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Pre-Calculus 12 > Chapter 1: Function Transformations > Flashcards

Chapter 1: Function Transformations Flashcards

1
Q

What are function transformations?

A

Taking functions and changing/altering them by:

  • moving them up/down (vertical translation)
  • moving them left/right (horizontal translation)
  • stretch (horizontal/vertical)
  • reflection
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2
Q

What are horizontal and vertical translations?

A

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

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3
Q

A horizontal translation affects which variable?

A

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
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4
Q

A vertical translation affects which variable?

A

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
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5
Q

What is vertex form and what does each variable mean?

A

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)

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6
Q

What is mapping notation?

A

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)

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7
Q

What does y = f(x) imply?

A

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

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8
Q

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

A

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

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9
Q

What is the form of a reflection?

A

y=-f(-x)

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10
Q

What is the mapping notation for a reflection?

A

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

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11
Q

A horizontal reflection affects which variable?

A

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

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12
Q

A vertical reflection affects which variable?

A

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

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13
Q

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

A

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

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14
Q

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

A

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

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15
Q

What is the form of a stretch?

A

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

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16
Q

What is the mapping notation for a stretch?

A

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

17
Q

A vertical stretch affects which variable?

A

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)

18
Q

A horizontal stretch affects which variable?

A

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)

19
Q

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

A

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

20
Q

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

A

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

Then you would describe your transformations accordingly.

21
Q

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

A

Horizontal and vertical translations

22
Q

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

A

Reflection transformations

23
Q

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

A

Stretch transformations

24
Q

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

A

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

25
Q

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

A

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!!

26
Q

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

A

y=af(b(x-h))+k

a = vertical stretch/reflection
b = horizontal stretch/reflection
h = horizontal translation
k = vertical translation
27
Q

How many transformations could you have on one graph?

A

You can have up to SIX transformations!!

  • Vertical reflection
  • Horizontal reflection
  • Vertical stretch
  • Horizontal stretch
  • Vertical translation
  • Horizontal translation
28
Q

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

A

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

29
Q

What is the definition of an inverse relation or function?

A

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.

30
Q

What is the mapping notation for an inverse relation?

A

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

31
Q

How do you identify transformations on a graph just by looking at it?

f(x) -> g(x)

A
  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!
32
Q

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

A
  • x and y are switched

- there has been a reflection over line y=x

33
Q

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

A

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.

34
Q

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

A

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.

35
Q

What is the general form of restricted domains?

A

Generally, your restricted domains will revolve around your “p” value because it deals with the x-values.

Therefore:

x<p>p</p>

36
Q

What happens to your domain and range in inverses?

A

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

37
Q

What must you remember about your notation when writing inverses?

A

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

38
Q

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

A

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!!!

Pre-Calculus 12 (3 decks)

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