 9: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/9.jpg)
QUADRATIC FUNCTION
![Page 10: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/10.jpg)
QUBIC FUNCTION
![Page 11: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/11.jpg)
QUBIC FUNCTION
![Page 12: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/12.jpg)
RECIPROCAL FUNCTION
![Page 13: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/13.jpg)
RECIPROCAL FUNCTION
![Page 14: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/14.jpg)
• Using calculator to complete the tables
• Using the scale given to mark the points
on the x-axis and y-axis
• Plotting all the points using the scale given
![Page 15: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/15.jpg)
COMP
![Page 16: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/16.jpg)
CALC MemoryExample 1Calculate the result
for Y = 3X – 5,
when X = 4, and when X = 6
)3X
- 5
CALC
3X – 5
4 = 7
CALC 6 = 13
ALPHA
![Page 17: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/17.jpg)
CALC MemoryExample 2Calculate the result
for Y = X2 + 3X – 12,
when X = 7, and when X = 8
) 3X
x2 +ALPHA
)
X
- 21
CALC
X2 + 3X – 12
7 = 58CALC 8 = 76
ALPHA
![Page 18: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/18.jpg)
CALC MemoryExample 3Calculate the result
for Y = 2X2 + X – 6,
when X = 3, and when X = -3
)
3
X
x2 +ALPHA
)
X
- 6
CALC
2X2 + X – 6
3 = 15CALC (-) = 9
ALPHA
2
![Page 19: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/19.jpg)
CALC MemoryExample 4Calculate the result
for Y = -X3 + 2X + 5,
when X = 2, and when X = -1
)
1
X
x2
+ 2ALPHA
)
X
+ 5
-X3 + 2X + 5 2 = 1CALC (-) = 4
ALPHA
(-)SHIFT x3
CALC
![Page 20: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/20.jpg)
Example 5Calculate the result
for Y = 6 when X = -3,
X
and when X = 0.5
)
XALPHA
3CALC (-) = -2
6
CALC 0 . 125 =
ab/c 6┘x
![Page 21: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/21.jpg)
Example 6Calculate the result
for Y = 6 when X = -3,
X
and when X = 0.5
)
XALPHA
3CALC (-) = -2
6
CALC 0 . 125 =
x-1 6x-1
![Page 22: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/22.jpg)
Y = -2X2 + 40
X 0 0.5 1 1.5 2 3 3.5 4
Y
Y = X3 – 3X + 3
X -3 -2 -1 0 0.5 1 1.5 2
Y
Y = -16
X
X -4 -3 -2 -1 1 2 3 4
Y
40 39.5 38 35.5 32 22 15.5 8
-15 1 5 3 16.25 1.875 51
4 5.33 8 16 -16 -8 -5.33 -4
![Page 23: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/23.jpg)
2(4)2 + 5(4) – 1 = 51
Using Calculator
2 ( )4 x2 + 5 ( 4 )
- 1 =
![Page 24: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/24.jpg)
(-2)3 - 12(-2) + 10 = 26
Using Calculator
2( ) x2
+ 1 0
(-) 1
=
SHIFT - 2
2( )(-)
2( ) 3
+ 1 0
(-) 1
=
V
- 2
2( )(-)
OR
![Page 25: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/25.jpg)
Using Calculator
6
(-3)= -2
3( )6 ÷ (-) =
![Page 26: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/26.jpg)
x -5 -4 -3 -2 -1 0 1 2 3
y 15 8 0 -1 0 3 15
y = x2 + 2x
y = (
-3
) 2 + 2 (
-3
) = 3
y = (
2
) 2 + 2 (
2
) = 8
3
8
![Page 27: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/27.jpg)
x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4
y 1 1.6 8 -8 -4 -1.25 -1
4
2
y = -4
( )
=
y = -4
( )
=
-1
4 -2-2
y = -4
x
Completing the table of values
-1 2
![Page 28: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/28.jpg)
2.4 3.0 3.9
x-axis scale : 2 cm to 2 units
Marking the points on the x-axis and y-axis
2 4
![Page 29: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/29.jpg)
-4 -2
-3.6 -3.0 -2.1
x-axis scale : 2 cm to 2 units
![Page 30: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/30.jpg)
10
15
12
14.5
13.5
y-axis scale : 2 cm to 5 units
10.75
![Page 31: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/31.jpg)
-20
-15
-18
-15.5
-16.5
y-axis scale : 2 cm to 5 units
-19.25
![Page 32: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/32.jpg)
The x-coordinate and y-coordinate
![Page 33: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/33.jpg)
xA
B
x
x
x
C
D
(-3,2)
(2,0)
(4,-3)
(0,-4)
Ex
(4,4)
Fx
(-7,-2)
![Page 34: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/34.jpg)
-5
-1 1
x
x
x
x
A
B
C
D
Ex (-0.5,2)
(-1,-3)
(0,3)
(0.3,1.5)
(0.5,-1.5)
![Page 35: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/35.jpg)
-2
-1 1
xx
x
x
A
B
CD (-1,-1.2)
(0,1.2)
(0.3,0.6)
(0.2,-1)
![Page 36: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/36.jpg)
-10
-1 1
x
x
x
x
A
B
C
D(-1,-6)
(0,5)
(0.3,3)
(0.5,-3)
![Page 37: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/37.jpg)
2.1 A Drawing the Graphs
Construct a table for a chosen range of x values, for example
-4 ≤ x ≤ 4
Draw the x-axis and the y-axis and suitable scale for each axis
starting from the origin
Plot the x and y values as coordinate pairs on the Cartesian Plane
Join the points to form a straight line (using ruler) or smooth curve
(using French Curve/flexible ruler) with a sharp pencil
Label the graphs
To draw the graph of a function, follow these steps;
![Page 38: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/38.jpg)
2.1 A Drawing the Graphs
Draw the graph of y = 3x + 2
for -2 ≤ x ≤ 2
solution
x
y
-2 0
-4 2 8
0-2-4 2 4
-2
-4
2
4
6
8
x
y
GRAPH OF A LINEAR FUNCTION
8
3 + 2
22
![Page 39: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/39.jpg)
Draw the graph of y = x2 + 2x for -5 ≤ x ≤ 3
solution
x -5 -4 -3 -2 -1 0 1 2 3
y 15 8 0 -1 0 3 15
y = x2 + 2x
3 83
+ 22
-3 -3
![Page 40: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/40.jpg)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3
y = x2 + 2x
GRAPH OF A QUADRATIC FUNCTION
![Page 41: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/41.jpg)
Draw the graph of y = x3 - 12x + 3 for -4 ≤ x ≤ 4
solution
x -4 -3 -2 -1 0 1 2 3 4
y -13 12 19 14 3 -8 -13 -6 19
y = x3 - 12x + 3
-13
- 123
-4 -4 + 3
![Page 42: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/42.jpg)
0 1 2 3 4-1-2-3-4
-5
-10
-15
5
10
15
20
25
y
x
x
x
x
x
x
x
x
x
x
GRAPH OF A CUBIC FUNCTION
![Page 43: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/43.jpg)
Draw the graph of y = -4 for -4 ≤ x ≤ 4.
x
solution
y = -4
x
x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4
y 1 1.6 4 8 -8 -4 -2 -1.25 -14
-4
-1
![Page 44: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/44.jpg)
1 2 3 4-1-2-3-4 0
y
x
2
4
6
8
-2
-4
-6
-8
X
X
X
X
X
X
X
X
X
GRAPH OF A RECIPROCAL FUNCTION
![Page 45: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/45.jpg)
xx
USING FRENCH CURVE
![Page 46: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/46.jpg)
x
x
x
x
USING FRENCH CURVE
![Page 47: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/47.jpg)
x
x
x
USING FRENCH CURVE
![Page 48: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/48.jpg)
-10
-15-20
![Page 49: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/49.jpg)
![Page 52: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/52.jpg)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3-3.5
5
y = 11
-4.4 2.5
2.1 B Finding Values of Variable from a Graph
y = x2 + 2x
Find
(a) the value of
y when
x = -3.5
(b) the value of
x when
y = 11
solution
From the graph;
(a)y = 5
(b) X = -4.4, 2.5
![Page 53: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/53.jpg)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3-3.5
5
2.1 B Finding Values of Variable from a Graph
y = x2 + 2x
![Page 54: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/54.jpg)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3
5
11
2.1 B Finding Values of Variable from a Graph
y = x2 + 2x-4.4 2.5
![Page 55: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/55.jpg)
x =1.5
![Page 56: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/56.jpg)
x
y
0 1 2 3 4
8
6
4
2
-2
-4
-6
-8
-4 -3 -2 -1
xx
x
x
x
x
x
x
-2.2
-1.2
1.8
3.4
( a ) y = -2.2
( b ) x = -1.2
Find
(a) the value of
y when
x = 1.8
(b) the value of
x when
y = 3.4
solution
y = -4
x
Values obtained from
the graphs are
approximations
Notes
![Page 57: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/57.jpg)
2.1 C Identifying the shape of a Graph from
a Given Function
LINEAR a
y
x
y = x
0
b
x
y = -x + 2
0
2
y
![Page 58: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/58.jpg)
2.1 C Identifying the shape of a Graph from
a Given Function
QUADRATIC a
y
x
y = x2
0
bx
y = -x2
0
y
![Page 59: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/59.jpg)
2.1 C Identifying the shape of a Graph from
a Given Function
CUBIC a
y
x
y = x3
0
b
x
y = -x3 + 2
0
2
y
![Page 60: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/60.jpg)
2.1 C Identifying the shape of a Graph from
a Given Function
RECIPROCAL a
y
x0
b
x0
y
y = 1
x
y = -1
x
![Page 61: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/61.jpg)
2.1 D Sketching Graphs of Function
• Sketching a graph means drawing a graph without
the actual data
• When we sketch the graph, we do not use
a graph paper, however we must know the important
characteristics of the graph such as its general form
(shape), the y-intercept and x-intercept
• It helps us to visualise the relationship of the variables
![Page 62: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/62.jpg)
EXAMPLE y = 2x + 4
4
-2 0
y
x
find the x-intercept of
y = 2x + 4.
Substitute y = 0
2x + 4 = 0
2x = -4
x = -2
Thus, x-intercept = -2
find the y-intercept of
y = 2x + 4.
Substitute x = 0
y = 2(0) + 4
y = 4
Thus, y-intercept = 4
draw a straight line that
passes x-intercept and y-intercept
y = 2x + 4
A Sketching The Graph of A Linear Function
![Page 63: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/63.jpg)
B Sketching The Graph of A Quadratic Function
EXAMPLE y = -2x2 + 8
a < 0
the shape of the graph is
y-intercept is 8
find the x-intercept of
y = -2x2 + 8.
Substitute y = 0-2x2 + 8 = 0
-2x2 = -8x2 = 4
Thus, x-intercept = -2 and 2
x0
y
-2 2
8
![Page 64: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/64.jpg)
B Sketching The Graph of A Cubic Function
EXAMPLE y = -3x3 + 5
a < 0
the shape of the graph is
y-intercept is 5
x0
y
5
![Page 65: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/65.jpg)
2.2 The Solution of An Equation By Graphical
Method
Solve the equation x2 = x + 2
Solution
x2 = x + 2
x2 - x – 2 = 0
(x– 2)(x + 1) = 0
x = 2, x = -1
![Page 66: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/66.jpg)
2
y
1
3
4
0-1 1-2 2 x
y = x2
y = x + 2
A
B
• Let y = x + 2
and y = x2
• Draw both
graphs on
the same
axes
• Look at the
points of
intersection:
A and B.
Read the
values of the
coordinates
of x.
x = -1 and
x = 2
Solve the equation x2 = x + 2 by using the Graphical Method
![Page 67: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/67.jpg)
2.2 The Solution of An Equation By Graphical
Method
12. (a) Complete Table 1 for the equation y = x2 + 2x by writing down the values of y
when x = -3 and 2.
x -5 -4 -3 -2 -1 0 1 2 3
y 15 8 0 -1 0 3 15
(b) By using scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis,
draw the graph of y = x2 + 2x for -5 ≤ x ≤ 3.
(c) From your graph, find
(i) the value of y when x = -3.5,
(ii) the value of x when y = 11.
(d) Draw a suitable straight line on your graph to find a value of x which satisfies
the equation of x 2 + x – 4 = 0 for -5 ≤ x ≤ 3.
TABLE 1
![Page 68: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/68.jpg)
2.2 The Solution of An Equation By Graphical
Method
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve each of the following equations.
a) x2 - 5x - 3 = 4
b) x2 - 5x - 3 = 2x + 4
c) x2 - 5x - 2 = x + 4
d) x2 - 5x - 10 = 0
e) x2 - 7x - 2 = 0
EXAMPLE 1
![Page 69: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/69.jpg)
solution
a) x2 - 5x - 3 = 4 x2 - 5x - 3 = y 4
Therefore, y = 4 is the suitable straight line
b) x2 - 5x - 3 = 2x + 2 x2 - 5x - 3 = y 2x + 2 y
Therefore, y = 2x + 2 is the suitable straight line
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve the equation: x2 - 5x - 3 = 4
y
![Page 70: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/70.jpg)
solution
c) x2 - 5x - 2 = x + 4
- 1
-1 on both sides
Therefore, y = x + 3 is the suitable straight line
x2 - 5x - 2 = x + 4 - 1
x2 - 5x - 3 = x + 3 x + 3
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve the equation: x2 - 5x – 2 = x + 4
![Page 71: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/71.jpg)
solution
d) x2 - 5x - 10 = 0 Rearrange the equation
Therefore, y = 7 is the suitable straight line
x2 - 5x = 10
x2 - 5x = 10 - 3- 3
x2 - 5x - 3 = 7 7
-3 on both sides
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve the equation: x2 - 5x - 10 = 0
![Page 72: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/72.jpg)
solution
e) x2 - 7x - 2 = 0 Rearrange the equation
Therefore, y = 2x - 1 is the suitable straight line
x2 = 7x + 2
x2 = 7x + 2 - 5x - 3- 5x - 3
x2 - 5x - 3 = 2x -12x -1
-5x - 3 on both sides
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve the equation: x2 - 7x - 2 = 0
![Page 73: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/73.jpg)
Alternative Method
Since a straight line is needed, we used to eliminate the term, x2.
The following method can be used
y = x2 - 5x - 3 1
0 = x2 - 7x - 2 2
1 - 2 y-0 = -5x - (-7x) - 3 - ( -2)
y = 2x - 1
e
The graph y = x2 - 5x - 3 is drawn. Determine the suitable straight line
to be drawn to solve the equation: x2 - 7x - 2 = 0
![Page 74: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/74.jpg)
2.2 The Solution of An Equation By Graphical
Method
The graph y = 8 is drawn. Determine the suitable straight line
x
to be drawn to solve each of the following equations.
a) 4 = x + 1
x
b) -8 = -2x - 2
x
EXAMPLE 2
![Page 75: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/75.jpg)
solution
4 = x + 1
xMultiply both sides by 2a
We get 8 = 2x + 2
x
Therefore, y = 2x + 2 is the suitable straight line
2x + 2
The graph y = 8 is drawn. Determine the suitable straight line
x
to be drawn to solve each the equation: 4 = x + 1
x
![Page 76: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/76.jpg)
solution
-8 = -2x - 2
x
Multiply both sides by -1b
We get 8 = 2x + 2
x
Therefore, y = 2x + 2 is the suitable straight line
2x + 2
The graph y = 8 is drawn. Determine the suitable straight line
x
to be drawn to solve each the equation: - 8 = -2x - 2
x
![Page 77: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/77.jpg)
2.2 The Solution of An Equation By Graphical
Method
12. (a) Complete Table 1 for the equation y = x2 + 2x by writing down the values of y
when x = -3 and 2.
x -5 -4 -3 -2 -1 0 1 2 3
y 15 8 0 -1 0 3 15
(b) By using scale of 2 cm to 1 unit on the x-axis and 2 cm to 2 units on the y-axis,
draw the graph of y = x2 + 2x for -5 ≤ x ≤ 3.
(c) From your graph, find
(i) the value of y when x = -3.5,
(ii) the value of x when y = 11.
(d) Draw a suitable straight line on your graph to find a value of x which satisfies
the equation of x 2 + x – 4 = 0 for -5 ≤ x ≤ 3.
TABLE 1
![Page 78: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/78.jpg)
12. (a)
x -5 -4 -3 -2 -1 0 1 2 3
y 15 8 0 -1 0 3 15
y = x2 + 2x
y = (-3 ) 2 + 2 (-3 ) = 3
y = ( 2 ) 2 + 2 ( 2 ) = 8
8
solution
3
![Page 79: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/79.jpg)
12. (a)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3
![Page 80: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/80.jpg)
12. (c)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3-3.5
5
y = 11
-4.4 2.5
Answer:
(i) y = 5.0
(ii) x = -4.4
x = 2.5
![Page 81: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/81.jpg)
12. (d)
x
x
x
xx
x
x
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3
y = x2 + 2x + 00 = x2 + x - 4-
y = x + 4
x 0 -4
y 4 0
x
x1.5-2.5
Answer:
(d) x = 1.5
x = -2.5
![Page 82: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/82.jpg)
![Page 83: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/83.jpg)
ax2 + bx + c = 0
x2 + x – 4 = 0
a = 1 b = 1 c = -4
MODE EQN
1 1Unknowns ?
2 3Degree?
2 3
2 a ? 1 = b ? 1 = c ?
(-) 4 x1 = 1.561552813 = x2 = -2.561552813
Press 3x
=
![Page 84: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/84.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 A Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
How can we determine whether a given point satisfies
y = 3x + 1, y < 3x + 1or y > 3x + 1 ?
![Page 85: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/85.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 A Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
Let us consider the point (3,5). The point can only satisfies one of the
following relations:
(a) y = 3x + 1 (b) y < 3x + 1 (c) y > 3x + 1
y 3x + 1
5 3(3) + 1
5 10
=
<
>
<
Since the y-coordinate of the point (3,5) is 5, which is less than 10,
we conclude that y < 3x + 1 . Therefore, the point (3,5) satisfies the relation
y < 3x + 1
<
![Page 86: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/86.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 A Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
Determine whether the following points satisfy y = 3x - 1, y < 3x - 1 or
y > 3x - 1.
(a) (1,-1) (b) (3,10) (c) (2,9)
EXAMPLE
![Page 87: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/87.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 A Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
For point (1,-1)
When x = 1, y = 3(1) - 1 = 2
Since the y-coordinate of the point (1,-1) is -1, which is less than 2,
we conclude that y < 3x - 1 . Therefore, the point (1,-1) satisfies the relation
y < 3x - 1
solution a
![Page 88: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/88.jpg)
2.3REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES
2.3 A Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
For point (3,10)
When x = 3, y = 3(3) - 1 = 8
Since the y-coordinate of the point (3,10) is 10, which is greater than 8,
we conclude that y > 3x - 1 . Therefore, the point (3,10) satisfies the relation
y > 3x - 1
solution b
![Page 89: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/89.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 A
For point (-1,-4)
When x = -1, y = 3(-1) - 1 = -4
Since the y-coordinate of the point (-1,-4) is -4, which is equal to -4,
we conclude that y = 3x - 1 . Therefore, the point (-1,-4) satisfies the relation
y = 3x - 1
solution c
Determining Whether a Given Point Satisfies y = ax + b,
y > ax + b or y < ax + b
![Page 90: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/90.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 B
Determining The Position of A Given Point Relative to
y = ax + b
All the points satisfying y < ax + b are below the graph
All the points satisfying y = ax + b are on the graph
All the points satisfying y > ax + b are above the graph
![Page 91: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/91.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 B
Determining The Position of A Given Point Relative to
y = ax + b
-2-4
2
4
6
8
x
y
20
-2
-4
4-6
-8
6
P(4,8)
Q(4,2)
y < xy > x
The point P(4,8) lies
above the line y = x.
This region is represented
by y > x
The point Q(4,2) lies
below the line y = x.
This region is represented
by y < x
Q(4,4)
The point Q(4,4) lies
on the line y = x.
This region is represented
by y = x
![Page 92: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/92.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 B
Determining The Position of A Given Point Relative to
y = ax + b
-2-4
2
4
6
8
x
y
20
-2
-4
4-8
-8
8
P(-8,6)
Q(4,4)
y < 3x + 2y > 3x + 2
The point P(-8,6) lies
above the line y = 3x + 2.
This region is represented
by y > 3x + 2
The point Q(4,4) lies
below the line y = 3x + 2.
This region is represented
by y < 3x + 2
Q(2,8)
![Page 93: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/93.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 C Identifying The Region Satisfying y > ax + b or y < ax + b
-2-4
2
4
6
8
x
y
20
-2
-4
4-8
-8
8
Determine whether the
shaded region in the graph
satisfies y < 3x + 2 or
y > 3x + 2
EXAMPLE
solution
The shaded region is
below the graph, y = 3x + 2.
Hence, this shaded region
satisfies y< 3x + 2
![Page 94: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/94.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 D Shading The Regions Representing Given Inequalities
Symbol Type of
Line
< or > Dashed
Line
≤ or ≥ Solid line
The type of line to be drawn depends
on inequality symbol
The table above shows the
symbols of inequality and the
corresponding type of line
to be drawn
HoT TiPs
The dashed line indicates that all points
are not included in the region. The solid
line indicates that all points on the line
are included
![Page 95: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/95.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 D Shading The Regions Representing Given Inequalities
0x
y
b
0x
y
b
0x
y
b
0x
y
b
y > ax + b
a > 0 y < ax + b
a > 0
y ≥ ax + b
a > 0 y ≤ ax + b
a > 0
![Page 96: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/96.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 D Shading The Regions Representing Given Inequalities
0x
y
by >ax + b
a < 0
0x
y
b y ≥ ax + b
a < 0
x
y
x
y
y ≤ ax + b
a < 0 b
0
y < ax + b
a < 0
0
b
![Page 97: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/97.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 D Shading The Regions Representing Given Inequalities
0
y
a
x >a
a > 0
x
y
a
x > a
a < 0
x
y
x
y
x ≤ a
a < 0 x ≤ a
a > 0
0a
x0
0 a
![Page 98: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/98.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES
2.3 EDetermine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
2
0 2 3-3
1
EXAMPLEShade the region that satisfies
3y < 2x + 6, 2y ≥ -x + 2 and x ≤ 3.
X = 3
![Page 99: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/99.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES
y
x
2
0 2 3-3
1
X = 3
Shade the region that satisfies
3y < 2x + 6, 2y ≥ -x + 2 and x ≤ 3.
![Page 100: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/100.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES
y
x
2
0 2 3-3
1
X = 3A
Region A satisfies 2y ≥ -x + 2, 3y < 2x + 6, and x ≤ 3
![Page 101: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/101.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 E Determine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
2
0 2 3-3
1
X = 3A
Region A satisfies
2y ≥ -x + 2,
3y < 2x + 6, and x ≤ 3
![Page 102: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/102.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 E Determine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
2
0 2 3-3
1
X = 3A
Region A satisfies
2y > -x + 2,
3y ≤ 2x + 6, and x < 3
![Page 103: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/103.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 E Determine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
3
0
x = 3
Region B satisfies
y ≥ -x + 3,
y < x , and x ≤ 3
B
![Page 104: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/104.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 E Determine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
3
0
x = 3
Region B satisfies
y > -x + 3,
y ≤ x , and x < 3
B
![Page 105: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/105.jpg)
2.3 REGION REPRESENTING INEQUALITIES IN TWO
VARIABLES2.3 E Determine The Region which Satisfies Two or More
Simultaneous Linear Inequalities
y
x
3
0
x = -3
Region C satisfies
y > -x + 3,
y ≤ -2x , and x >-3
-3
C
![Page 106: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/106.jpg)
x
x
x
xx
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3x
1.5-2.5
y = x2 + 2x
y = x + 4
xx
x
Region A satisfies
y ≥ x2 + 2x,
y ≤ x + 4,
and x ≥ 0
A
![Page 107: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/107.jpg)
x
x
x
xx
x
x
y
x0 1 2
2
4
6
8
10
-1-3-4-5
16
14
12
-2 3x
1.5-2.5
y = x2 + 2x
y = x + 4
xx
x
Region A satisfies
y ≥ x2 + 2x,
y ≤ x < 4,
and x ≥ 0
A
![Page 108: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/108.jpg)
Shade the region that satisfies
y ≤ 2x + 8, y ≥ x, and y < 8
y
x0
y = 8
y = x
y = 2x + 8
SPM Clone
y ≤ 2x + 8
y ≥ x
y < 8
![Page 109: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/109.jpg)
Shade the region that satisfies
y ≤ 2x + 8, y ≥ x, and y < 8
y
x0
y = 8
y = x
K1
y = 2x + 8 3
SPM Clone
P2
![Page 110: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/110.jpg)
3.y
x0
y = 8
y = x
K2
y = 2x + 8
2
![Page 111: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/111.jpg)
3.y
x0
y = x
K1
y = 2x + 8 1
![Page 112: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/112.jpg)
y
xO
y = 2x6
On the graphs provided, shade the region which satisfies
the three inequalities x < 3, y ≤ 2x – 6 and y ≥ -6
[3 marks]
y = 6
![Page 113: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/113.jpg)
y
xO
y = 2x6
y = 6
Solution:
x = 3
K3
x-intercept = -(-6 ÷2) = 3
x < 3, y ≤ 2x – 6 and y ≥ -6
![Page 114: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/114.jpg)
On the graphs provided, shade the region which satisfies
the three inequalities y ≤ x - 4, y ≤ -3x + 12 and y > -4
[3 marks]
y
x
y = 3x+12
O
y = x4
![Page 115: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/115.jpg)
y
x
y = 3x+12
O
y = x4
y = 4
Solution:
K3
y-intercept =-4
y ≤ x - 4, y ≤ -3x + 12 and y > -4
![Page 116: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/116.jpg)
SPM 2003 PAPER 2
REFER TO QUESTION
NO. 12
![Page 117: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/117.jpg)
12. ( a )
x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4
y 1 1.6 8 -8 -4 -1.25 -1
4
2
K1K1
y = -4
( )
=
y = -4
( )
=
-1
4 -2-2
![Page 118: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/118.jpg)
1 2 3 4-1-2-3-4 0
y
x
2
4
6
8
-2
-4
-6
-8
X
X
X
X
X
X
X
X
X
K1
K1N1
K1N1
12(b)
![Page 119: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/119.jpg)
12. ( a )
x -4 -2.5 -1 -0.5 0.5 1 2 3.2 4
y 1 1.6 8 -8 -4 -1.25 -14 2 K1K0
![Page 120: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/120.jpg)
1 2 3 4-1-2-3-4 0
y
x
2
4
6
8
-2
-4
-6
-8
X
X
XX
X
X
X
X
X
K1
K1N1
K1 N0
12(b)
![Page 121: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/121.jpg)
x
y
0 1 2 3 4
8
6
4
2
-2
-4
-6
-8
-4 -3 -2 -1
xx
x
x
x
x
x
12. ( c )
x
-2.2
-1.2
1.8
3.4
( i ) y = -2.2
( ii ) x = -1.2
P1
P1
![Page 122: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/122.jpg)
12. (d)
x
y
0 1 2 3 4
8
6
4
2
-2
-4
-6
-8
-4 -3 -2 -1
xx
x
x
x
x
x
x
y = -2x - 3
-2.4
0.8
K1K1
x = - 2.4
x = 0.8
N1
4 = 2x + 3
x
- 4 = -2x - 3
x
![Page 123: graphs of functions 2](https://reader031.vdocuments.net/reader031/viewer/2022013115/5597ddd01a28ab58388b45fa/html5/thumbnails/123.jpg)