Math Unit 3 Test Flashcards
Relation:
Relation: any relationship between two variables (ie. x and y)
Function:
Function: a special type of relation where each value of x corresponds with only one value of y. [ie. All values of x are unique]
All functions are ?, but not all relations are ?
Relations
Functions
Function: only ONE ? for each ?
y for each x
Relation: MULTIPLE ?’s for an ?
y’s for an x
Function: For every value of x there is only ? value of y. The x values are ? repeated Note: it does not matter the y values are repeated
One
Not
Relation: For the x value of -1, there are three values of y (4,6,7); therefore, it is ? a function. The x values of -1 is repeated
Not
Function: ? x values repeated
No
Relation: The x value 1 is ?
Repeated
To determine if a graph is the graph of a function, principle:
for every x value there should be only one y value.
Vertical Line Test:
if you draw an imaginary vertical through the graph at any position, it will only ever cross the relation once. If it crosses more than one time, it is not a function.
y = 3x - 1, ?, ?
Linear, Function
x^2 + y^2 = 25, ?, ?
Circle, Relation
y = -2x2 + 5x - 1, ?, ?
Quadratic, Function
Function Notation
sub the (x) in the equation for the x and wherever else x is
Domain -
all possible input values for the relation - the x values
Range -
all possible output values for the relation - the y values
{(-2, 2), (-3, 3), (-4, 4), (-3, 5), (-1, 6)} -
D: {-4,-3,-2,-1} R: {2, 3, 4, 5, 6}
If it’s a line -
D:{xER} R:{yER}
If it’s a parabola -
D:{xER} R:{yER l y≥?}
If it’s a circle -
D:{xER l ? ≥ x ≥ ? } R:{yER l ? ≥ y ≥ ? }
In (not function notation) y = a(x - h)^2 + k (a is…
the direction of opening, and vertical stretch/compression)
In (not function notation) y = a(x - h)^2 + k (h is…
a horizontal translation left or right)
In (not function notation) y = a(x - h)^2 + k (k is…
a vertical translation up or down
The “a” = Direction of Opening
a ＞0 parabola opens UP
a ＜0 parabola opens DOWN
The “a” = Stretch/Compression
- l a l ＞1 stretched by a factor ‘a’ (multiply y values by ‘a’)
- 0＜ l a l＜1 compressed by a factor of ‘a’ (multiply y values by ‘a’)
The “h” = Horizontal Translation
(x-h)^2 translates ‘h’ units to the right
(x+h)^2 translates ‘h’ units to the left
The “k” = Vertical Translation
+k translates up ‘k’ units
-k translates down ‘k’ units
In (function notation) y = af [k (x - d)] + c (a is…
a = Direction of Opening (reflection of the x-axis) & Vertical Stretch/Compression
In (function notation) y = af [k (x - d)] + c (k is…
k = Reflection of y-axis & Horizontal Stretch/Compression
In (function notation) y = af [k (x - d)] + c (d is…
d = Horizontal translation left/right
In (function notation) y = af [k (x - d)] + c (c is…
c = Vertical translation up/down
Vertical Reflection (of the x-axis) - y = af(x)
If a < 0, all y values become NEGATIVE, all x values remain the same
Horizontal Reflection (of the y-axis) - y = f(kx)
If k < 0, all x values become NEGATIVE, all y values remain the same
Horizontal Translation (Left or Right) - y = f(x-d)
All x values move to the Left or Right, all y values remain the same
y=f(x-d) →Right y=f(x+d)→Left
Vertical Translation (Up or Down) - y = f(x)+c
All y values move Up or Down, all x values remain the same
y=f(x)+c →Up y=f(x)-c →Down
In y = af [k (x - d)] + c
a = Direction of Opening (reflection of the x-axis) & Vertical Stretch/Compression
k = Reflection of y-axis & Horizontal Stretch/Compression
d = Horizontal translation left/right
c = Vertical translation up/down
Vertical Stretch - y = 2√x
Multiply all Y values by 2
Vertical Compression - y = ½√x
Multiply all Y values by ½
a is either the ? or the ? factor
vertical stretch or the vertical compression factor
If a > 1, there is a ?
vertical stretch
If 0 < a < 1 , there is a
vertical compression
If A present -
multiply the y values of the parent function y=f(x) by A
Vertical stretch (if a is present …
multiply the y values by a)
Horizontal Compression √2x
Multiply all X values by ½
Horizontal Stretch √½x
Multiply all x values by 2
k is either the ? or the ? factor
Horizontal stretch or the horizontal compression
If k > 1, there is a…
horizontal compression
If 0 < k < 1, there is a…
horizontal stretch
If K present -
multiply the x-values of the parent function y=f(x) by 1/k the reciprocal of ‘k’)
The “a”
- Vertical Reflection (Direction of Opening)
a ＜0 reflection of the x-axis - Vertical Stretch/Compression
l a l ＞1 stretched by a factor ‘a’ (multiply y values by ‘a’)
0＜ l a l＜1 compressed by a factor of ‘a’ (multiply y values by ‘a’)
The “k
- Horizontal Reflection
k ＜0 reflection of the y-axis - Horizontal Stretch/Compression
l k l ＞1 compressed (multiply x values by the reciprocal of ‘k’→ ½)
0＜ l k l＜1 stretched (multiply x values by the reciprocal of ‘k’ → ½)
In any function y=af(kx):
- a represents the vertical stretch/compression and reflection Action: multiply y-values by ‘a’
- k represents the horizontal stretch/compression and reflection Action: multiply x-values by ‘1/k’ (reciprocal of ‘k’)
Mapping Notation Steps
- Create a Table of Values using a minimum of 3 key points from the parent function y = f(x)
- Apply the transformations to the x-values and y-values in the table to create a new Table of Values
- Create the equations to use on the table of values using your a, k, d, and c
- Graph the new transformed graph
- Mapping Notation Formula
(1/kx + d, ay + c)
Mapping Notation Formula
(1/kx + d, ay + c)
The reciprocal function
f(x) = 1/x
Values of f(x) = 1/x
-3 and -⅓
-2 and -½
-1 and -1
-½ and -2
-⅓ and -3
0 and DNE
⅓ and 3
½ and 2
1 and 1
2 and ½
3 and ⅓
Asymtopes;
An asymptote is a line that a graph approaches without touching. Similarly, horizontal asymptotes occur because y can come close to a value, but can never equal that value.
When f(x) =1/x…
Vertical asymptote at x = 0, and a horizontal asymptote at y=0
the y = a [k(x - h)] + c in f(x) = 1/x
y = a [1/k (x - d)] + c
So when C gets moved the horizontal asymptote changes too!
y = (1/x) + 3 - horizontal asymptote y = 3 and vertical x = 0
So when H gets moved the vertical asymptote changes too!
y = (1/x - 2) - vertical asymptote x = 2 and horizontal y = 0
Determining Transformed Functions from Graphs
Steps:
- If it opens down, a is negative
- If there’s a reflection in the y axis, k is negative too
- The vertex (x,y) determines the d (=x) and c (=y)
- To figure out the a-value is how many moves down/up between each point so over 1 down 2 is a value of -2
- To figure out the k-value is how many moves over between each point so over 4 down 1 is a k-value of 4
- Plug the identified values into the equation y = af [k(x - d)] + c
Inverse Relations Definition:
A function and its inverse relation can be described as the “DO” and the “UNDO” functions
Generally speaking, the process of finding an inverse is simply…
the swapping of the x and y coordinates.
In Inverse Relations - The ? and ? obviously will also be interchanged
The domain and range obviously will also be interchanged
This newly formed inverse will be a relation, but may not necessarily be a ?.
Function
If the function is f(x) , then the inverse of the function is denoted…
f -1(x) .
Let’s graph the function and its inverse:
1) First plot the points of the original function. (or use mapping notation to change a
parent function and then plot those points)
2) For each point, swap the x and y values, then plot the inverted point
3) Draw a mirror line to check
Find the inverse of a relation using the equation
Steps:
- Write the original equation as y=_____
- Switch the x-value and y-value in the equation
- Solve the new equation for y
- Graph the new equation if required
Parent Function x^2
X Y
(0,0)
(1,1)
(-1,1)
(2,4)
(-2,4)
Parent Function √x
X Y
(0,0)
(1,1)
(4,2)
(9,3)
Parent Function 1/x
X Y
(- 3, -⅓)
(-2, -½)
(-1, -1)
(-½, -2)
(-⅓, -3)
(0, DNE)
(⅓ , 3)
(½, 2)
(1,1)
(2, ½)
(3, ⅓)
