Transformations of Functions SECTION 2.7 Transformations of Functions Learn the meaning of transformations. Use vertical or horizontal shifts to graph functions. Use reflections to graph functions. Use stretching or compressing to graph functions. 1 2 3 4

TRANSFORMATIONS If a new function is formed by performing certain operations on a given function f , then the graph of the new function is called a transformation of the graph of f.

Example: y = |x| + 2 Parent function (y = |x|) shown on graph in red. The transformation of the parent function is shown in blue. It is a shift up (or vertical translation up) of 2 units.)

Example: y = x - 1 Parent function (y = x) shown on graph in red. The transformation of the parent function is shown in blue. It is a shift down (or vertical translation down) of 1 unit.

Parent Functions – The simplest function of its kind Parent Functions – The simplest function of its kind. All other functions of its kind are Transformations of the parent.

Translation (Shift) A vertical translation is made on a function by adding or subtracting a number to the function. Example: y = x + 3 (translation up) Example: y = x² - 5 (translation down) A translation up is also called a vertical shift up. A translation down is also called a vertical shift down.

Vertical Shifting Vertical Translation For b > 0, the graph of y = f(x) + b is the graph of y = f(x) shifted up b units; the graph of y = f(x)  b is the graph of y = f(x) shifted down b units.

The graph of Is like the graph of SHIFTED 3 units up

The graph of Is like the graph of SHIFTED 2 units down

EXAMPLE 1 Graphing Vertical Shifts Let Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.

EXAMPLE 1 Graphing Vertical Shifts Solution Make a table of values.

Graph the equations. The graph of y = |x| + 2 is the graph of y = |x| EXAMPLE 1 Graphing Vertical Shifts Solution continued Graph the equations. The graph of y = |x| + 2 is the graph of y = |x| shifted two units up. The graph of y = |x| – 3 is the shifted three units down.

VERTICAL SHIFT Let d > 0. The graph of y = f (x) + d is the graph of y = f (x) shifted d units up, and the graph of y = f (x) – d is the graph of y = f (x) shifted d units down.

The effect of the transformation on the graph Replacing function with function – number SHIFTS the basic graph number units down Replacing function with function + number SHIFTS the basic graph number units up

Let f (x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2. EXAMPLE 2 Writing Functions for Horizontal Shifts Let f (x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2. A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide. Describe how the graphs of g and h relate to the graph of f.

EXAMPLE 2 Writing Functions for Horizontal Shifts

EXAMPLE 2 Writing Functions for Horizontal Shifts

All three functions are squaring functions. EXAMPLE 2 Writing Functions for Horizontal Shifts Solution All three functions are squaring functions. a. g is obtained by replacing x with x – 2 in f . The x-intercept of f is 0. The x-intercept of g is 2. For each point (x, y) on the graph of f , there will be a corresponding point (x + 2, y) on the graph of g. The graph of g is the graph of f shifted 2 units to the right.

b. h is obtained by replacing x with x + 3 in f . EXAMPLE 2 Writing Functions for Horizontal Shifts Solution continued b. h is obtained by replacing x with x + 3 in f . The x-intercept of f is 0. The x-intercept of h is –3. For each point (x, y) on the graph of f , there will be a corresponding point (x – 3, y) on the graph of h. The graph of h is the graph of f shifted 3 units to the left. The tables confirm both these considerations.

HORIZONTAL SHIFT The graph of y = f (x – c) is the graph of y = f (x) shifted |c| units to the right, if c > 0, to the left if c < 0.

Horizontal Shifting Horizontal Translation For d > 0, the graph of y = f(x  d) is the graph of y = f(x) shifted right d units; the graph of y = f(x + d) is the graph of y = f(x) shifted left d units.

The effect of the transformation on the graph Replacing x with x – number SHIFTS the basic graph number units to the right Replacing x with x + number SHIFTS the basic graph number units to the left

The graph of Is like the graph of SHIFTED 2 units to the right

Vertical shifts Horizontal shifts Moves the graph up or down Impacts only the “y” values of the function No changes are made to the “x” values Horizontal shifts Moves the graph left or right Impacts only the “x” values of the function No changes are made to the “y” values

The values that translate the graph of a function will occur as a number added or subtracted either inside or outside a function. Numbers added or subtracted inside translate left or right, while numbers added or subtracted outside translate up or down.

Recognizing the shift from the equation, examples of shifting the function f(x) = Vertical shift of 3 units up Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

Points represented by (x , y) on the graph of f(x) become If the point (6, -3) is on the graph of f(x), find the corresponding point on the graph of f(x+3) + 2

Combining a vertical & horizontal shift Example of function that is shifted down 4 units and right 6 units from the original function.

Sketch the graph of the function Graphing Combined Vertical and Horizontal Shifts EXAMPLE 3 Sketch the graph of the function Solution Identify and graph the parent function

The graph of Is like the graph of SHIFTED 3 units to the left

Solution continued Graphing Combined Vertical and Horizontal Shifts EXAMPLE 3 Solution continued Translate 2 units to the left Translate 3 units down

REFLECTION IN THE x-AXIS The graph of y = – f (x) is a reflection of the graph of y = f (x) in the x-axis. If a point (x, y) is on the graph of y = f (x), then the point (x, –y) is on the graph of y = – f (x).

REFLECTION IN THE x-AXIS

REFLECTION IN THE y-AXIS The graph of y = f (–x) is a reflection of the graph of y = f (x) in the y-axis. If a point (x, y) is on the graph of y = f (x), then the point (–x, y) is on the graph of y = f (–x).

REFLECTION IN THE y-AXIS

EXAMPLE 4 Combining Transformations Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|. Solution Step 1 Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|.

EXAMPLE 4 Combining Transformations Solution continued Step 2 Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

EXAMPLE 4 Combining Transformations Solution continued Step 3 Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

Stretching or Compressing a Function Vertically EXAMPLE 5 Let Sketch the graphs of f, g, and h on the same coordinate plane, and describe how the graphs of g and h are related to the graph of f. Solution x –2 –1 1 2 f(x) g(x) 4 h(x) 1/2

Solution continued Stretching or Compressing a Function Vertically EXAMPLE 5 Solution continued

Stretching or Compressing a Function Vertically EXAMPLE 5 Solution continued The graph of y = 2|x| is the graph of y = |x| vertically stretched (expanded) by multiplying each of its y–coordinates by 2. The graph of |x| is the graph of y = |x| vertically compressed (shrunk) by multiplying each of its y–coordinates by .

VERTICAL STRETCHING OR COMPRESSING The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is A vertical stretch away from the x-axis if a > 1; 2. A vertical compression toward the x-axis if 0 < a < 1. If a < 0, the graph of f is first reflected in the x-axis, then vertically stretched or compressed.

The point (-12, 4) is on the graph of y = f(x) The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x). g(x) = f(x-2) g(x)= 4f(x) g(x) = f(½x) g(x) = -f(x) (-10, 4) (-12, 16) (-24, 4) (-12, -4)