Download - Quadratic Transformations
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QUADRATIC TRANSFORMATIONSMrs. Aldous, Mr. Beetz & Mr. ThauvetteIB DP SL Mathematics
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You should be able to… Identify and describe the following
transformations: translations, reflections and stretches
Find the equation of the image function following one or more transformations (combinations of transformations on functions)
Sketch the image of a function under a transformation
Give a full geometric description of the transformation(s) that map a function or its graph onto its image
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Function Transformations
The function can have the following forms
Each parameter (a, b, c, d) and ‘-’ sign have a different effect on the graph of the ‘parent’ or ‘base’ function
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Geogebra Transformations
Open Geogebra and click on the ‘slider’ option
Click in the coordinate plane to create a slider Change your Interval
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Geogebra Transformations
Repeat this 4 times Should have 4 sliders
on your screen
Must have 4 sliders first Cannot complete
task without have the sliders first
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Geogebra Transformations In the ‘input’ bar
on bottom left, type the following Examine the graph
that shows up and key features
Now, type the following in the input bar
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Geogebra Transformations Use the slider for ‘a’ and explore what
happens as the value of ‘a’ changes
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Transformations Take the function you had for the
‘Functions Gallery” activity On Geogebra, explore the different forms a
function can take on and come up with a rule for each
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Communicate Your Understanding If , describe the difference in the
graphs of and . If , describe the difference in the
graphs of and . The graph of is shown. Describe
how the coordinates of the points on each of the following graph
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Communicate Your Understanding Identify the combination of
transformations on that results in the given function
(a) (b)
Describe how you would graph the function
Describe how you would graph the function
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Transformations WorksheetThe graph of is shown below.
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The graph of is shown below.
Draw the required graph.(a) (c)
(b)
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This function will stretch the graph of f(x) vertically away from the x-axis by a factor of 2. As such, all points (x, y) will be mapped onto (x, 2y).
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This function will stretch the graph of f(x) horizontally away from the y-axis by a factor of 2. As such, all points (x, y) will be mapped onto (2x, y).
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This function will translate the graph of f(x) horizontally to the right by 3 units. As such, all points (x, y) will be mapped onto (x + 3, y). Notice that the shape of the graph does not change.
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The graph of is shown below.
(d) The point A(3, –1) is on the graph of f. The point A’ is the corresponding point on the graph of . Find the coordinates of A’.
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This function will reflect the graph of f(x) in the x-axis, translate it to the left by 1 unit and down by 2 units. As such, all points (x, y) will be mapped onto (x – 1, –y – 2). Therefore, (3, –1) (3 – 1, –(–1) – 2) (2, –1). So A’ has coordinates (2, –1).
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You should know…
A translation is described by a vector , which
shifts a graph horizontally by units and vertically by units without changing the shape of the graph A function of the form
translates the graph of by the vector . All points (x, y)
are mapped onto (x + p, y + q)
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You should know… A function of the form represents a
vertical stretch by a scale factor of . When is greater than 1 or less than –1 the graph moves away from the x-axis and it moves towards the x-axis when is between –1 and 1; all points (x, y) are mapped onto (x, ay) and the shape of the graph is changedStret
chMathemati
calForm
Effect
Vertical
If a > 1, then expand the graph vertically by a factor of ‘a’.If 0 < a < 1, then compress the graph vertically by a factor of ‘a’.
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You should know… A function of the form represents a
horizontal stretch by a scale factor of . The graph moves away from the y-axis when b is a value between –1 and 1 and it moves towards the y-axis when b is greater than 1 or less than –1; all points (x, y) are mapped onto and the shape of the graph is changed
Stretch
MathematicalForm
Effect
Horizontal
If b > 1, then compress the graph horizontally by a factor of 1/b.If 0 < b < 1, then expand the graph horizontally by a factor of 1/b.
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You should know… A function of the form
represents a reflection in the x-axis such that all points (x, y) are mapped onto (x, –y); the shape of the graph is unchanged
A function of the form represents a reflection in the y-axis such that all points (x, y) are mapped onto ( –x, y); the shape of the graph is unchanged
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Be prepared… Horizontal transformations such as
stretches can be tricky. Remember that a horizontal stretch by a factor of b > 1 “stretches” the graph towards the y-axis.