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Combining Transformations

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Transforming Functions

Vocabulary

Word Definition
____________ Also called translation or slide; a transformation applied to the graph of a function which does not change the shape of the graph, only the location
____________ a result of adding a constant term to the value of a function; moves up or down
Horizontal Shift _________________________________________________________________

Reflection

_________________________________________________________________
____________ transformation which results in the width of a graph being increased or decreased; the result of the co-efficient of the term being between 0 and 1.
Compression _________________________________________________________________

What transformations must be applied to y = x^{2} , in order to graph g(x) = 3(x - 4)^{2} + 2 ?

Practice

Answer the following questions:
  1. If a function is multiplied by a coefficient, what will happen to the graph of the function?
  2. What does multiplying x by a number greater than one create?
  3. What happens when we multiply x by a number between 0 and 1
  4. In order to obtain a reflection over the y axis what do we have to do to x?
  5. How do we obtain a reflection over the x-axis?
  6. Write a function that will create a horizontal compression of the following: f(x) = x^2 + 3
  7. Write a function that will horizontally stretch the following: f(x) = x^2 - 6
  8. Rewrite this function f(x) = -\sqrt{x} to get a reflection over the x-axis.
  9. Rewrite this function f(x) = \sqrt{x} to get a reflection over the y-axis.
Graph each of the following using transformations. Identify which transformations are used.
  1. f(x) = |x-3| + 4
  2. h(x) = \sqrt{x + 7}
  3. g(x) = \frac{1}{x - 5}
  4. f(x) = -3x^3
  5. h(x) = (x - 7)^3 + 4
  6. f(x) = \frac{1}{4}(x - 9)^2 + 5
  7. f(x) = 3\sqrt{x + 2} - 6
  8. f(x) = \frac{3}{4(x + 5)} + \frac{4}{5}
Answer the following questions:
  1. What part of the function g(x) = -(f(x) + 1) = -(x^3 + 1) shifts the graph of f(x)vertically?
  2. What part of the function g(x) = -(f(x) + 1) reflects the graph of f(x) across the x-axis?
  3. What is different between the functions g(x) = -(x^3 + 1.0) and h(x) = -x^3 + 1.0 that changes the appearance of the graph?
  4. Write a function g(x) whose graph looks like the graph of f(x) = |x| reflected across the x-axis and shifted up 1 unit. g(x) =
  5. How do you transform the graph of: f(x) = x^3 so that it looks like the graph of:f(x) = 4x^3 + 6 
    1. Stretch it by a factor of ¼ and shift it up 6 units. 
    2. Stretch it by a factor of 6 and shift it left 4 units. 
    3. Stretch it by a factor of 4 and shift it down 6 units. 
    4. Stretch it by a factor of 4 and shift it up 6 units.

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