U1 A2- Transformations Flashcards
Translations- what does h do
and what type of translation does this result in
Change the x value
Horizontal translation
Translations- what does k do
And what type of translation does this result in
Change the y values
Vertical translation
Translations- how do you know how the y value is affected
Determine k value by isolating for y
Translations- what are the k and h values in this equation
y=(x-5)+2
k=2
h=5
Translations- k>0, graph is a translated ______
Up
Translations- k<0, graph is translated _____
Down
Translations- how to determine the k value
Isolate for y
Translations- how to determine the h value
Always opposite of what you see in equation
Since there’s no way to isolate for x
Translations- h>0, graph is a translated ______
Right
Since that’s the direction of the positive values
Translations- h<0 graph is a translated ______
Left
Since that’s the direction of the negative values
Translations- Identify the k and h values in the expression
y-4=(x+3)
k= 4
h= -3
Steps for determining a translation
1) determine k and h values
2) associate direction (vert/hor and up/down or left/ right) with each value
Mapping rule tells you how to ____ a point/graph based on what’s _____
Transform
Occuring
Mapping rule is ____ related to the words of the transformation description
Directly
Translations- what is the mapping rule for k
(x,y) —-> (x,y + k)
Translations- what is the mapping rule for h
(x,y) —-> (x+h, y)
Reflections- you can’t just flip the graph over, you have to flip it ____ the line
About
Reflections- What are the x axis and y axis also called
x axis: y=0
y axis: x=0
Reflections: what is the mapping rule for reflecting a graph over the x axis
y —> -y
Reflections: how do you determine the new expression for reflection about the x axis
1) y —> -y
2) isolate for y
Invariant points - points that
___ ____ after a transformation occurs
Don’t change
Reflections- Which points are invariant for a reflection about the x axis
(And why)
X-intercepts
In the mapping rule, y becomes -y, but you can have -0
Reflections: what is the mapping rule for reflecting a graph over the y axis
x —-> -x
Reflections: how do you determine the new expression for reflection about the x axis
1) replace every x with (-x)
Since x —-> -x
Reflections- Which points are invariant for a reflection about the y axis
(And why)
y-intercepts
In the mapping rule, x becomes -x, but you can have -0
Reflection: reflecting over the x axis is a _____ reflection (direction )
Vertical
Reflection: reflecting over the y axis is a _____ reflection (direction )
Horizontal
Stretches: in a vertical stretch what variable changes
y
Stretches: how to determine the vertical stretch factor
Isolate for y
Stretches: what is the mapping rule for a vertical stretch factor
(x,y)—-> (x, ay)
Stretches: how to determine the horizontal stretch factor
Take the reciprocal
b) —-> (1/b
Stretches: what is the mapping rule for a horizontal stretch factor
(x,y)——> (1/b x, y)
How to get a table of values from graph in calc
1) hit 2nd
2) hit graph
3) hit table
Stretches: what are the invariant points for a vertical stretch
(And why)
X-intercepts
You can multiply any stretch factor by 0 and still get 0
REMEMBER HOW YOU DESCRIBE THE TRANSFORMATION IN WORDS SHOULD BE DIRECTLY HOW THE _____ ____ IS WRITTEN
Mapping rule
Stretches: what is the vertical stretch factor in this
y—-> 5y
(1/5)
Have to divide both sides by 5 (to isolate for y)
Stretches: what variable does a horizontal stretch change
x
Stretches: what are the invariant points for a horizontal stretch
(And why)
Y-intercepts
Can multiply any stretch factor by 0 and you still get 0
Stretches: what is the horizontal stretch factor in this
x—> 3x
(1/3)
Because that’s the reciprocal of 3
Stretches: if a mapping rule is already given do we have to isolate for y and/or take the reciprocal of x
(And why)
No
A mapping rule tells us DIRECTLY what happens
Stretches: if the stretch factor is GREATER than 1 there’s a(n) ________
Expansion
Stretches: if the stretch factor is BETWEEN 0 AND 1 there’s a(n) ________
Compression compression
What are the three types of reflections we deal with
- about the x-axis
- about the y-axis
- about the line y=x
Reflections about the line y=x:
What is the mapping rule
(x,y)—-> (y,x)
Reflections about the line y=x: what does this mean
y=f^-1(x)
INVERSE
So you have to switch x and y
Reflections about the line y=x:
How to determine the new equation
1) apply the mapping rule
(x,y)—->(y,x)
2) isolate for y again
Reflections about the line y=x: what are the invariant points
(And why)
Points of the graph that lie on the line of y=x
Even if you switch the x and y of these points they’re still the same
Reflections about the line y=x: IMPORTANT THING TO REMEMBER if there’s a square root in you final answer
There are actually two different equations:
+) and (-
Reflections about the line y=x: what is the vertical line test
If a vertical line drawn anywhere passes through the graph MORE THAN ONCE it’s NOT a function
Reflections about the line y=x: what does the vertical line test tell us
If a function is a function or not
Reflections about the line y=x: since x and y switch, ____ and ____ switch with each other to become the characteristics of the transformed graph
Domain
Range
Reflections about the line y=x:
What are the domain and range of the function y=f^-1(x)
y=f(x):
D: [-6,8]
R: [-2,6]
D: [-2,6]
R: [-6,8]
(Domain and range just switch for transformation)
Reflections about the line y=x: is a graph of a circle a function
(And why)
No, it fails the vertical line test
Vertical line would pass through the graph TWICE
What order should you always perform transformations in
and what is it this similar to
1) Stretches (multiply)
2) Reflections
3) Translations
BEDMAS
Generally speaking, the order of transformations is only important if two transformations occur in the ____ direction and a _____ is involved
(And example)
Same
Translation
(Horizontal stretch followed by a horizontal translation)
Does the Oder in which you perform transformations in matter
YES
What is the form of a function that has combined transformations compared to the original
y= af[b(x-h)]+k
How to determine the new graph/ point from an equation in the correct form
1) isolate for y if needed
2) factor out the “b” term if not already factored out
3) determine a mapping rule for each variable present (except x and y), go from left to write, one at a time
4) combine all of the mapping rules into one
5) apply that to the point/ graph
Combining transformations: when you’re writting out mapping rules is a (-) sign included with a rule or is it its own
A (-) SIGN IS ITS OWN RULE
How to determine an equation based off of an original graph and a transformed graph
1) determine if there’s a vert or hor stretch (and write mapping rule)
2) draw new graph
3) determine if there’s a reflection (and write mapping rule)
4) draw new graph
5) determine if there are translations (and write mapping rules)
6) draw new graph (this should exaclty match transformed graph now)
7) put mapping rules into equation using the correct form
Combining transformations: if x is ever being multipled directly by a # in a binomial (2x-1) what do you HAVE to do
Factor out the 2
The graph of y= root f(x): What is the domain and range of root x
(And why)
D: [0, infinite)
R: [0, infinite)
Can’t square root a negative number, so value has to be above 0
How to get a decimal into a fraction using calc
Hit math, frac
The graph of y= root f(x): what is the domain and range of this function when
a and b>0
(And why)
D: [0, infinite]
R: [0,infinite]
(Both values being greater than 0 means there are no negative values, so no reflections, therefore the graph cannot go below 0 on either)
The graph of y= root f(x): what are the invariant points
And why
y=0 (x intercept)
And
y=1
(If you take the square root of each of these values you STILL get the same answer)
The graph of y= root f(x): if a graph is transformed into this, what part of the graph disappears
(and why)
The negative part of y
Can’t square root a negative value
The graph of y= root f(x): what is the mapping rule
(x,y)—-> (x, root y)
The graph of y= root f(x): does the x co-ordinate ever change
No
The graph of y= root f(x): steps for graphing this
1) draw some new points at convienent spots (and connect them using straight lines) using the mapping rule (x,y)—-> (x, root y)
2) draw horizontal lines at y=0 and uy=1 (these are the invariant points)
3) mark invariant points
4) take negative y parts out of the graph
5) connect the dots using the rule that if y>1 you draw a curve below the line, and if 0
The graph of y= root f(x): rules for drawing curves on graph
if y>1 the curve will be drawn ____ the line
Below
The graph of y= root f(x): the only transformations that affect domain and range are _______ and _______
Reflections
Translations
Point of discontinuity: equation ____ be simplified further
Can
Vertical asymptote: equation _____ be simplified further
Can’t
Always state restrictions on the ____ function, not the factored one
Original
Often, a rational expression will also exhibit a ______ asymptote in addition to a discontinuity
Horizontal
Horizontal asymptotes are determined by _____
Inspection
Horizontal asymptote: how many situations are there
3
Horizontal asymptote: if the degree of the leading coefficient in the denominator is LARGER than the degree of the numerator, the horizontal asymptote is always ____
y=0
Horizontal asymptote: if the degree of the leading coefficient in the denominator is EQUAL TO the degree of the numerator, the horizontal asymptote is always be the ___ of the coefficients of the ____ degree
Ratio
Highest
Horizontal asymptote: if the degree of the leading coefficient in the denominator is LESS than the degree of the numerator, there is ____ horizontal asymptote
No
What is the horizontal asymptote in this equation:
x^2)/(3x^2-x-5
(1/3)
Take the leading coefficients of the highest degree in the num and denom
The graph of
y= (x^2 + bx + c)/(x-d)
is a straight line with a point of discontinuity at (2,5). What is the value of c
1) x=2 is the restriction so the bottom must me (x-2)
2) since it’s a discontinuity something has to cancel out for the expression to be simplified so one of the top factors is (x-2)
3) cancel out the (x-2) on the top and bottom and you’re left with y=(x-a)
4) y=5 and x=2 so plug those values in to solve for a, you get 3, which means the other factor is (x+3)
5) foil the top factors out so you have it in the correct form, c=(-6)
How do you find the y value of a point of discontinuity
Sub the restricted c value into the simplified function
How to find the domain of a rational function
State restrictions on the denominator
How to find range of a rational function
Sub x value(s) that are discontinuities into equation to determine y- value(s) range
How to determine an equation from the graph of a rational function (graph with swoopy lines)
1) determine any discontinuities (POD or vert asymptote) and find bottom factor(s) from that, can also find top factor if stuff has to cancel
2) determine horizontal asymptotes and make sure equation lies up with that corresponding situation
3) if there are x intercepts you can find a factor in the numerator
4) find a point on the graph and sub in those x and y values to find the coefficient
What does the x intercept tell us when we’re finding an equation from a graph
A factor in the numerator
How to find the inverse of a rational function
1) swap x and y
2) isolate as usual and if you get stuck:
- FOIL
- MAKE SURE LIKE TERMS THAT YOU HAVE TO ISOLATE ARE ON THE SAME SIDE
- FACTOR SOMETHING OUT
How to find the new domain/range when a function is transformed when you’re given the old points for domain/range
ONLY apply the y transformations to the range points
ONLY apply the x transformations to the domain points
What type of function is the one where you have to draw curves above and below the line on the graph
y= root x
If transformations are given to you in word form what order do you use when you put them into an equation
Exactly the SAME order that was given to you in words
The graph of y= root f(x): rules for drawing curves on graph
if y is greater than 0 but less than 1 the curve will be drawn _____ the line
(and why)
Above
When you square root a number less than 1 you end up with a bigger number than what you started with
When determine vertical stretch factor what do you always have to make sure of
y is ISOLATED
How do you determine the domain of a composite function
Have to put the two functions together first before you state the restrictions
