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

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).

VERTICAL TRANSLATIONS So the graph 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?

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 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).

HORIZONTAL TRANSLATIONS So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift 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?

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

DILATION: 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.

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. 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. Above is the graph of What would 2f(x) look like? What would 4f(x) look like?

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?

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 over 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?

There is one last transformation we want to look at. 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) Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value. Above is the graph of What would f(-x) look like? (This means we are going to take the negative of x before putting in the function)

Summary of Transformations So Far **Do reflections and dilations BEFORE vertical and horizontal translations** If a > 1, then vertical dilation or stretch by a factor of a If 0 < a < 1, then vertical dilation or compression by a factor of a If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a) vertical translation of k f(-x) reflection about y-axis horizontal translation of h (opposite sign of number with the x)

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

There is one more Transformation we need to know. Do reflections and dilations BEFORE vertical and horizontal translations If a > 1, then vertical dilation or stretch by a factor of a If 0 < a < 1, then vertical dilation or compression by a factor of a If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a) vertical translation of k f(-x) reflection about y-axis horizontal translation of h (opposite sign of number with the x) horizontal dilation by a factor of b

Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified.