Transformations of Functions and Graphs We will be

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

VERTICAL TRANSLATIONS As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function. Above is the graph of What would f(x) + - 31 look like? (This would mean taking all the function values andand subtracting from them). the function values adding 13 to 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 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+2) looklike? (Thiswouldmeantakingall allthe xx f(x-1) look values and adding 2 to 1 them putting them inthem the function). subtracting frombefore them before in the function).

HORIZONTAL TRANSLATIONS shift right 3 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). Above is the graph of What would f(x+1) look like? What would f(x-3) look like? So shift along the x-axis by 3

up 3 We could have a function that is transformed or translated both vertically AND horizontally. 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 (green) graph hashas ay value a y value that is 2 is 4 times as much as the original (blue) graph's y value. Above is the graph of What would 2 f(x) look like? What would 4 f(x) look like? 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.

What if the value of a was positive but less than 1? Notice for any x on the graph, the new (green) (red) graph has a ya y graph has 1/2 as much as the value that is 1/4 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? 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.

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

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. 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) 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)

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