Math 3Name______

5-6 ActivityVertical Stretches & Compressions

  • I can transform functions through horizontal translations, vertical translation, vertical reflections, and vertical stretches/compressions both graphically and algebraically
  • I can apply knowledge of horizontal translations, vertical translations, vertical

reflections, and vertical stretches/compressions to problem situations

1.In each graph below, a parent function is shown (dotted line). Draw a sketch the graph of the given dilated function.

a.b.c.

d.e.f.

g.

2a. Describe what happens to the graph of when . ______

______

2b. Describe what happens to the graph of when . ______

______


3.The next diagram shows the graph of the absolute value parent function .

The next diagram shows two variations of the parent function .

a.Complete a copy of the following table of values for and for a sample of values of the independent variable x.

x / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4
f (x) / 4 / 3 / 2 / 1 / 0 / 1 / 2 / 3 / 4
g (x)
x / -4 / -3 / -2 / -1 / 0 / 1 / 2 / 3 / 4
f (x) / 4 / 3 / 2 / 1 / 0 / 1 / 2 / 3 / 4
h (x)

b.How are the values of related to the values of ?

The values of are the values of multiplied by ______.

c.How are the values of related to the values of ?

The values of are the values of multiplied by ______.

d.What function rule (equation) seems to generate the values for ?

4.Multiplying a function by a number is called a vertical dilation. There are two types of vertical dilations: vertical stretches and vertical compressions.

A vertical stretch occurs when the parent function is multiplied by what type of number?

______

A vertical compression occurs when the parent function is multiplied by what type of number?

______

5.To the right is a graph of .

If, sketch

Suggestion: plot 6 important points on the graph
and dilate them by a factor of 2.

6.On the grid, plot points and draw a curve for:

a.

x / -4 / -2 / 0 / 2 / 4
f(x)

Based on what you have learned, sketch the below

graphs WITHOUT using a table or making a graph

on your calculator.

b.

c.

7a.Plot points and draw a curve for:

i.

ii.

Suggestion: plot 5 important points on the graph
and dilate them by the specified dilation factor.

7b.How are the graphs of y = cos(x), and related to each other? Be specific in terms of maxima, minima, and zeroes.

8a.In general, if the graph of has the point and, what point onwill be the image of ?

8b.When we dilate a function, does the x-value or the y-value get multiplied by the dilation factor?

9.The graph of has the following characteristics:

  • Zeros at
  • y-intercept of
  • local minimum atand local maximum at

If , find the following:

Zeroes: ______y-intercept: ______

Local minimum: ______local maximum: ______

10.In general, if , explain how the graphs of andrelated? Be specific in terms of maxima, minima, zeroes, y-intercept, etc.

11.12.

Parent function: Parent function:

13.The graph of has been vertically stretched/compressed. The transformed graph contains the point . What is the equation of the transformed graph?

14.Compare the two functions pictured.

The graphcontains the point.

What are the coordinates of the image of on h(x)?

How do the x-coordinates of a point on

and its image compare?

How do the y-coordinates of a point onand its image compare?

What is the equation for?______

15.Fill in the table. Conjecture an equation for and. Check your equation on your calculator.

How are the y-coordinates of y = cos(x) related to the

y-coordinates of f(x) and g(x) in the tables and graphs above?

______

______

16.In earlier courses, you have used quadratic functions to model the relationship between time and distance of moving objects (soccer balls, pumpkins, etc.) as they respond to gravity. The rules for those functions were all derived from the simplest quadratic. For instance, if you drop a ball from the top of a 60-foot tall building, its position at any time t seconds later will be given by .

The diagram below shows a graph of.

16a.Use the below table to help sketch and label graphs of , , and

x / / / / / 0 / 1 / 2 / 3 / 4

16b.Study the following sequences of transformations to see which would lead to a correct graph for .

In each case, provide algebraic reasoning that justifies your answer.

i.Stretch the graph of vertically by a factor of 16, then reflect across the x-axis, and then translate up 60 units.

ii.Stretch the graph ofvertically by a factor of 16, then translate up 60, and then reflect across the x-axis

iii.Reflect across the x-axis, then translate up 60 units, and then stretch vertically by a factor of 16.

17.

Summary:How do you vertically stretch/compress a graph? Algebraically? Graphically? How is vertically stretching/compressing similar to and different from vertically translating?