1. Basic Transformations of Functions and Graphs
We will be looking at simple functions and seeing how various modifications to the functions transform them.

2. VERTICAL TRANSLATIONS
Above is the graph of
As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function.
VERTICAL TRANSLATIONS
What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).
What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them).

3. VERTICAL TRANSLATIONS
So the graph of f(x) + k, where k is any real number is the graph of f(x) but vertically shifted by k. If k is positive it will shift up. If k is negative it will shift down.
VERTICAL TRANSLATIONS
Above is the graph of
What would f(x) + 2 look like?
What would f(x) - 4 look like?

4. HORIZONTAL TRANSLATIONS
Above is the graph of
As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but in the opposite way of the sign of the number.
What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).
What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function).

5. HORIZONTAL TRANSLATIONS
So the graph f(x-h), where h is any real number is the graph of f(x) that is horizontally shifted by h. Notice f(x-h) yields a shift to the right. (If you set what is in parenthesis = 0 & solve it will tell you how to shift the graph along x axis).
shift right 3
Above is the graph of
What would f(x+1) look like?
So shift along the x-axis by 3
What would f(x-3) look like?

6. We could have a function that is transformed or translated both vertically AND horizontally.
up 3
left 2
Above is the graph of in
Blue
What would the graph of look like?

7. DILATIONS
and
If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number.
Let's try some functions multiplied by non-zero real numbers to see this.

8. So the graph a⦁f(x), where a is any real number greater than 1, is the graph of f(x) but vertically stretched or dilated by a factor of a.
Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.
Above is the graph of
What would 2f(x) look like?
What would 4f(x) look like?

9. What if the value of a was positive but less than 1?
So the graph a f(x), where a is 0 < a < 1, is the graph of f(x) but vertically compressed or dilated by a factor of a.
Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value.
Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value.
Above is the graph of
What would 1/2 f(x) look like?
What would 1/4 f(x) look like?

10. What if the value of a was negative?
So the graph - f(x) is a reflection about the x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function across the x-axis)
Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.
Above is the graph of
What would - f(x) look like?

11. Summary of Transformations So Far
If a > 1, then vertical stretch by a factor of a
If 0 < a < 1, then vertical compression by a factor of a
If a < 0, then reflection about the x-axis also exists
vertical translation of k
horizontal translation of h (opposite sign of what’s in parenthesis)

12. Acknowledgement
Shawna Haider from Salt Lake Community College
Tamara Raymond from WCSD Assessment Department