Transformations of Functions and their Graphs Ms. P

Linear Transformations Translations (shifts) Reflections Dilations (stretches or shrinks) We examine the mathematics:  Graphically  Numerically  Symbolically  Verbally These are the common linear transformations used in high school algebra courses.

Translations

How do we get the flag figure in the left graph to move to the position in the right graph?

Translations This picture might help.

Translations How do we get the flag figure in the left graph to move to the position in the right graph? Here are the alternate numerical representations of the line graphs above

Translations How do we get the flag figure in the left graph to move to the position in the right graph? This does it! =

Translations Alternately, we could first add 1 to the y-coordinates and then 3 to the x-coordinates to arrive at the final image.

Translations What translation could be applied to the left graph to obtain the right graph? y = ???

Translations Following the vertex, it appears that the vertex, and hence all the points, have been shifted up 1 unit and right 3 units. Graphic Representations:

Translations Numerically, 3 has been added to each x-coordinate and 1 has been added to each y coordinate of the function on the left to produce the function on the right. Thus the graph is shifted up 1 unit and right 3 units. Numeric Representations:

Translations To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units.

Translations To find the symbolic formula for the graph that is seen above on the right, let’s separate our translation into one that shifts the function’s graph up by one unit, and then shift the graph to the right 3 units. The graph on the left above has the equation y = x 2. To translate 1 unit up, we must add 1 to every y-coordinate. We can alternately add 1 to x 2 as y and x 2 are equal. Thus we have y = x 2 + 1

Translations We verify our results below: The above demonstrates a vertical shift up of 1. y = f(x) + 1 is a shift up of 1 unit that was applied to the graph y = f(x). How can we shift the graph of y = x 2 down 2 units?

Translations We verify our results below: The above demonstrates a vertical shift down of 2. y = f(x) - 2 is a shift down 2 unit to the graph y = f(x) Vertical Shifts If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward. Did you guess to subtract 2 units?

Vertical Translation Example Graph y = |x| 11 2 22 |x||x|x

Aside: y = |x| on the TI83/84

Vertical Translation Example Graph y = |x| 11 42 22 |x|+2|x||x|x

Vertical Translation Example Graph y = |x| 11 12 22 |x| -1|x||x|x

Example Vertical Translations y = 3x 2

Example Vertical Translations y = 3x 2 y = 3x 2 – 3 y = 3x 2 + 2

Example Vertical Translations y = x 3

Example Vertical Translations y = x 3 y = x 3 – 3 y = x 3 + 2

Translations Vertical Shift Animation:

Translations The vertex has been shifted up 1 unit and right 3 units. Starting with y = x 2 we know that adding 1 to x 2, that is y = x 2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units? Getting back to our unfinished task:

Translations Starting with y = x 2 we know that adding 1 to x 2, that is y = x 2 +1 shifts the graph up 1 unit. Now, how to we also shift the graph 3 units to the right, that is a horizontal shift of 3 units? We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula?

Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula? xy=x xy=x x+3y=x

Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula? x+3y=x So, let’s try y = (x + 3) ??? Oops!!!

Translations We need to add 3 to all the x-coordinates without changing the y-coordinates, but how do we do that in the symbolic formula? x+3y=x So, let’s try y = (x - 3) ??? Hurray!!!!!!

Translations Vertical Shifts If k is a real number and y = f(x) is a function, we say that the graph of y = f(x) + k is the graph of f(x) shifted vertically by k units. If k > 0 then the shift is upward and if k < 0, the shift is downward. Horizontal Shifts If h is a real number and y = f(x) is a function, we say that the graph of y = f(x - h) is the graph of f(x) shifted horizontally by h units. If h follows a minus sign, then the shift is right and if h follows a + sign, then the shift is left.

Example Horizontal Translation Graph g(x) = |x| 11 2 22 |x||x|x

Example Horizontal Translation Graph g(x) = |x + 1| 11 12 22 |x + 1||x||x|x

Example Horizontal Translation Graph g(x) = |x - 2| 11 42 22 |x - 2||x||x|x

Horizontal Translation y = 3x 2

Horizontal Translation y = 3x 2 y = 3(x+2) 2 y = 3(x-2) 2

Horizontal Shift Animation

Summary of Shift Transformations To Graph:Shift the Graph of y = f(x) by c units y = f(x) + cUP y = f(x) - cDOWN y = f(x + c)LEFT y = f(x - c)RIGHT

Translations – Combining Shifts Investigate Vertex form of a Quadratic Function: y = x 2 + bx + c y = x 2 vertex: (0, 0) y = (x – 3) vertex: (3, 1) Vertex Form of a Quadratic Function (when a = 1): The quadratic function: y = (x – h) 2 + k has vertex (h, k).

Translations Compare the following 2 graphs by explaining what to do to the graph of the first function to obtain the graph of the second function. f(x) = x 4 g(x) = (x – 3) 4 - 2

Warm-up If 0 < x < 1, rank the following in order from smallest to largest:

Warm-up

Reflections

How do we get the flag figure in the left graph to move to the position in the right graph?

Reflections How do we get the flag figure in the left graph to move to the position in the right graph? The numeric representations of the line graphs are:

Reflections So how should we change the equation of the function, y = x 2 so that the result will be its reflection (across the x-axis)? Try y = - (x 2 ) or simply y = - x 2 (Note: = - 4 while (-2) 2 = 4)

Reflection: Reflection: (across the x-axis) The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x).

Example Reflection over x-axis f(x) = x 2

Example Reflection over x-axis f(x) = x 2 f(x) = -x 2

Example Reflection over x-axis f(x) = x 3

Example Reflection over x-axis f(x) = x 3 f(x) = -x 3

Example Reflection over x-axis f(x) = x + 1

Example Reflection over x-axis f(x) = x + 1 f(x) = -(x + 1) = -x - 1

More Reflections Reflection in x-axis: 2 nd coordinate is negated Reflection in y-axis: 1 st coordinate is negated

Reflection: Reflection: (across the x-axis) The graph of the function, y = - f(x) is the reflection of the graph of the function y = f(x). Reflection: (across the y-axis) The graph of the function, y = f(-x) is the reflection of the graph of the function y = f(x).

Example Reflection over y-axis f(x) = x 2

Example Reflection over y-axis f(x) = x 2 f(-x) = (-x) 2 = x 2

Example Reflection over y-axis f(x) = x 3

Example Reflection over y-axis f(x) = x 3 f(-x) = (-x) 3 = -x 3

Example Reflection over y-axis f(x) = x + 1

Example Reflection over y-axis f(x) = x + 1 f(-x) = -x + 1

Dilations

How do we get the flag figure in the left graph to move to the position in the right graph? Dilations (Vertical Stretches and Shrink)

Dilations (Stretches and Shrinks) Definitions: Vertical Stretching and Shrinking The graph of y = af(x) is obtained from the graph of y = f(x) by a). shrinking the graph of y = f ( x) by a when a > 1, or b). stretching the graph of y = f ( x) by a when 0 < a < 1. Vertical StretchVertical Shrink

Example Vertical Stretching/Shrinking y = |x|

Example: Vertical Stretching/Shrinking y = |x| y = 0.5|x| y = 3|x|

Vertical Stretching / Shrinking Animation

What is this? Base Function y = |x| y = ???? y = -2|x -1| + 4

Warm-up Explain how the graph of can be obtained from the graph of.

How do we get the flag figure in the left graph to move to the position in the right graph? Dilations (Horizontal Stretches and Shrink)

Horizontal Stretching / Shrinking Animation

Procedure: Multiple Transformations Graph a function involving more than one transformation in the following order: 1. Horizontal translation 2. Stretching or shrinking 3. Reflecting 4. Vertical translation Multiple Transformations

Graphing with More than One Transformation Graph -|x – 2| + 1 First graph f(x) = |x|

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right.

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2|

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

Graphing with More than One Transformation Graph f(x) = -|x – 2| + 1 First graph f(x) = |x| 1. Perform horizontal translation: f(x) = |x-2| The graph shifts 2 to the right. 2. There is no stretch 3. Reflect in x-axis: f(x) = -|x-2| 4. Perform vertical translation: f(x) = -|x-2| + 1 The graph shifts up 1 unit.

Questions?

Time for worksheet

Other Transformation: Shears (x, y) (x+y, y)

Can we Apply this Shear to y = x 2 ? Look at a line graph first! Apply the shear: (x, y) (x+y, y)

Can we Apply this Shear to y = x 2 ? Apply the shear: (x, y) (x+y, y)

Can we Apply this Shear to y = x 2 ? Apply the shear: (x, y) (x+y, y)

Yes we CAN Apply this Shear to y = x 2. Apply the shear: (x, y) (x+y, y) BUT…Can we write the symbolic equation in terms of x and y?

Shear Example Apply the shear: (x, y) (x+y, y) to y = x 2 Parametrically we have: x = t + t 2 Our job is to eliminate t. y = t 2 We will use the substitution method. Now substitute t back into the x equation and we have.

Shears Horizontal Shear for k a constant (x, y ) (x+ky, y) Vertical Shear for k a constant (x, y ) (x, kx+y)

Other Linear Transformations? Rotations