quadratic transformations

QUADRATIC TRANSFORMATIONSMrs. Aldous, Mr. Beetz & Mr. ThauvetteIB DP SL Mathematics

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

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

Geogebra Transformations

Open Geogebra and click on the ‘slider’ option

Click in the coordinate plane to create a slider Change your Interval

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

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

Geogebra Transformations Use the slider for ‘a’ and explore what

happens as the value of ‘a’ changes

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

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

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

Transformations WorksheetThe graph of is shown below.

The graph of is shown below.

Draw the required graph.(a) (c)

(b)

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).

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).

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.

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’.

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).

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)

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

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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’.

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.

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

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.