introduction this chapter focuses on sketching graphs we will also be looking at using them to solve...
TRANSCRIPT
![Page 1: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/1.jpg)
![Page 2: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/2.jpg)
Introduction
• This Chapter focuses on sketching Graphs
• We will also be looking at using them to solve Equations
• There will also be some work on Graph transformations
![Page 3: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/3.jpg)
![Page 4: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/4.jpg)
Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
A cubic equation will take one of the following shapes
For any x3
For any -x3
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
( 2)( 1)( 1)y x x x
If y = 0
0 ( 2)( 1)( 1)x x x
So x = 2, 1 or -1(-1,0) (1,0) and (2,0)
If x = 0
(0 2)(0 1)(0 1)y
So y = 2
(0,2)
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
( 2)( 1)( 1)y x x x
(-1,0) (1,0) (2,0) (0,2)
x
y
2
2
-1
1
If we substitute in x = 3, we get a value of y = 8. The curve must be increasing after this
point…
![Page 7: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/7.jpg)
Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
( 2)(1 )(1 )y x x x
If y = 0
0 ( 2)(1 )(1 )x x x
So x = 2, 1 or -1(-1,0) (1,0) and (2,0)
If x = 0
(0 2)(1 0)(1 0)y
So y = -2
(0,-2)
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
( 2)(1 )(1 )y x x x
(-1,0) (1,0) (2,0) (0,-2)
x
y
2
-2
-1
1
If we substitute in x = 3, we get a value of y = -8. The curve must be decreasing after this
point…
![Page 9: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/9.jpg)
Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
2( 1) ( 1)y x x
If y = 0
20 ( 1) ( 1)x x So x = 1 or -1
(-1,0) and (1,0)
If x = 0
2(0 1) (0 1)y So y = 1
(0,1)
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
2( 1) ( 1)y x x
(-1,0) (1,0) (0,1)
x
y
1
-1
1
If we substitute in x = 2, we get a value of y = 3. The curve must be increasing after this
point…
‘repeated root’
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
3 22 3y x x x
If y = 0
0 ( 3)( 1)x x x So x = 0, 3 or -1
(0,0) (3,0) and (-1,0)
If x = 0
0(0 3)(0 1)y So y = 0
(0,0)
2( 2 3)y x x x
( 3)( 1)y x x x
Factorise
Factorise fully
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Sketching CurvesSketching Cubics
You need to be able to sketch equations of the form:
This involves finding the places where the graph crosses the axes, in the same way you do when sketching a Quadratic.
4A
3 2y ax bx cx d
( )( )( )y x a x b x c
or
ExampleSketch the graph of the function:
3 22 3y x x x
(0,0) (3,0) (-1,0)
x
y
0-1
3
If we substitute in x = 4, we get a value of y = 20. The curve must be increasing after this
point…
![Page 13: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/13.jpg)
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Sketching Curves
Sketching CubicsYou need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
4B
ExampleSketch the graph of the function:
3y x
x
yy = x3
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Sketching Curves
Sketching CubicsYou need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
4B
ExampleSketch the graph of the function:
3y x
x
yy = x3
y = -x3
A cubic with a negative ‘x3’ will be reflected in the x-axis
‘Whatever you get for x3, you now have the negative of
that..’
5
-5
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Sketching Curves
Sketching CubicsYou need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
4B
ExampleSketch the graph of the function:
3( 1)y x
x
yy = x3
When a value ‘a’ is added to a cubic, inside a bracket, it is a
horizontal shift of ‘-a’
‘I will now get the same values for y, but with values of x that
are 1 less than before’
y = (x + 1)3
1
When x = 0:
3(0 1)y 1y
y-intercept
-1
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Sketching Curves
Sketching CubicsYou need to be able to sketch and interpret cubics that are variations of y = x3
This will be covered in more detail in C2. You can still plot the graphs in the same way we have seen before. This topic is offering a ‘shortcut’ if you can understand it.
4B
ExampleSketch the graph of the function:
3(3 )y x
x
yy = x3
y = (3 - x)3
27
When x = 0:
3(3 0)y 27y
y-intercept
3(3 )y x 3( 3)y x
Reflected in the x-
axis
Horizontal shift, 3 to the right
3
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Sketching CurvesThe Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
ky
x
ExampleSketch the graph of the function 1
yx
and its asymptotes.
124-4-2-1y
10.50.25-0.25-0.5-1x x
y
y = 1/x
4C
You cannot divide by 0, so you get no value at this point
These are where the
graph ‘never reaches’, in this case the
axes…
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Sketching CurvesThe Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
ky
x
ExampleSketch the graph of the function 3
yx
and its asymptotes.
x
y
y = 1/x
4C
y = 3/x
The curve will be the same, but further out…
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Sketching CurvesThe Reciprocal Function
You need to be able to sketch the ‘reciprocal’ function. This takes the form:
Where ‘k’ is a constant.
ky
x
ExampleSketch the graph of the function 1
yx
and its asymptotes.
x
y
y = 1/x
4C
y = -1/x
The curve will be the same, but reflected in
the x-axis
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Sketching CurvesSolving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
ExampleOn the same diagram, sketch the
following curves:
4D
( 3)y x x 2 (1 )y x x and
x
y
( 3)y x x Quadratic ‘U’ shapeCrosses through 0 and 3
0 3
( 3)y x x
2 (1 )y x x Cubic ‘negative’ shapeCrosses through 0
and 1. The ‘0’ is repeated so just
‘touched’
1
2 (1 )y x x
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Sketching CurvesSolving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
ExampleOn the same diagram, sketch the
following curves:
4D
( 3)y x x 2 (1 )y x x andFind the co-ordinates of the points of
intersection
These will be where the graphs are equal…
x
y
0 3
( 3)y x x
1
2 (1 )y x x
2( 3) (1 )x x x x 2 2 33x x x x 3 3 0x x 2( 3) 0x x
Expand bracketsGroup
together
Factorise
0x 2 3 0x 2 3x
3x
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Sketching CurvesSolving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
ExampleOn the same diagram, sketch the
following curves:
4D
( 3)y x x 2 (1 )y x x andFind the co-ordinates of the points of
intersection
These will be where the graphs are equal…2( 3) (1 )x x x x
2 2 33x x x x 3 3 0x x 2( 3) 0x x
Expand bracketsGroup
together
Factorise
0x 2 3 0x 2 3x
3x
( 3)y x x
x=-√3 x=0 x=√3
( 3)y x x ( 3)y x x ( 3)y x x
3( 3 3)y 0(0 3)y
0y 3 3 3y
3( 3 3)y
3 3 3y
(0,0)(-√3 , 3+3√3) (√3 , 3-3√3)
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Sketching CurvesSolving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
ExampleOn the same diagram, sketch the
following curves:
4D
2 ( 1)y x x 2
yx
and
x
y
2 ( 1)y x x Cubic ‘positive’ shapeCrosses through 0
and 1. The ‘0’ is repeated.
02
yx
Reciprocal ‘positive’ shape
Does not cross any axes
1
y = 2/x
![Page 27: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/27.jpg)
Sketching CurvesSolving Equations and Sketching
You need to be able to sketch 2 equations on a set of axes, as well as solve equations based on graphs.
How does the graph show there are 2 solutions to the equation..
ExampleOn the same diagram, sketch the
following curves:
4D
2 ( 1)y x x 2
yx
and
x
y
0 1
y = 2/x2 2( 1) 0x x
x
2 2( 1)x x
x
2 2( 1) 0x x
x
Set equations equal, and re-
arrange
And they cross in 2 places…
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Sketching CurvesMore Transformations
You have seen that a curve with the following function:
Will be transformed horizontally ‘-a’ units.
A curve with this function:
Will be transformed vertically ‘a’ units
4E
( )f x a
( )f x a
f(x)f(x + 2)
f(x) + 2
x
y
2 units left
2 units up
f(x + 2) The x values reduce by 2 for the same y values
f(x) + 2 The y values from the original function increase by 2
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Sketching CurvesMore Transformations
Sketch the following functions:
f(x) = x2
Standard curve Label known points
g(x) = (x + 3)2
Moved 3 units left Work out new ‘key points’
h(x) = x2 + 3 Moved 3 units up Work out new ‘key points’
4E
x
y f(x)
0
x
y g(x)
-3x
y h(x)
3
9
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Sketching CurvesMore Transformations
Given that:
i) f(x) = x3
Sketch the curve where y = f(x - 1). State any locations where the graphs crosses the axes.
f(x) = x3
f(x – 1) = (x – 1)3
So for this curve, when x = 0, y = -1It therefore crosses at y = -1
4E
f(x)
0x
y
f(x – 1)
1-1
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Sketching CurvesMore Transformations
Given that:
i) g(x) = x(x – 2)
Sketch the curve where y = g(x + 1). State any locations where the graphs crosses the axes.
g(x) is a positive quadratic crossing at 0 and 2.
g(x) = x(x – 2)
g(x + 1) = (x + 1)(x + 1 – 2) g(x + 1) = (x + 1)(x – 1)
So for this curve, when x = 0, y = -1It therefore crosses at y = -1
4E
g(x)
0x
y
1-1 2
g(x + 1)
x’s replaced with ‘x + 1’
-1
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Sketching CurvesMore Transformations
Given that:i) h(x) = 1/x
Sketch the curve where y = h(x) + 1. State any locations where the graphs crosses the axes and the equations of any asymptotes.
h(x) is a positive reciprocal graph
h(x) = 1/x
h(x) + 1 = 1/x + 1
The asymptotes are: x = 0 (the y-axis)
y = 1
It will cross the x-axis at -1 since this value will make the equation = 0
4E
h(x)
x
y
1
-1
h(x) + 1
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Sketching CurvesEven more Transformations
You also need to be able to perform transformations of the form:
this is a horizontal stretch of 1/a.
You also need to know:
this is a vertical stretch by factor ‘a’
4F
( )f ax
( )af x
(2 )y f x( )y f x‘We will get the same y values, using half the x
values’
This is because the x values get multiplied by 2
before the y values are worked out
2 ( )y f x( )y f x‘We will get y values twice as big, using the same x
values’
This is because when we work out the y values,
they are doubled after
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Sketching CurvesEven more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = f(2x)
Sketch the original curve, working out key points.
If x = 0
If y = 0
4F
x
y f(x)
-3 3
9
29y x
(3 )(3 )y x x
9y
29y x
0 (3 )(3 )x x
(0,9)
(3,0) (-3,0)
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Sketching CurvesEven more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = f(2x)
Substitute ‘2x’ in place of ‘x’
If x = 0
If y = 0
4F
x
y f(x)
-3 3
9
x
y f(2x)
-1.5 1.5
929 (2 )y x
(3 2 )(3 2 )y x x
9y
29 4y x
0 (3 2 )(3 2 )x x
(0,9)
(-1.5,0)
(1.5,0)
29 4y x
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Sketching CurvesEven more Transformations
Given that f(x) = 9 – x2, sketch the curve with equation;
a) y = 2f(x)
f(x), the original equation, is doubled..
If x = 0
If y = 0
4F
x
y f(x)
-3 3
9
x
y 2f(x)
-3 3
18
29y x
2(3 )(3 )y x x
18y
22(9 )y x
0 2(3 )(3 )x x
(0,18)
(3,0) (-3,0)
22(9 )y x
![Page 39: Introduction This Chapter focuses on sketching Graphs We will also be looking at using them to solve Equations There will also be some work on Graph transformations](https://reader036.vdocument.in/reader036/viewer/2022081519/56649f415503460f94c6064b/html5/thumbnails/39.jpg)
Summary
• We have learnt the shapes of several different curves
• We have learnt how to apply transformations to those curves
• We have also looked at how to work out the ‘key points’