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Standard Form of Stretches of Functions

Parameters:

 a a0 for all six functions vertical stretch factor → reflection in x-axis if a < 0 and k = 0 zeros of transformed function af(x) will be a times the zeros of f(x) Functions 1, 3, 6: does not change domain or range Functions 2, 4, 5: changes range, but not domain a < 0: y < k a > 0: y > k b b0 for all six functions If f[4x-12], rewrite as: f[(4(x-3)] Changing b does not change range Function 5 changes domain b < 0: x < h b > 0: x > h Others: does not change domain 1/b horizontal stretch factor → reflection in y-axis and b < 0 and h = 0 zeros of transformed function f(bx) will be 1/b times the zeros of f(x) a and b are both < 0 reflection in the origin Description of Stretch/ Reflection Vertical If | a | > 1, then a(x): stretch by a factor a (expansion) If | a | < 1, then a(x): stretch by a factor a (compression) and reflection in x-axis. Horizontal If | b | > 1, then (bx): stretch by a factor 1/b (compression) If | b | < 1, then (bx): stretch by a factor 1/b (expansion) and reflection in the y-axis. Algebraic Description of Stretch/ Reflection Vertical y y/a: Replace the y in the original function with y/a. Horizontal x (bx).: Replace the x in the original function with bx (x, y) (1/bx, ay) blue data point red data point mapping notation to find location of data points after a horizontal/vertical stretch. Notice the green rectangle and purple lines that connect to the x- and y-axis. The length of the vertical purple and green line is a times the length of the vertical purple line. The length of the horizontal purple and green line is 1/b times the length of the horizontal purple line.
• stretch: transformation of geometric figure in which every x-coordinate, y-coordinate, or both is multiplied by the same factor.

Interactive Activity

• Click on the applet to activate
• Click one of the 6 function icons.
• Move the parameter sliders for a and b.
• Study the effects of each parameter with respect to the original equation (grey graph). Determine the parameter that effects:
• vertical stretch (expansion/compression)
• horizontal stretch (expansion/compression)
• reflection in the x-axis
• reflection in they-axis
• Drag the blue data point from the far left to the far right. Observe the movement of the red data point to see how the transformation can be used to map the location of corresponding data points on the transformed function. The location(ordered pair) of the blue and red data point is given at the bottom of the screen.
• Click the red or blue data point to update the red data point if you use the up/down arrow to move one of the sliders.
• Select SET FUNTION to explore the effect of the a and b parameters on other functions.
 1. y = a (bx) or y/a = bx (linear) 2. y = a [b(x)]2 or y/a = [b(x)]2 (quadratic) 3. y = a [b(x)]3 or y/a = [b(x)]3 (cubic) 4. y = a |bx| or y/a = |bx| (absolute value) not included in Math 30P 5. or (radical/square root) 6. or (radical/cube root)

Vertical Stretch

In all 6 functions, a is the vertical stretch factor and parameter..

 transformed function Note: a0 for all six functions y = af(x) or y y/a y = -f(x), a < 0 or y -y Creates reflection in the x-axis when a < 0. reflection in the origin when a < 0 and b < 0 vertical stretch factor a a0 for all six functions zeros of transformed function af(x) will be a times the zeros of f(x) Domain - set of possible x-values changing a does not change domain Function 5: x > 0 Others: xR Range - set of possible y-values changing a does not change domain Functions 2, 4, 5: y > 0 Others: yR Description of Vertical Stretch/ Reflection If | a | > 1, then a(x): stretch by a factor a (expansion) If | a | < 1, then a(x): stretch by a factor a (compression) and reflection in x-axis. Algebraic Description of Vertical Stretch/ Reflection y y/a: Replace the y in the original function with y/a. (x, y) (x, ay) blue data point red data point mapping notation to find location of data points after a vertical stretch. Notice the green rectangle and purple lines that connect to the x- and y-axis. The length of the vertical purple and green line is a times the length of the vertical purple line.

Interactive Activity

• Click on the applet to activate
• Click one of the 6 function icons.
• Move the parameter sliders for a. Note the original equation remains and a second graph is drawn according to the parameters selected.
• Drag the blue data point from the far left to the far right. Observe the movement of the red data point to see how the transformation can be used to map the location of corresponding data points on the transformed function. The location(ordered pair) of the blue and red data point is given at the bottom of the screen.
• Click the red or blue data point to update the red data point if you use the up/down arrow to move one of the sliders.
• Select SET FUNTION to explore the effect of the b parameter on other functions.
 1. y = a (bx) or y/a = bx (linear) 2. y = a [b(x)]2 or y/a = [b(x)]2 (quadratic) 3. y = a [b(x)]3 or y/a = [b(x)]3 (cubic) 4. y = a |bx| or y/a = |bx| (absolute value) not included in Math 30P 5. or (radical/square root) 6. or (radical/cube root)

Horizontal Stretch/Reflection

In all 6 functions, b is the horizontal stretch parameter. 1/b is the horizontal stretch factor.

In all 6 functions reflection about the y-axis occurs when b < 0.

 transformed function If f[4x-12], rewrite as: f[(4(x-3)] Note: b0 for all six functions y = f(bx) or x bx y = f(-x), b < 0 or x -x about y-axis when b < 0. Creates reflection in the y-axis when b < 0. reflection in the origin when a < 0 and b < 0 horizontal stretch factor 1/b horizontal stretch factor, reflection when b < 0 zeros of transformed function f(bx) will be 1/b times the zeros of f(x) horizontal stretch parameter b b0 for all six functions If f[4x-12], rewrite as: f[(4(x-3)] Battle of the Opposites in bx 0 < b < 1 compresses the graph b > 1 expands the graph Domain - set of possible x-values Function 5 changes domain b < 0: x < h b > 0: x > h Others do not change domain xR Range - set of possible y-values changing b does not change range yR Description of Horizontal Stretch/ Reflection If | b | > 1, then (bx): stretch by a factor 1/b (compression) If | b | < 1, then (bx): stretch by a factor 1/b (expansion) and reflection in the y-axis. Algebraic Description of Horizotal Stretch/ Reflection x (bx): Replace the x in the original function with bx (x, y) (1/bx, y) blue data point red data point mapping notation to find location of data points after a horizontal stretch. Notice the green rectangle and purple lines that connect to the x- and y-axis. The length of the horizontal purple and green line is 1/b times the length of the horizontal purple line.

Interactive Activity

• Click on the applet to activate
• Click one of the 6 function icons.
• Move the parameter sliders for a. Note the original equation remains and a second graph is drawn according to the parameters selected.
• Drag the blue data point from the far left to the far right. Observe the movement of the red data point to see how the transformation can be used to map the location of corresponding data points on the transformed function. The location(ordered pair) of the blue and red data point is given at the bottom of the screen.
• Click the red or blue data point to update the red data point if you use the up/down arrow to move one of the sliders.
• Select SET FUNTION to explore the effect of the a parameter on other functions.
 1. y = a (bx) or y/a = bx (linear) 2. y = a [b(x)]2 or y/a = [b(x)]2 (quadratic) 3. y = a [b(x)]3 or y/a = [b(x)]3 (cubic) 4. y = a |bx| or y/a = |bx| (absolute value) not included in Math 30P 5. or (radical/square root) 6. or (radical/cube root)

A copy of the above applet is provided to study the remainder of the notes for descriptions of stretches.

 1. y = a (bx) or y/a = bx (linear) 2. y = a [b(x)]2 or y/a = [b(x)]2 (quadratic) 3. y = a [b(x)]3 or y/a = [b(x)]3 (cubic) 4. y = a |bx| or y/a = |bx| (absolute value) not included in Math 30P 5. or (radical/square root) 6. or (radical/cube root)

## Stretches about lines other than x- and y-axis

The stretches you have seen so far have been the simplest cases­—stretching the function about either the x-axis or the y-axis. Functions can be stretched about other lines as well.

Before computers were used to graph equations, these kinds of stretches were drawn by hand using a couple of simple ideas. To draw a vertical stretch by hand, you first measure the distance from the line to the graph. Then you multiply this distance by the stretch factor and plot the new point the calculated distance from the line. Repeat these steps until the new graph is complete. Horizontal stretches can be drawn in a similar manner.

Example 1

Examine the following graphs. Here, the graph of y = g(x) is the graph of y = f(x) stretched vertically by a factor of 2 about the line y = 5.

Notice that all points on the line y = 5 are invariant (unchanged) under this transformation. Also, notice that the vertex has moved to twice the distance from the line y = 5. The vertex of f(x), (1,3), is 2 units from y = 5; and the vertex of g(x), (1, 1), is 4 units from y = 5.

Example 2

xamine the following graphs. Here the graph of y = k(x) is the graph of y = h(x) stretched by a factor of about the line x = -3.

Notice that the points on the line x = -3 are invariant under this transformation. Also, notice that the vertex of k(x) is 1/3 the distance from x = -3 compared to the distance of the vertex of h(x). The vertex of y = h(x), (3,-3), is 6 units from x = -3; and the vertex of k(x), (-1, -3), is 2 units from x = -3.

Example 3

The graph of is stretched by a factor of 2 about the line y = 5.

1. Determine the equation of the resulting function.
2. At which points will the graphs intersect each other?

### Solution

1. The following steps show how to determine the equation of the transformed graph. The graphs given allow you to see what happened to the graph at each step.

Step 1: Move the graph of y = f(x) down 5 units so the line it is stretched about, y = 5, is on the x-axis.

Step 2: Stretch the new function vertically by a factor of 2.

Step 3: Move the new function up 5 units so the line it was stretched about is back in its original position.

The equation defines the stretched function. The following shows the original function and the end function stretched about the line y = 5.

2. First, find the possible x-values of the points of intersection.

Now, find the corresponding y-values.

The two graphs intersect each other at

Parts of this work has been adapted from a Math 30 Pure learning resource originally produced and owned by Alberta Education

Comments to:  Jim Reed - Homepage