When performing multiple transformations, it is very easy to make a small error. This is especially true when you try to do every step mentally. Point notation is a useful tool for concentrating your efforts on a single point and helps you to avoid making small mistakes.
What would
Using Function Notation and Point Notation
A transformation can be written in function notation and in point notation. Function notation is very common and practical because it allows you to graph any function using the same basic thought process it takes to graph a parabola in vertex form.
Another way to graph a function is to transform each point one at a time. This method works well when a table of
Essentially, it takes each coordinate
This notation is called point notation. The new
The new
The function notation and point notation representations of the transformation "Horizontal shift right three units, vertical shift up 4 units" are
Notice that the operations with the
Apply the transformation above to the following table of points.


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Notice that point notation greatly reduces the mental visualization required to keep all the transformations straight at once.
Examples
Example 1
Earlier, you were asked what the function
Example 2
Convert the following function in point notation to words and then function notation.
Horizontal stretch by a factor of 3 and then a horizontal shift right one unit. Vertical reflection over the
Example 3
Convert the following function notation into words and then point notation. Finally, apply the transformation to three example points.
Vertical reflection across the
Example 4
Convert the following function notation into point notation and apply it to the included table of points


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The
Example 5
Convert the following point notation to words and to function notation and then apply the transformation to the included table of points.


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This problem is different because it seems like there is a transformation happening to the original left point. This is an added layer of challenge because the transformation of interest is just the difference between the two points. Notice that the
Review


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5. \begin{align*}k(x)\rightarrow k(x3)\end{align*}
Convert the following functions in point notation to function notation.
6. \begin{align*}(x,y) \rightarrow \left ( \frac{1}{2} x+3,y4 \right )\end{align*}
7. \begin{align*}(x,y) \rightarrow (2x+4,y+1)\end{align*}
8. \begin{align*}(x,y) \rightarrow (4x,3y5)\end{align*}
9. \begin{align*}(2x,y) \rightarrow (4x,  y+1)\end{align*}
10. \begin{align*}(x+1,y2) \rightarrow (3x+3, y+3)\end{align*}
Convert the following functions in function notation to point notation.
11. \begin{align*}f(x) \rightarrow 3f(x2)+1\end{align*}
12. \begin{align*}g(x) \rightarrow 4g(x1)+3\end{align*}
13. \begin{align*}h(x) \rightarrow \frac{1}{2} h(2x+2)5\end{align*}
14. \begin{align*}j(x) \rightarrow 5j \left ( \frac{1}{2} x2 \right )1\end{align*}
15. \begin{align*}k(x) \rightarrow \frac{1}{4} k(2x4)\end{align*}
Review (Answers)
To see the Review answers, open this PDF file and look for section 1.3.