Algebra 2 1-8a Exploring Transformations Translations

Vocabulary Transformation – Changing a graph’s size, shape or position Translation – Shifting a graph horizontally or vertically

Big Idea We are going to explore graphing the absolute value function, from this basic format y = |x – h| + k

Calculator Exploration Work with a graphing calculator Turn it on Select the graphing icon

Calculator Exploration Add a graph f1(x) = |x| You may click and drag the graph to be lower on the screen |x| is entered abs(x) This can also be found in the function list (open book key)

Calculator Exploration

Add two more graphs f2(x) = |x| + 4 f3(x) = |x| - 3 CTRL – G Opens the f(x) line on the screen

Calculator Exploration How does increasing “k” affect the graph’s location? Arrow over the graph to “see” which formula goes with which graph

Quick Questions Did the graph shape change? Does the graph point in the same direction? How much did the graph move up? How much did the graph move down?

Calculator Exploration Open a new graphing page “Home”, “icon”

Calculator Exploration Add three different graphs f4(x) = |x| f5(x) = |x + 3| f6(x) = |x - 4|

Calculator Exploration How does Changing “h” affect the graph?

Quick Questions Did the graph shape change? Does the graph point in the same direction? How much did the graph move left? How much did the graph move right?

Think Deep If adding to “k” moves positively along the y- axis, why does add to “h” move negatively along the x-axis?

Think Deep If adding to “k” moves positively along the y- axis, why does add to “h” move negatively along the x-axis? Think “inputs” and “outputs”

Think Deep This “slide” is the input (x-axis) through our projector, the Smartboard screen is our output (y-axis). If I could add to the output, it would raise the picture on the screen. (y-axis) If I could add to the input, it would move the screen to the right (without moving the projector). So the graph appears to move to the left.

Think Deep Math is always consistent, so line up the constants and the variables like such y – k = |x – h|

Mirror y – k = |x – h| When we add to the output, the function moves up When we subtract from the output, the function moves down

Summary y – k = |x – h| When we add to the input, the function moves to the left When we subtract from the input, the function moves to the right

Summary

Example Graph the following equations (without the calculator) y=|x| – 8 y=|x – 5| y=|x + 4| + 3

Practice

Calculator A few more calculator basics Click and drag on the axis to zoom Click and drag on the background to move it all around

Algebra Exploring Transformations Reflections

Vocabulary Review Square Root Function – The parent function graph resulting from graphing

Big Idea We are going to explore graph reflections based on the parent function

Calculator Exploration Work with a graphing calculator Turn it on Select the graphing icon

Calculator Exploration Add a graph f1(x) = 0.5x+2

Calculator Exploration Add one more graph f2(x) = -f1(x) CTRL – G Opens the f(x) line on the screen

Question What is the equation for y 2, in terms of x?

Calculator Exploration Change the first graph as follows f1(x) = – 2x – 4 CTRL – G Opens the f(x) line on the screen

Question Now what is the equation for y 2, in terms of x?

Calculator Exploration Change the first graph again f1(x) = x CTRL – G Opens the f(x) line on the screen

Question In general how are the two graphs of y=f(x) and y= – f(x) related?

Calculator Exploration Again change the graph f1(x) = 0.5x+2

Calculator Exploration Also change f2(x) f2(x) = f1(-x) CTRL – G Opens the f(x) line on the screen

Questions What is the equation for y 2, in terms of x? How do the two graphs compare?

Calculator Exploration Change the first graph as follows f1(x) = – 2x – 4 CTRL – G Opens the f(x) line on the screen

Question Now what is the equation for y 2, in terms of x? How do the two graphs compare?

Calculator Exploration Change the first graph again f1(x) = x CTRL – G Opens the f(x) line on the screen

Question Now what is the equation for y 2, in terms of x? How do the two graphs compare?

Calculator Exploration Change the first graph again f1(x) = (x – 3) 2 + 2

Question In general how are the two graphs of y=f(x) and y= – f(x) related?

Calculator Exploration Again change the graph

Conjecture How will the following graphs look? f2(x) = – f1(x) f2(x) = f1(– x) f2(x) =– f1(– x)

Quick Questions What shape does the square root function look most like? How can it look like the whole shape? Why is it on its side? How could you write a function so the shape was symmetrical?

Concept Summary

Quick Questions Moves 3 to the right Moves 3 to the left Moves up 2 Moves down 2

Example Write an equation for each of these graphs.

Example Answers

Practice

Answers

Practice

Summary f(x) = –f(x) When y is replaced with negative y, the function is reflected in the x-axis

Summary f(x) = f(–x) When x is replaced with negative x, the function is reflected in the y-axis

Summary f(x) = –f(–x) When both functions are replaced with negative variables, the output is reflected in both axis

Practice

Homework Pages 63 – 66 6, 7, 14 – 20, 52, 55 – 61