1. Transforming Equations
MOVE THE MONSTER
Transforming Equations
We talked about parent functions yesterday, now we will look at how to manipulate functions using a super special parent function.

2. Monster Function
Represents all the basic forms a function can take like some of the parent functions you saw yesterday.
This guy! The Monster function.

3. Objective: I will apply transformations to points or sets of points
Objective: I will apply transformations to points or sets of points. Essential Questions: What changes in equations yield certain transformations?

4. f(x) ( -6, ) ( -4, ( -2, ( 0, ( 2, ( 4, ( 6, PARENT D: { } R: { }
( x , f(x) )
( -6, )
( -4,
( -2,
( 0,
( 2,
( 4,
( 6,
First, let’s look at the graph to find our values for f(x).
What is f(-6)?
What is the domain for f(x)?
What is the range for f(x)?
Classify the transformation:
PARENT
D: { }
R: { }

5. -f(x) D: R: ( x , f(x) ) ( x’, - f(x)) ( -6, -1 ) ( -4, 3 ) ( -2, 1 )
( -6,
-1 )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Alright, let’s fill in the table and play connect the dots.
What is the negative of f(-6)? Here is the value for f(-6). What is it’s negative?
Then we can plot these points for x and –f(x).
Are we changing x? NO Fill in those same points for x’
Classify the transformation: D: R:

6. -f(x) D: R: ( x , f(x) ) ( x’, - f(x)) ( -6, -1 ) ) ( -4, 3 ) ( -2,
( -6,
-1 ) )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
What happened to the graph? Flipped, reflected over the x-axis. So it is a reflection.
What is the domain of the new green graph? The same.
What is the range of the new graph? Different. So it’s a vertical reflection.
Classify the transformation: D: R:

7. -f(x) D: {-6 ≤ x ≤ 6} R: {-5 ≤ y ≤ 1} ( x , f(x) ) ( x’, - f(x)) ( -6,
( -6,
-1 )
-1)
( -4,
3 )
-3 )
( -2,
1 )
-1 )
( 0,
5 )
-5 )
( 2,
2 )
-2 )
( 4,
4 )
-4 )
( 6,
0 )
Classify the transformation:
Vertical Reflection
(across the x-axis)
D: {-6 ≤ x ≤ 6}
R: {-5 ≤ y ≤ 1}

8. f(-x) D: R: ( x , f(x) ) ( x’, f(-x)) ( -6, -1 ) ( -4, 3 ) ( -2, 1 )
( -6,
-1 )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Fill in the table and play connect the dots.
Here is x, what is negative x? 6. So what’s the value at f(6)?
We can fill out the table this way to get a graph.
Are we making changes to x? YES Are we making changes to y? NO so fill in those values for f(-x)
Classify the transformation: D: R:

9. f(-x) D: R: ( x , f(x) ) ( x’, f(-x)) ( -6, -1 ) ( -4, 3 ) ( -2, 1 )
( -6,
-1 )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
How are we changing x?
Plug in those values. Now we can plot the points.
Classify the transformation: D: R:

10. f(-x) D: R: ( x , f(x) ) ( x’, f(-x)) ( -6, -1 ) ( 6, ( -4, 3 ) ( 4,
( -6,
-1 )
( 6,
( -4,
3 )
( 4,
( -2,
1 )
( 2,
( 0,
5 )
( 0,
( 2,
2 )
( -2,
( 4,
4 )
( -4,
( 6,
0 )
( -6,
What happened to the graph? Flipped, reflected over the y-axis. So this time it’s a horizontal reflection.
What is the domain of the new green graph?
What is the range of the new green graph?
Classify the transformation: D: R: 10

11. Horizontal Reflection
f(-x)
( x , f(x) )
( x’, f(-x))
( -6,
-1 )
( 6,
( -4,
3 )
( 4,
( -2,
1 )
( 2,
( 0,
5 )
( 0,
( 2,
2 )
( -2,
( 4,
4 )
( -4,
( 6,
0 )
( -6,
Do you guys see how this is reflected across the y-axis? The long leg is on the right now; the yellow eye is on the left now.
Alright, next kind of transformation [next slide].
Classify the transformation:
Horizontal Reflection
(across the y-axis)
D: {-6 ≤ x ≤ 6}
R: {-1 ≤ y ≤ 5}

12. f(x-2) D: R: ( x , f(x) ) ( x’, f(x-2)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Are we changing x or y? X so your y values will be the same. What’s a new x value?
Let’s fill in this row for x=4. What is f(4-2)? F(2). So what’s the value at f(2)? 2.
Let’s fill in one more for x=-4. What is f(-4-2)? F(-6). So what’s the value at f(-6)? -1.
Now we can plot these points.
Classify the transformation: D: R:

13. f(x-2) D: R: ( x , f(x) ) ( x’, f(x-2)) ( -6, -1 ) ( -4, -1 ) ( -4,
( -6,
-1 ) (
-4, -1 )
( -4,
3 )
-2, 3
( -2,
1 ) 0, 1
( 0,
5 ) 2, 5
( 2,
2 ) 4, 2
( 4,
4 ) 6, 4
( 6,
0 ) 8,
What happened to the graph? It shifted to the right.
What is the domain? How is it different?
What is the range?
Remember x lies. X says, “I’m being moved in the negative direction.” when the graph actually moves to the right. What happens inside the parentheses is a lie.
Classify the transformation: D: R:

14. f(x-2) D: {-4 ≤ x ≤ 8} R: {-1 ≤ y ≤ 5} ( x , f(x) ) ( x’, f(x-2))
( -6,
-1 ) (
-4, -1 )
( -4,
3 )
-2, 3
( -2,
1 ) 0, 1
( 0,
5 ) 2, 5
( 2,
2 ) 4, 2
( 4,
4 ) 6, 4
( 6,
0 ) 8,
Classify the transformation:
Horizontal translation
to the right. (x is a liar)
D: {-4 ≤ x ≤ 8}
R: {-1 ≤ y ≤ 5}

15. f(x+2) D: R: ( x , f(x) ) ( x’, f(x+2)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Are we changing the y-values here? NO. Those values will be the same. What is changing? The x. How?
So what will happen to this graph if x is a liar? X is saying, “I’m moving in the positive direction.” But x lies, so x is actually moving which way?
LEFT.
Classify the transformation: D: R:

16. f(x+2) D: R: ( x , f(x) ) ( x’, f(x+2)) ( -6, -1 ) ( -1 ) ( -4, 3 ) 3
( -6,
-1 ) ( -1 )
( -4,
3 ) 3
( -2,
1 ) 1
( 0,
5 ) 5
( 2,
2 ) 2
( 4,
4 ) 4
( 6,
0 )
So what will happen to this graph if x is a liar? X is saying, “I’m moving in the positive direction.” But x lies, so x is actually moving which way?
LEFT.
What is the domain of the new green graph?
What is the range?
How is that different? The x values shifted to the left by two. (Before it was {-6 ≤ x ≤ 6}
Classify the transformation: D: R:

17. Horizontal translation
f(x+2)
( x , f(x) )
( x’, f(x+2))
( -6,
-1 ) (
-8, -1 )
( -4,
3 )
-6, 3
( -2,
1 )
-4, 1
( 0,
5 )
-2, 5
( 2,
2 ) 0, 2
( 4,
4 ) 2, 4
( 6,
0 ) 4,
Do you guys see that? Do you see how the graph and the domain shifted to the left?
Classify the transformation:
Horizontal translation
to the left. (x is a liar)
D: {-8 ≤ x ≤ 4}
R: {-1 ≤ y ≤ 5}

18. f(x)-2 D: R: ( x , f(x) ) ( x’, f(x)-2)) ( -6, -1 ) ( ) ( -4, 3 )
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
While x is a liar, f(x) tells the truth. So here, where does f(x) say it’s going?
In the negative direction, down, two units. f(x) is the same as y.
Classify the transformation: D: R:

19. f(x)-2 D: R: ( x , f(x) ) ( x’, f(x)-2)) ( -6, -1 ) ( -6, ) ( -4, 3 )
( -6,
-1 ) (
-6, )
( -4,
3 )
-4,
( -2,
1 )
-2,
( 0,
5 ) 0,
( 2,
2 ) 2,
( 4,
4 ) 4,
( 6,
0 ) 6,
We moved the graph down, now we can fill in the table.
What’s another way to fill in the table? By subtracting two from f(x).
What is the domain of the new green function?
What is the range?
Which one is different from the original? The y-values, the range. (the original was -1<y<5)
Classify the transformation: D: R:

20. Vertical translation downward.
f(x)-2
( x , f(x) )
( x’, f(x)-2))
( -6,
-1 ) (
-6, -3 )
( -4,
3 )
-4, 1
( -2,
1 )
-2, -1
( 0,
5 ) 0, 3
( 2,
2 ) 2,
( 4,
4 ) 4, 2
( 6,
0 ) 6, -2
What happened to the range values? They shifted by a negative two =-3, 5-2=3
Classify the transformation:
Vertical translation downward.
(f(x) tells the truth)
D: {-6 ≤ x ≤ 6}
R: {-3 ≤ y ≤ 3}

21. f(x)+2 D: R: ( x , f(x) ) (x’, f(x)+2)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Now which direction will we move?
Classify the transformation: D: R:

22. f(x)+2 D: R: ( x , f(x) ) (x’, f(x)+2)) ( -6, -1 ) ( -6, ) ( -4, 3 )
( -6,
-1 ) (
-6, )
( -4,
3 )
-4,
( -2,
1 )
-2,
( 0,
5 ) 0,
( 2,
2 ) 2,
( 4,
4 ) 4,
( 6,
0 ) 6,
Now which direction will we move? UP
Is that vertical or horizontal? It’s happening to f(x), our y values, so that’s vertical.
Classify the transformation: D: R:

23. Vertical translation upward.
f(x)+2
( x , f(x) )
(x’, f(x)+2))
( -6,
-1 ) (
-6, 1 )
( -4,
3 )
-4, 5
( -2,
1 )
-2, 3
( 0,
5 ) 0, 7
( 2,
2 ) 2, 4
( 4,
4 ) 4, 6
( 6,
0 ) 6, 2
Classify the transformation:
Vertical translation upward.
(f(x) tells the truth)
D: {-6 ≤ x ≤ 6}
R: {1 ≤ y ≤ 7}

24. f(2x) D: R: ( x , f(x) ) (x’, f(2x)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Here is something happening to x. X is saying, “I’m being multiplied by 2.” But actually what will happen to the graph? What is the opposite of multiplying by 2? Dividing by 2.
Classify the transformation: D: R:

25. f(2x) D: {-3 ≤ x ≤ 3} R: {-1 ≤ y ≤ 5} ( x , f(x) ) (x’, f(2x)) ( -6,
( -6,
-1 ) (
-3, -1 )
( -4,
3 )
-2, 3
( -2,
1 )
-1, 1
( 0,
5 ) 5
( 2,
2 ) 2
( 4,
4 ) 4
( 6,
0 )
Classify the transformation:
Horizontal Compression
(by a factor of b from 1/b in the x direction; x is still a liar)
D: {-3 ≤ x ≤ 3}
R: {-1 ≤ y ≤ 5}
What happened to the x-values? The Domain? What happened to the range?
The coefficient is your factor of change.

26. f( x) 1 2 D: R: ( x , f(x) ) (x’, f(½x)) ( -6, -1 ) ( ) ( -4, 3 )
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Again, here is x saying, “I’m being divided in half!” What will actually happen?
X gets stretched by a factor of b = 2!
Classify the transformation: D: R:

27. f( x) 1 2 D: {-12 ≤ x ≤ 12} R: {-1 ≤ y ≤ 5} ( x , f(x) ) (x’, f(½x))
( -6,
-1 ) (
-12, -1 )
( -4,
3 )
-8, 3
( -2,
1 )
-4, 1
( 0,
5 ) 0, 5
( 2,
2 ) 4, 2
( 4,
4 ) 8, 4
( 6,
0 )
12,
Classify the transformation:
Horizontal stretch.
(stretched by in the x direction by a factor of c from 1/b; x is still a liar)
D: {-12 ≤ x ≤ 12}
R: {-1 ≤ y ≤ 5}
What happened to the domain? Changed by a factor of c from 1/b = 1/2. b=2
What happened to the range? Nothing.

28. 2f(x) D: R: ( x , f(x) ) (x’, 2f(x)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Now what’s happening? Is something happening to f(x) or to x here? F(x)
Does f(x) lie or tell the truth? Tell the truth.
So what will happen to the graph?
Will the domain change or will the range change?
Classify the transformation: D: R:

29. (stretched in the y direction by a factor of b from 1/b)
2f(x)
( x , f(x) )
(x’, 2f(x))
( -6,
-1 ) (
-6, -2 )
( -4,
3 )
-4, 6
( -2,
1 )
-2, 2
( 0,
5 ) 0, 10
( 2,
2 ) 2, 4
( 4,
4 ) 4, 8
( 6,
0 ) 6,
Classify the transformation:
Vertical stretch.
(stretched in the y direction by a factor of b from 1/b)
D: {-6 ≤ x ≤ 6}
R: {-2 ≤ y ≤ 10}
The range changed because we have something happening to the f(x), to the y-values.

30. f(x) 1 2 D: R: ( x , f(x) ) (x’, ½f(x)) ( -6, -1 ) ( ) ( -4, 3 ) ( -2,
( -6,
-1 ) ( )
( -4,
3 )
( -2,
1 )
( 0,
5 )
( 2,
2 )
( 4,
4 )
( 6,
0 )
Last one, what will happen? Is x changing or is f(x) changing?
How? Is it moving one direction or another? No. It’s being cut in half.
Will the domain change or the range? The Range. By how much?
Classify the transformation: D: R:

31. (compressed in the y direction by a factor of b from 1/b)
f(x) 1 2
( x , f(x) )
(x’, ½f(x))
( -6,
-1 ) (
-6, -½ )
( -4,
3 )
-4,
1.5
( -2,
1 )
-2, .5
( 0,
5 ) 0,
2.5
( 2,
2 ) 2, 1
( 4,
4 ) 4, 2
( 6,
0 ) 6,
Classify the transformation:
Vertical compression.
(compressed in the y direction by a factor of b from 1/b)
D: {-6 ≤ x ≤ 6}
R: {-1/2 ≤ y ≤ 5/2}
By half.

32. MOVE THE MONSTER 1) Graph: f(x-3)+2 2) Graph: –f(x+1)
We talked about parent functions yesterday, now we will look at how to manipulate functions using a super special parent function.