Basic transformations of graphs

Сентябрь 13, 2022

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Математика
Basic transformations of graphs

Содержание

2. Lecture overview: learning outcomes At the end of this lecture, you should be able to: 1.5.1
3. 1.5.1 : Sketch the graph of cubic function given in factorized form. Foundation Year Program 2017-18
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6. Foundation Year Program 2017-18 2A
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8. Foundation Year Program 2017-18 Transformations We will now consider four elementary transformations and some simple combined
9. 1.5.2. Horizontal translation Foundation Year Program 2017-18 f(x) → f(x + a) Shape is unchanged. Moves
10. 1.5.3. Vertical translation Foundation Year Program 2017-18 Shape is unchanged. Moves upwards if a>0. Moves downwards
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15. 1.5.4. Vertical stretch Foundation Year Program 2017-18 Roots unchanged Axis of symmetry unchanged Intercept at (0,
16. 1.5.5 Horizontal stretch Foundation Year Program 2017-18 f(x) = x2 - 4 h(x) = (0.5x)2 -
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20. 1.5.6 Simple combined transformations Foundation Year Program 2017-18
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29. In this lecture, we covered 1.5.1 Sketch the graph of a cubic function given in factorized
30. Next lecture 1.6 Simultaneous equations Foundation Year Program 2017-18
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Lecture overview: learning outcomes
At the end of this lecture, you should

be able to:
1.5.1 Sketch the graph of a cubic function given in factorized form.
1.5.2 Apply a horizontal translation to a given curve.
1.5.3 Apply a vertical translation to a given curve.
1.5.4 Apply a vertical stretch to a given curve.
1.5.5 Apply a horizontal stretch to a given curve.
1.5.6 Apply simple combined transformations to a given curve.

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2017-18

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1.5.1 : Sketch the graph of cubic function given in factorized

form.

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2A

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Transformations
We will now consider four elementary transformations and some

simple combined transformations.
f(x) → f(x+a) Translation (x-axis)
f(x) → f(x)+a Translation (y-axis)
f(x) → f(ax) Scaling (stretching)(x-axis)
f(x) → af(x) Scaling (stretching)(y-axis)

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1.5.2. Horizontal translation
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f(x) → f(x + a)
Shape is unchanged.
Moves

to the left if a>0.
Moves to the right if a<0.

h(x) =(x+2)2 - 4

f(x) = x2 - 4

g(x) =(x-2)2 - 4

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1.5.3. Vertical translation
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Shape is unchanged.
Moves upwards if a>0.
Moves downwards

if a<0.

h(x) = x2 - 8

g(x) = x2

f(x) = x2 - 4

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1.5.4. Vertical stretch
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Roots unchanged
Axis of symmetry unchanged
Intercept at (0,

c) is transformed to (0, ac)
If a < 0, the branches of transformed curve are reoriented from upward to downward or vice versa from original.

f(x) = x2 - 4

h(x) = -0.5(x2 – 4)

g(x) = 2(x2 – 4)

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1.5.5 Horizontal stretch
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f(x) = x2 - 4
h(x) = (0.5x)2

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Even power. Location of roots changes if |a|≠ 1. Axis of symmetry unchanged.

g(x) = (2x)2 - 4

f(x) = x3 – 3x2 – x + 2

g(x) = (1.2x)3 – 3(1.2x)2 –(1.2x) + 2

h(x) = (0.8x)3 – 3(0.8x)2 –(0.8x) + 2

Odd power.

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1.5.6 Simple combined transformations
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In this lecture, we covered
1.5.1 Sketch the graph of a cubic

function given in factorized form.
1.5.2 Apply a horizontal translation to a given curve.
1.5.3 Apply a vertical translation to a given curve.
1.5.4 Apply a vertical stretch to a given curve.
1.5.5 Apply a horizontal stretch to a given curve.
1.5.6 Apply simple combined transformations to a given curve.

Foundation Year Program

2017-18

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