Section 2.6 part 1 Dilations and the Absolute-Value Family (4.6 in textbook)

Vocabulary

Dilations

Scale Factor

Rigid Transformations

Nonrigid Transformations

Vertical Dilation

Horizontal Dilation

Summary

  • Given the graph of y = f(x), the graph of

or y = bf(x)

is a ______dilation by a factor of b. In other words, each original y value of the parent function is multiplied by b for the transformed graph, where the x value stays the same.

  • Given the graph of y = f(x), the graph of

is a ______dilation by a factor of a. In other words, each original x value of the parent function is multiplied by a for the transformed graph, where the y value stays the same.

Investigation: Make My Graph with Dilations and Absolute Values

Step 1Make an accurate graph of the parent function y = |x| on the coordinate plane below.

Make a TableOn the Calculator: y = |x|

x / y
-2
-1
0
1
2

Step 2Solve each equation for y. Then, on your calculator graph the following equations. Then complete the following table

Equation / Equation in y = form / Verbal Description / Vertical/Horizontal Dilation / Scale factor / Point where (1,1) was transformed

Step 3Use your findings from step 2 and your knowledge of transformations to write the equation of each absolute value. Check your equation on your calculator.

a. b.

c. d.

  1. f.

Step 4State what a, b, h, k, and the absolute value barsdo to the graphs.

Step 5 Now apply what you have learned to other families of functions. Write the equation of the following graphs.

a.b.

c. d.

e. f.

Try some examples

Example: Describe what happens to the graph of y = f(x) when it is transformed into

Example: Write an equation for the function that results from translating y = |x|, 3 units left and vertically dilating by a factor of 2.

Name: ______Date: ______Period: ___

Discovering Advanced Algebra

Lesson 4.6

We are learning a new parent function today, the Absolute Value Function, y = │x│.

Step 1: Graph the parent function y = │x│ by completing the table below and graphing the points. Then describe the shape of the graph.

x / y
-3
-2
-1
0
1
2
3

Step 2: Translating Absolute Value Functions work the same was as Linear, Quadratic, and Square root functions. Make a conjecture of what the general equation of a translated Absolute Value Function using h and k might look like.

Step 3: Let’s practice! Using the equation from step 2, write an absolute value equation that produces the congruent graph. Check your equation on the calculator.

Step 4: Let’s try another transformation. What do you think would happen to the graph of the parent function y = │x│ if it was changed to y = - │x│? Sketch your prediction below. Use the calculator to see if your prediction was correct.

Now to introduce something new…dilations.

Step 5: For each equation below, complete the table and use the points to sketch the graph. Then write a sentence that compares the new graph you drew, with the graph of the parent function from step 1.

(a) y/2 = │x│Solve for y.

x / y
-3
-2
-1
0
1
2
3

(b)y = │x/3│

x / y
-3
-2
-1
0
1
2
3

(d)y = │x/2│

x / y
-3
-2
-1
0
1
2
3

Step 6: What happens to the parent function when…

a) the number being multiplied is outside of the absolute value?

b) the number being multiplied is inside the absolute value?

Step 7: Just as translation rules can be applied throughout all the parent functions we have studied, so too can the rules for dilations. Below each of the parent functions have been graphed. Follow the directions under each graph to complete the activity.

(a) Quadratic (y = x2)(b) Square Root (y = √x)

Graph the following in different colorsGraph the following in different colors and

and circle the equation in the same color.circle the equation in the same color.

y = 2x2 y = 2√x

y = -(x/3)2y = 3√(x/3)

Name: ______

Exit 2.6

The graphs to the left are the graphs of the parent functions. Write the equations of the transformed graphs.

Name: ______

Exit 2.6

The graphs to the left are the graphs of the parent functions. Write the equations of the transformed graphs.


Name: ______Homework 2.6

Absolute Value and Dilations (Section 4.6 in textbook)