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Transformations of curves |
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CONTENTS |
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Once you have registered, you can work through the slides one by one.
The workout comprises a series of sides that guide you systematically through the topic concept by concept, skill by skill. The slides may be used with or without the support of a tutor. The methodology is based on problem-solving that advances in logical succession by concept and difficulty. The student is presented with a problem or series of questions, and the next slide presents the fully-worked solution. To use the material you must sign-in or create an account.
blacksacademy.net comprises a complete course in mathematics with resources that are comprehensive.
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SAMPLE FROM THE WORKOUT |
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SLIDE 1 - QUESTION 1 |
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SLIDE 2 - SOLUTION |
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SAMPLE FROM THE LIBRARY |
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QUESTION [difficulty 0.4] |
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SOLUTION |
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DEPENDENCIES |
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CONCEPTS |
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LEV. |
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Vertical translation in the completed square
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806.5 |
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Vertical translation problem of piecewise linear graph
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806.7 |
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Vertical translation of parabola
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806.9 |
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Vertical translation of hyperbola
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807.1 |
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Vertical translation of cubic
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807.3 |
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Horizontal translation in completed square
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807.5 |
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Horizontal translation in negative direction
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807.7 |
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Horizontal translation of any function
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807.9 |
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Horizontal translation of piecewise linear graph
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808.0 |
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Combination of horizontal and vertical translation
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808.4 |
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Moving the parabola by translation of min / max point
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808.6 |
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Finding translations of y = x^2 from completed square
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808.7 |
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Horizontal translation of hyperbola
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808.9 |
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Scale factor and scaling
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809.4 |
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Scalings of the parabola
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809.5 |
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Negative scale factor
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809.6 |
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Completed square form: 3 transformations of y = x^2
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810.0 |
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Transformations of a graph not order invariant
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810.4 |
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Scalings of piecewise linear graphs
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810.8 |
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Scaling of hyperbola
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811.4 |
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RAW CONTENT OF THE WORKOUT |
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Transformations of graphs
SLIDE 1
Consolidation – completing the square
Put into completed square form, and sketch the parabola .
SLIDE 2
SLIDE 3
and
? Sketch on the same graph the functions
? Compare the graphs of and .
How, if at all, does the shape of the parabola differ in the graphs of the two functions?
What difference or differences are there between the graphs of and ?
? What is the effect of adding 2 to to obtain on the parabola ?
SLIDE 4
The parabolas and have exactly the same shape. The parabola itself has not be altered by change from to . The only difference is that has been shifted upwards by . The effect of adding 2 to and obtaining is a vertical translation upwards of the parabola by .
SLIDE 5
Vertical translation
The effect of adding to the function is a vertical translation of the parabola by .
? What is the function obtained by adding to ?
? If , what is the effect on the graph of of adding ?
SLIDE 6
? Adding to we obtain the function
? If and then the graph of
is a vertical translation downwards by .
SLIDE 7
The diagram shows the graph of a function . Onto the same diagram, sketch the graphs of ? and ? and in each case state the coordinates of the vertex.
SLIDE 8
?
?
SLIDE 9
The diagram shows the graph of
? Complete the square on
? The function has no roots. Explain how the graph shows this.
? has one root. Find .
? Add to the sketch, state the coordinates of the maximum of and find .
SLIDE 10
?
? The graph of does not cut the x-axis, and so has no roots.
?
? ,
SLIDE 11
The diagram shows the graph of
? Add to the diagram a sketch of the graph .
? State the asymptotes of
? . Find .
SLIDE 12
Asymptotes of
SLIDE 13
The diagram shows the graph of the cubic function
? State the coordinates of the intercept of on the y-axis.
? State a root and corresponding factor of .
? The function is such that it has exactly two roots. Find .
? Add to the sketch. State the coordinates of the minimum, maximum and intercept of .
SLIDE 14
? The intercept of lies at .
? cuts the x-axis at . A root is and a factor is .
? If has exactly two roots, the graph of must be shifted down so that the minimum touches the x-axis. Vertical translation by .
? . The max, min and intercept of are all shifted down by 41 to , and intercept respectively.
SLIDE 15
The diagram shows the graph of . On a copy of the diagram draw the graph of . What is the relationship between the two graphs?
SLIDE 16
The graph of is the horizontal translation of the graph of in the positive direction by .
SLIDE 17
The diagram shows the graph of . On a copy of the diagram draw the graph of . Describe the transformation of the graph of into the graph of .
SLIDE 18
The graph of is the horizontal translation of the graph of in the negative direction by .
SLIDE 19
Horizontal translation
The graph of the function is the horizontal translation by . It shifts the graph of by .
SLIDE 20
The diagram shows the graph of . On a copy of the diagram draw the graphs of .
SLIDE 21
SLIDE 22
The diagram shows the graph of . On a copy of the diagram draw the graphs of .
SLIDE 23
SLIDE 24
The diagram shows the graph of . On a copy of the diagram draw the graph of .
SLIDE 25
SLIDE 26
The parabola is obtained from the parabola by a horizontal translation by and a vertical translation by . The minimum of is at the origin. The minimum of lies at .
SLIDE 27
Sketch the graph of . Mark onto the sketch the minimum. What is the transformation that takes to ?
SLIDE 28
The graph of is obtained from the graph of by a horizontal translation of and a vertical translation of .
SLIDE 29
The diagram shows the graph of
? Add to the diagram a sketch of the graph .
? State the asymptotes of .
? Complete the following mapping diagram for .
SLIDE 30
The asymptotes of are and .
SLIDE 31
The graph of is obtained from the graph of by a horizontal translation of and a vertical translation of .
SLIDE 32
? Sketch the graph of .
? How is the graph of related to the graph of ?
? What are the asymptotes of ?
? Add to your diagram a sketch of the graph of .
? . Find .
SLIDE 33
The graph of is obtained from the graph of by a horizontal translation of and a vertical translation of . The asymptotes of are and .
. Given , then .
SLIDE 34
On the same diagram draw the graphs of and . How does the scale factor of 2 transform the graph of ?
SLIDE 35
The y¬-coordinate of is, for any given value of x, twice the value of the y-coordinate of . The effect of the scale factor of is to make the parabola of steeper.
SLIDE 36
On the same diagram draw the graphs of and . How does the scale factor of transform the graph of ?
SLIDE 37
The y¬-coordinate of is, for any given value of x, half the value of the y-coordinate of . The effect of the scale factor of is to make the parabola of less steep.
SLIDE 38
On the same diagram draw the graphs of and . How does the negative scale factor of transform the graph of ?
SLIDE 39
The effect of the negative scale factor of is to reflect the parabola of in the x-axis.
SLIDE 40
The quadratic function has completed square form .
By completing the square on state the relation between the coefficients, in and the coefficients in .
SLIDE 41
By equating coefficients in , we obtain
SLIDE 42
How may the graph of be obtained from the graph of by three transformations? Make a diagram showing a sketch of both and to illustrate the transformations.
SLIDE 43 – SOLUTION AND EXPLANATION
The graph of is obtained from the graph of by the following three transformations.
? A horizontal translation by .
? A scaling by scale factor .
? A vertical translation by .
SLIDE 44
The graph of is subject to
? (1) A horizontal translation by , (2) a scaling by scale factor , followed by (3) a vertical translation by .
? (1) A horizontal translation by , followed by (2) a vertical translation by , (3) a scaling by scale factor .
Determine in terms of the outcome if each. State in each case, the coordinates of the minimum point of the resultant transformed parabola. What does this example teach us about the order in which transformations are made?
SLIDE 45
? (1) A horizontal translation by , (2) a scaling by scale factor , followed by (3) a vertical translation by .
? (1) A horizontal translation by , followed by (2) a vertical translation by , (3) a scaling by scale factor .
While the axis of symmetry of the transformed graph is the same in both cases, the vertical translation is not. In the first case, the min is at and in the second case, it is at .
Hence, the order in which the transformations are made makes a difference to the final graph.
SLIDE 46
In each case the function is the result of three transformations of the parabola . State in order what those three transformations are executed.
?
?
SLIDE 47
?
This is the completed square, and is the result of the following transformations in order:
(1) a horizontal translation by
(2) a scaling by scale factor
(3) a vertical translation by
?
(1) a scaling by scale factor ?
(2) a horizontal translation by ?
(3) a vertical translation by ?
SLIDE 48
The graph of the function is shown above. Onto the same diagram, draw the graphs of
? ?
SLIDE 49
? ?
SLIDE 50
The graph of the function is shown above. Draw on to the diagram a sketch of .
SLIDE 51
SLIDE 52
The graph of the function is shown above. is subject to (1) a horizontal translation by , (2) a scaling by scale factor and (3) a vertical translation by . The result is . Find in terms of and add to the diagram a sketch of its graph.
SLIDE 53
(1) a horizontal translation by ?
(2) a scaling by scale factor ?
(3) a vertical translation by ?
SLIDE 54
The diagram shows the graph of .
Sketch onto the diagram the graph of .
SLIDE 55
is the vertical scaling of by scale factor 2.
SLIDE 54
The diagram shows the graph of .
Sketch onto the diagram the graph of .
SLIDE 55
is obtained from by (1) a vertical scaling by factor . This is equivalent to a reflection in the x-axis. This is followed by (2) a vertical translation by . The asymptotes of are and .
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