Graphing Techniques:

Graphing Techniques: Transformations Transformations: Review Transformations We will be looking at functions from our library of functions and seeing how various modifications to the functions transform them. Transformations Transformations Transformations Transformations Transformations

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) - 3 look like? (This would mean taking all the function values and subtracting 3 from them). What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them).

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+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function). 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).

HORIZONTAL TRANSLATIONS So the graphf(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 graphf(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?

What would the graph of 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

and If we multiply a function by a non-zero real number it has the affect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number. DILATION Let's try some functions from our library of 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 af(x), whereais 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 would2f(x) look like? What would4f(x) look like?

What if the value of a was positive but less than 1? So the graph af(x), whereais 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 (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. Above is the graph of What would1/2 f(x) look like? What would1/4 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 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 wasfrom our library of functions. Graph using transformations moves up 1 reflects about the x -axis moves right 2

There is one more Transformation we need to know. Do reflections 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

Vertical Dilation Now complete the Changes of Scale and Origin for Graphs Booklet to explore this idea further and to consolidate all graphical transformations.

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 to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

Graphing Techniques: