Chapter 1: Function Transformations

Question 1

What are function transformations?

Answer 1

Taking functions and changing/altering them by:

• moving them up/down (vertical translation)
• moving them left/right (horizontal translation)
• stretch (horizontal/vertical)
• reflection

Question 2

What are horizontal and vertical translations?

Answer 2

Sliding the graph to the left/right along the x axis (horizontal) or up/down along the y axis (vertical)

Question 3

A horizontal translation affects which variable?

Answer 3

HT involves the “x” variable; changes the x value, meaning that it shifts the graph left/right

General form: y = f(x-h)

• If h>0 (positive), move to the RIGHT
• If h<0 (negative), move to the LEFT
• *RULE: Take the opposite sign of what’s with x

Question 4

A vertical translation affects which variable?

Answer 4

VT involves the “y” variable; changes the entire function (y value), meaning that it shifts the graph up/down

Actual form: y-k = f(x-h)
**BUT y is typically seen isolated
Therefore, the GENERAL FORM: y = f(x-h) + k
- If k>0 (positive), move UP
- If k<0 (negative), move DOWN

Question 5

What is vertex form and what does each variable mean?

Answer 5

QUADRATIC FUNCTION - Vertex form: y = a(x-p)^2 + q

Where (x,y) are coordinates
a = stretch/compression
p = AOS
(p,q) = vertex

**NOTE: You take the opposite sign of p when trying to determine the vertex.

Ex) y = (x+2)^2 - 5
Vertex = (-2, -5)

Question 6

What is mapping notation?

Answer 6

Mapping notation is writing describing the translations as coordinates.

FORM: (x,y) -> (x+h, y+k)

HT: (x,y) -> (x+h, y)
VT: (x,y) -> (x, y+k)

Question 7

What does y = f(x) imply?

Answer 7

y=f(x) means that this function is RANDOM and cannot be calculated for its values. It is unlike a cubic or quadratic function.

Question 8

What is one important rule you must remember for the h value (horizontal translation)?

Answer 8

When you’re taking the value for h (or p in a quadratic function, vertex form) you need to remember to take the OPPOSITE SIGN for that value.

Ex) y=(x-1) actually means h=1
y=(x+9)-6 actually means h=-9

Question 9

What is the form of a reflection?

Answer 9

y=-f(-x)

Question 10

What is the mapping notation for a reflection?

Answer 10

(x,y) -> (-x,-y)

Question 11

A horizontal reflection affects which variable?

Answer 11

HR involves the “x” variable. When there is a -1 multiplied to the b value, the graph is reflected horizontally across the y-axis.

Question 12

A vertical reflection affects which variable?

Answer 12

VR involves the “y” variable. When there is a -1 multiplied to the a value, the graph is reflected vertically across the x-axis.

*Hence: In vertex form, when you see -a, your parabola is turned upside down (or right side-up) across the x-axis

Question 13

If you’re given a line x=0, what kind of reflection is this?

Answer 13

x=0 means that no matter which y-values you pick, x will always be 0.

Thus, if you were to graph a series of points with x=0, you would get a VERTICAL LINE.

AKA: a horizontal reflection

Question 14

If you’re given a line y=0, what kind of reflection is this?

Answer 14

y=0 means that no matter which x-values you pick, y will always be 0.

Thus, if you were to graph a series of points with y=0, you would get a HORIZONTAL LINE.

AKA: a vertical reflection

Question 15

What is the form of a stretch?

Answer 15

y=af(bx) or ay=f(bx)

Question 16

What is the mapping notation for a stretch?

Answer 16

(x,y) -> (x/|b|, ay)

Question 17

A vertical stretch affects which variable?

Answer 17

VS involves the “y” variable. The y-values are multiplied by a factor of |a| and the x-values remain the same.

The graph will be stretched on the y-axis (tall/short)

Question 18

A horizontal stretch affects which variable?

Answer 18

HS involves the “x” variable. The x-values are multiplied by a factor of 1/|b| and the y-values remain the same.

The graph will be stretched on the x-axis (wide/slim)

Question 19

What is one important rule you must remember for the b value (horizontal stretch)

Answer 19

You always take the reciprocal of b, meaning that you take 1/b instead of just b.

*This is the same for (x-h) where you take the opposite sign for h

Question 20

Sometimes you’ll encounter ay = f(x), how do you describe the transformations?

Answer 20

You need to take the reciprocal of a or isolate y by dividing both sides by a.

Then you would describe your transformations accordingly.

Question 21

The equation: y=f(x-h)+k represents what kind of transformation?

Answer 21

Horizontal and vertical translations

Question 22

The equation: y=-f(-x) represents what kind of transformation?

Answer 22

Reflection transformations

Question 23

The equation: y=af(bx) represents what kind of transformation?

Answer 23

Stretch transformations

Question 24

Why do you only take the absolute value of a and b for stretches?

Answer 24

Because stretches refer to the change of shape in a graph. The negative value would be the reflection that changes the orientation.

Question 25

When combining transformations, what is the most important step you must take before going to graph or map?

Answer 25

Make sure your equation is in FACTORED FORM! There is a HUGE difference between y=-2f(3x+6)-4 and y=-2f(3(x+2))-4 **Your h value could be INCORRECT!! WATCH OUT!!

Question 26

What is the general form for combining transformations? What does each variable stand for?

Answer 26

y=af(b(x-h))+k ``` a = vertical stretch/reflection b = horizontal stretch/reflection h = horizontal translation k = vertical translation ```

Question 27

How many transformations could you have on one graph?

Answer 27

You can have up to SIX transformations!! - Vertical reflection - Horizontal reflection - Vertical stretch - Horizontal stretch - Vertical translation - Horizontal translation

Question 28

What must you remember about values affecting y and values affecting x?

Answer 28

Any time you have a value affecting the entire function (y), you can simply take the value as written. Ex) k and a BUT! If you have a value affecting the x-values, you have to take the OPPOSITE/RECIPROCAL!! Ex) h and b

Question 29

What is the definition of an inverse relation or function?

Answer 29

An inverse relation/function is a relation/function where the x and y are switched. In other words, if f(x) has a domain A and range B, the INVERSE, which would be f-1(x) would have a domain B and range A. The x and y values get swapped and the inverse is technically a REFLECTION across line y = x y = x would be the line where all x values are equal to the y values and vice versa.

Question 30

What is the mapping notation for an inverse relation?

Answer 30

(x,y) -> (y,x)

Question 31

How do you identify transformations on a graph just by looking at it? f(x) -> g(x)

Answer 31

1. Look for REFLECTIONS first! Has the graph been reflected across the y-axis (horizontal -b) or x-axis (vertical -a)? 2. Look for STRETCHES second! [HORIZONTAL STRETCH - b] - Count the units from left to right on the original graph and compare it to the units from left to right on the transformed graph. What must you multiply the units by to get from the original to the transformed? [VERTICAL STRETCH - a] - Count the units up and down on the original graph and compare it to the units from up and down on the transformed graph. What must you multiply the units by to get from the original to the transformed? 3. Look for TRANSLATIONS last! After the stretches have been applied (if any), has the graph been moved left/right/up/down? If so, by how many units? You can figure this out by using the mapping notation 4. Once you've determined a,b,h,k, you can start writing your transformed equation!

Question 32

How is f(x) related to the points of its inverse?

Answer 32

- x and y are switched | - there has been a reflection over line y=x

Question 33

What kind of function may need a RESRICTED DOMAIN and why?

Answer 33

Quadratic functions or anything that does NOT pass the horizontal line test will need a restricted domain or else the inverse will not be a function. This happens because when the graph is inverted, some x-values have more than one y-value, making it NOT a function.

Question 34

What is the purpose of a vertical line or horizontal line test?

Answer 34

The vertical line test tests for a function. Your graph cannot hit the vertical line more than once, or else it is NOT a function. For inverses, you can use the horizontal line test on the original graph. If the original graph touches the horizontal line more than once, the inverted graph will NOT be a function. Similarly, if the inverted graph does not pass the vertical line test, it is NOT a function.

Question 35

What is the general form of restricted domains?

Answer 35

Generally, your restricted domains will revolve around your "p" value because it deals with the x-values. Therefore: x

p

Question 36

What happens to your domain and range in inverses?

Answer 36

Your domain A and range B become domain B and range A.

Question 37

What must you remember about your notation when writing inverses?

Answer 37

Upon changing the function/relation to its inverse, you need to change y to y-1 or f(x) to f-1(x)

Question 38

Chapter 1 Test Mistake: When you're given a quadratic equation and asked to find the new equation after a HORIZONTAL REFLECTION, what would your equation look like? Example: y=(x+7)^2 +5

Answer 38

We are talking about a HORIZONTAL REFLECTION here - meaning that its b=-1 Therefore, it is only affecting the x-value. Your answer should be: y=(-x+7)^2 +5 NOT y=(-x-7)^2 +5 The "-" sign DOES NOT get distributed to everything in the brackets!!!