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Vertical and Horizontal Transformations

Shifts of parent functions produced by adding a constant term.

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Transforming Functions

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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 \begin{align*}y = x^{2}\end{align*} , in order to graph \begin{align*}g(x) = 3(x - 4)^{2} + 2\end{align*} ?

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: \begin{align*}f(x) = x^2 + 3\end{align*}
  7. Write a function that will horizontally stretch the following: \begin{align*}f(x) = x^2 - 6\end{align*}
  8. Rewrite this function \begin{align*}f(x) = -\sqrt{x}\end{align*} to get a reflection over the x-axis.
  9. Rewrite this function \begin{align*}f(x) = \sqrt{x}\end{align*} to get a reflection over the y-axis.
Graph each of the following using transformations. Identify which transformations are used.
  1. \begin{align*}f(x) = |x-3| + 4\end{align*}
  2. \begin{align*}h(x) = \sqrt{x + 7}\end{align*}
  3. \begin{align*}g(x) = \frac{1}{x - 5}\end{align*}
  4. \begin{align*}f(x) = -3x^3\end{align*}
  5. \begin{align*}h(x) = (x - 7)^3 + 4\end{align*}
  6. \begin{align*}f(x) = \frac{1}{4}(x - 9)^2 + 5\end{align*}
  7. \begin{align*}f(x) = 3\sqrt{x + 2} - 6\end{align*}
  8. \begin{align*}f(x) = \frac{3}{4(x + 5)} + \frac{4}{5}\end{align*}
Answer the following questions:
  1. What part of the function \begin{align*}g(x) = -(f(x) + 1) = -(x^3 + 1)\end{align*} shifts the graph of \begin{align*}f(x)\end{align*}vertically?
  2. What part of the function \begin{align*}g(x) = -(f(x) + 1)\end{align*} reflects the graph of \begin{align*}f(x)\end{align*} across the x-axis?
  3. What is different between the functions \begin{align*}g(x) = -(x^3 + 1.0)\end{align*} and \begin{align*}h(x) = -x^3 + 1.0\end{align*} that changes the appearance of the graph?
  4. Write a function \begin{align*}g(x)\end{align*} whose graph looks like the graph of \begin{align*}f(x) = |x|\end{align*} reflected across the x-axis and shifted up 1 unit. \begin{align*}g(x) =\end{align*}
  5. How do you transform the graph of: \begin{align*}f(x) = x^3\end{align*} so that it looks like the graph of:\begin{align*}f(x) = 4x^3 + 6\end{align*} 
    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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