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Quadratic Transformations

Quadratic Transformations. Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics. You should be able to…. Identify and describe the following transformations: translations, reflections and stretches

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Quadratic Transformations

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  1. Quadratic Transformations Mrs. Aldous, Mr. Beetz & Mr. ThauvetteIB DP SL Mathematics

  2. 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

  3. 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

  4. Geogebra Transformations • Open Geogebra and click on the ‘slider’ option • Click in the coordinate plane to create a slider • Change your Interval

  5. 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

  6. 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

  7. Geogebra Transformations • Use the slider for ‘a’ and explore what happens as the value of ‘a’ changes

  8. 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

  9. 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

  10. 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

  11. Transformations Worksheet The graph of is shown below.

  12. The graph of is shown below. Draw the required graph. (a) (c) (b)

  13. 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).

  14. 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).

  15. 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.

  16. 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’.

  17. 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).

  18. 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)

  19. 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 changed

  20. 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

  21. 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

  22. 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.

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