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

VERTICAL TRANSLATIONS Above is the graph of 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). As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function.

y = f(x) + k y = f(x) ̶ k Transformation k units Down k units UP

VERTICAL TRANSLATIONS

what is the transformation? y = f(x) +10 Parent function y = f(x) Up 10 units Down 9 units Up 5 units Down 7 units y = f(x)

Above is the graph of 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). As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but opposite way of the sign of the number. HORIZONTAL TRANSLATIONS 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).

y = f(x + h) y = f(x ̶ h) Transformation h units Right h units left For h>0, and

HORIZONTAL TRANSLATIONS shift right 3 shift left 1

what is the transformation? y = f(x+10) Parent function y = f(x) Left 10 units Right 9 units Left 5 units Right 7 units y = f(x)

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

what is the transformation? y = f(x+1)-6 Parent function y = f(x) Left 1 and down 6 Right 3 and up 2 Left 5 and up 7 Right 8 and down 1 y = f(x-8)-1 y = f(x)

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. DILATION:

Above is the graph of So the graph a f(x), where a is any real number GREATER THAN 1, is the graph of f(x) What would 2f(x) look like? What would 4f(x) look like? 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. The bigger a is. The narrower the graph is. vertically stretched by a factor of a.

Above is the graph of So the graph a f(x), where a is 0 < a < 1, is the graph of f(x) 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. 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. What if the value of a was positive but less than 1? What would 1/4 f(x) look like? What would 1/2 f(x) look like? The smaller a is. The wider the graph is. vertically Compressed by a factor of a.

y = a f(x) Transformation Stretched Vertically By factor of a a>1 0<a<1 Compressed Vertically By factor of a

VERTICAL TRANSLATIONS

3. Horizontal translation Procedure: Multiple Transformations (From left to right) 2. Stretching or shrinking 1. Reflecting 4. Vertical translation

what is the transformation? y = 5f(x+10)-6 vertically stretched by factor of 5, Left 10, down 6 vertically compressed by factor of ¼, Right 7, up 2

what is the transformation? vertically compressed by factor of 1/5, Left 6, down 7 vertically stretched by factor of ¼, Right 9, up 2

Above is the graph of 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 over the x-axis) What if the value of a was negative? What would - f(x) look like? 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 There is one last transformation we want to look at. Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value. What would f(-x) look like? (This means we are going to take the negative of x before putting in the function) So the graph f(-x) is a reflection about the y-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the y-axis)

Summary of Transformations So Far h >0 vertically stretched by a factor of a Up k units vertically compressed by a factor of a reflected across y-axis **Do reflections and dilations BEFORE vertical and horizontal translations** -f (x) If a > 1, If 0 < a < 1, reflected across x-axis f(-x) (opposite sign of number with the x) Left h units h <0Right h units k>0 k<0 Down k units

Graph using transformations We know what the graph would look like if it was from our library of functions. moves up 1 moves right 2 reflects about the x -axis