advanced functions 1.1. Flashcards
WHAT IS A RELEATION?
A relation is an expression that demonstrates the connection (relationship) between two variables: an independent variable and a dependent variable
AN EXPRESSION CAN IN THE FORM OF…?
SET OF ORDERED PARIS, MAPPING DIAGRAM, TABLE OF VALUES, GRPAH, AND DESCRIPTIUON IN WORDS OR A RULE
WHAT IS A FUNCTION
is a relation where each value of the independent variable x corresponds to only one value of the dependent variable y (no repeated x values, but it’s okay for the y values to be repeated).
HOW TO WRITE FUNCTION NOTATION
- WE REPLACE F(X) WITH Y
- F(X) MEANS “THE VALUE OF THE DEPENDANT VARIABLE FOR A SPECIFIC VALUE OF X
- IT MEansS THAT Y = F(X)
WHAT STEPS SHOUDL YOU TAKE TO MAKE AN EQUATION FOR THE FUNCTION?
DEFINE THE VARIABLES, WRITE THE EQUATIONS
WHAT IS THE DOMAIN
IS THE SET OF ALL POSSIBLE VALUES OF THE INDEPENDANT VARIABLE?
WHAT IS THE RANGE?
IS THE SET OF ALL POSSIBLE VALUES OF THE DEPENDANT VARAIBLE
HOW TO TELL IF THE RELATION IS THE FUNCTION?
- YOU CAN PERFROM THE VERTICAL LINE TEST
- DOES the DEPENDANT VARRIABLE MAPPED INTO TO DIFFERENT Y VALUES. X = 0 TWICE.
What are linear functions?
Slope y-intercept form: y=mx+b
Standard form: Ax+By=C or Ax+By+C=0
What are quadratic functions
Standard form: y = ax2 + bx + c
Vertex form: y = a(x – h)2+k
Factored form: y =a (x–s)(x– t)
What are some relations for Circles and are they functions?
Centre at the origin: r2 = x2+ y2 Centre at (a, b): r2= (x – a)2+ (y – b)2
- They are not functions because they do not pass the vertical line test.
IS THIS a FUNCTION?
This is a linear function because it is in the form y=mx+b
y=2–x
IS THIS a FUNCTION?
y=2−x can be rearranged to y=−x+2
This is a linear function because it is expressed in the form y=mx+b
y=5x2−9x+20
IS THIS a FUNCTION?
This is a quadratic function in standard form because it is expressed in the form
y=ax2+bx+c
y=−11(x+8)2+33
IS THIS a FUNCTION?
This is a quadratic function in vertex form because it is expressed in the form
y=a(x – h)2+k
y=33+9(x−4)2
IS THIS a FUNCTION?
HintBold text End: y=33+9(x−4)2 can be rearranged to y=9(x−4)2+33
This is a quadratic function in vertex form because it can be expressed in the form
y=a(x–h)2+k
y=−5(x−3)(x+4)
IS THIS a FUNCTION?
This is a quadratic function in factored form because it can be expressed in the form
y=a(x–s)(x–t)
25=x2+y2
IS THIS a FUNCTION?
This is not a function because it is a circle in the form r2=x2+y2
5x2+7y=8
IS THIS a FUNCTION?
HintBold text End: 5x2+7y=8 can be expressed as y=−57x2+87
This is a quadratic function in standard form: y=ax2+bx+c
For the function f(x)=x2+10x evaluate each of the following:
f(7)
f(2) + f(3)
2f(4)
f(7)=(7)2+10(7)
=49+70
f(2)+f(3)=[(2)2+10(2)]+[(3)2+10(3)]
=(4+20)+(9+30) =24+39 =63 ------ 2f(4) =2 [(4)2+10(4)] =2(16+40) =2(56) =112
is this a function
{(1, 4), (2, 8), (3,16), (4,32)}
The relation is a function because there is no repeated x – value.
is this a function
{(3,0), (2, 13), (3,-9), (5,11)}
The relation is not a function because x = 3 is repeated in the points (3,0) and (3,-9).
x y 1 2 2 5 3 10 4 17 5 26
IS this a function?
The relation is a function since there is no repeated x -value
y=x2+100x−200
Is this a function
The relation is a quadratic function in the form y=ax2+bx+c
y=2x−23
is this a function
The relation is a linear function in the form y=mx +b
625 = x2 + y2
Is this a function?
The relation is not a function because it is an equation of a circle in the form
r2=x2+y2
A friend is visiting from the U.S and he needs help figuring out how many Canadian dollars he is going to get for his US dollars. Given that the exchange rate for today is 1 US dollar to 1.3 Canadian dollars, make a basic equation that your friend could use to convert his US dollars to Canadian dollars (it should be a direct substitution equation).
Let y represent the amount of Canadian dollars and x the amount of US dollars
y=1.3x
After one week, your friend is heading back to the U.S and he has a few Canadian dollars to change back to U.S dollars. Given that the exchange rate for today is 1 US dollar to 1.3 Canadian dollars, make a basic equation that your friend could use to convert his Canadian dollars to U.S dollars (it should be a direct substitution equation).
Let x represent the amount of Canadian dollars and y the amount of US dollars
y=x1.3
Inverse Equations: What are they?
Inverse means opposite in position or effect.Bold text End
The inverse of a relation or a function is the opposite of the original relation, or function.
Using the layman’s language, it means switching the independent (x) values and the dependent (y) values.
Given f(x)=(x−1)2−2
a) Determine the equation of the inverse.
b) State the domain and range of the function and its inverse.
A) f(x)=(x−1)2−2 can be written as y=(x−1)2−2
x=x+2±x+2
√1±x+2
√(y−1)2−2=(y−1)2=y−1=y switch x and y add 2 to both sides of the equation apply the square root sign to both sides of the equation and remember to add ±to the square root sign, since the square root can be either positive or negative. add 1 to both sides of the equation.
The relation is not a function since every x -value results in two y values because of ± on the equation.
∴ the inverse is y= SQroot. 1±x+2
B)
f(x)=(x−1)2−2
The domain of any quadratic function is not restricted
D={x∈R}
The range is affected by the y-value of vertex and the direction of opening of the parabola
R={y∈R | y≥−2}
Since graph opens upwards, the y – value of the vertex is -2
Inverse
The domain of the original function becomes the range of the inverse function and the range of the original function becomes the domain of the inverse function (switch x and y)
D={x∈R | x≥−2}
R={y∈R}
Given f(x)=−2x+1
a) Determine f−1(x)
b) State the domain and range of f(x) and f−1(x) . Explain your thought process
y=−2x+1x=−2y+1x−1=−2yx−1−2=y−x+12=yf(x)=−2x+1f−1(x)=-x+12 switch x and y subtract 1 from both sides of the equation divide both sides of the equation by −2
B)
f(x)=−2x+1
The domain of any linear function is not restricted.
Therefore, D={x∈R}
The range of any linear function is not restricted
Therefore, R={y∈R}
f−1(x)=x−1−2
The domain of original function becomes the range of the inverse function and the range of the original function becomes the domain of the inverse function
D={x∈R}
R={y∈R}
a) Determine f−1(x) for each of the following functions. Explain your thought process
a) f(x)=2x−1
b) f(x)={(−3,1),(−2,0),(−1,−1),(0,−2)}
A)
Therefore, f−1(x)=2/x+1
B) f−1(x)={(1,−3),(0,−2),(−1,−1),(−2,0)}
HOW TO FIND THE INVERSE OF GRAPHS?
- MAP POINTS ON THE FUNCITONS, AND REVERSE BOTH X AND Y VALUES.
How would you sketch the graph, domain, and range of each function?
Quadratic: f(x)=x^2
Square root f(x)=√x
rational ; f(x)=1x
xample
x f(x) -2 f(−2)=(−2)2=4 -1 f(−1)=(−1)2=1 0 f(0)=(0)2=0 1 f(1)=(1)2=1 2 f(2)=(2)2=4
Example
x f(x) 0 f(0)=0–√=0 1 f(1)=1–√=1 4 f(4)=4–√=2 9 f(9)=9–√=3
x f(x) -2 f(−2)=1−2=−0.5 -1 f(−1)=1−1=−1 Vertical asymptote x=0 Horizontal asymptote y=0 1 f(1)=11=1 2 f(2)=12=0.5
WHAT FUNCTION IS THIS?
f(x)=a[k(x−d)]2+c.
QUAD FUNCTION
f(x)=a√k(x−d)+c.
WHAT FUNCTION IS THIS?
SQUARE ROOT
f(x)=a / k(x−d) +c.
WHAT FUNCTION IS THIS?
THIS IS A RATIONAL FUNCTIONAL
Description of changes
What happens to the graph when you move the slider to negative
A values?
The original graph is reflected across the X-axis
What happens to the graph when you move the slider to negative K - values?
GRAPH IS REFLECTED IN ACROSS THE Y AXIS
WHAT HAPPENS TO THE GRAPH WHEN YOU MOVE THE SLIDER TO A-VALUES THAT ARE GREATER THAN 1?
THE ORIGINAL GRAPH IS VERTICALLY STRETCHED BY A FACTOR OF A
WHAT HAPPENS TO THE GRAPH WHEN YOU MOVE THE SLIDER TO K-VALUES THAT ARE GREATER THAN 1
The original graph is horizontally compressed by a factor of 1/k
WHAT HAPPENS TO THE GRAPH WHEN YOU MOVE THE SLIDE R TO A-VALUES THAT ARE BETWEEN 0 AND 1?
The original graph is vertically compressed by a factor of A
What happens to the graph when you move the slider to k that are between 0 and 1?
The original graph is horizontally stretched by a factor of 1/k
What happens to the graph when you move the slider to positive C values?
The original graph moves upwards
What happens to the graph when you move the slider to negative C values?
The original graph moves downwards
What happens to the graph when you move the slider to positive d values?
The original graph moves to the right
What happens to the graph when you move the slider to negative D-values?
The original graph moves to the left
WHAT IS THE NAME OF THIS TRANSFORMATION?
f(x)+c
Vertical Translation
c > 0: Graph moves up c units
c < 0: Graph moves down c units
(x, y + c)
f(x−d)
SUMMARY OF TRANSFORMATION
Horizontal Translation
d > 0: Graph moves right d units
d < 0: Graph moves left d units
(x + d, y)
SUMMARY OF TRANSFORMATION
af(x)
Vertical Stretch
a > 1: Graph is stretched vertically by a factor of a.
0 < a < 1: Graph is compressed vertically by a factor of a.
(x, ay)
SUMMARY OF TRANSFORMATION f(kx)
Horizontal Stretch
k > 1: Graph is compress horizontally by a factor of 1/k
0 < k < 1: Graph is stretched horizontally by a factor of 1/k
SUMMARY OF TRANS
−f(x)
Vertical reflection
Graph is reflected across the Italic text startxItalic text End-axis
SUMMARY OF TRANS
f(−x)
Horizontal reflection
Graph is reflected across the Y-axis
How does transformations inside a function affect the x values
any trans done inside the function (K,D(, affects the x-values in the opposite manner
any trans done inside the function (A,c) affects the y-values in the opposite manner
what is the transformation statement?
and what is the order of Transformation?
(x,y) -> (x/k + d, ay + c)
The transformation statement is a statement that summarizes the effects of the parameters a, k, d and c on the x and y coordinates. It can be used to determine the coordinates of a transformed point, given a point on the parent function, or vice versa.
When describing transformations, start with stretches, compressions and reflections in any order, then describe translations last.
f(x)=−5(x−2)2+11 , ( -5, 25)
|state the parent function
write the general form
state the parameters
describe the transformations in the correct order
write the transformation statement
use the transformation statement to determine the coordinates of the given point (given point is on the parent function) on the transformed graph
CHECK WORK 1.3
f(x)=3 √-1/4x+16 , (49, 7)
state the parent function
write the general form
state the parameters
describe the transformations in the correct order
write the transformation statement
use the transformation statement to determine the coordinates of the given point (given point is on the parent function) on the transformed graph
CHECK WORK 1.3
f(x)=−2/6x+18 −5 , ( -2, −12 )
state the parent function
write the general form
state the parameters
describe the transformations in the correct order
write the transformation statement
use the transformation statement to determine the coordinates of the given point (given point is on the parent function) on the transformed graph
CHECK WORK 1.3
what is an x-intercept
an x –intercept is a point of contact between a graph and the x -axis. A parabola can cross or just touch the x -axis
x –intercepts are also called zeros
what are turning points?
A turning point is a point where a graph changes from increasing to decreasing or decreasing to increasing
what are optimum values?
If the quadratic function opens up, the graph has a minimum value
If the quadratic function opens down, the graph has a maximum value
what is the y-intercept?
The y-intercept of a quadratic function is the point where the graph crosses or touches the y-axis
what is the axis of symmetry?
The axis of symmetry is the imaginary line that divides the quadratic graph into two equal branches
The equation of the axis of symmetry is written in the form x=x -value of the vertex
What can we learn, given the vertex form of the equation of a quadratic function?
Vertex Form: f(x)=a(x−h)2+k
- To calculate the x –value of the vertex, set the expression in the bracket to zero and isolate x
- The y –value of the vertex is k , as observed in the equation
- The coordinates of the vertex are (x,y)=(h,k)
- If a and k have the same sign, the graph has no zeros
- If a and k have opposite signs, the graph has two zeros
- If k=0 , the graph has one zero
- If a>0 , the graph opens up (graph has a minimum)
- If a<0 , the graph opens down (graph has a maximum)
- To calculate the y –intercept, substitute x=0 and solve for y
For the given equation f(x)=−2(x−3)2−11 determine:
- The direciton of opening
- The number of zeros
- y and x intercept
- axis of symmetry
- Optimum value (max or min)
- Vertex
- Domain and Range
Downwards since a is negative
a and k have the same sign, therefore the graph has no zeros
