Functions
Function-a rule that assigns to each domain exactly one range
Vertical Line Test- a vertical line can only touch a point once going across a graph vertically
Domain- x values
Range- y values
To find out if a line is a function: use the vertical line test
Vertical Line Test- a vertical line can only touch a point once going across a graph vertically
Domain- x values
Range- y values
To find out if a line is a function: use the vertical line test
Finding the Domain and Range of a Function
Formula of a Function:
Slope of a function:
a function's slope is at a constant increase or decrease
-Lines are decreasing when slope is negative
-Lines are increasing when the slope is positive
Intercepts:
x-intercept-a point where the graph touches the x-axis
y-intercept- a point where the graph touches the y-axis
x-intercept-a point where the graph touches the x-axis
y-intercept- a point where the graph touches the y-axis
If you are given an equation:
To find the x-intercept: plug 0 in for the y values
To find the y-intercept: plug 0 in for the x values
To find the x-intercept: plug 0 in for the y values
To find the y-intercept: plug 0 in for the x values
Continuous/Discontinuity of a Function:
On a function---
o-not a point of a function
A full o- a point of a function
Continuous over the interval:
A graph is continuous if it does not contain endpoints
On a function---
o-not a point of a function
A full o- a point of a function
Continuous over the interval:
A graph is continuous if it does not contain endpoints
Jump Discontinuity:
Removable Discontinuity:
Infinite Discontinuity:
Symmetry
End Behavior
-the end behavior of a function: explains whether or not the function will continue in the positive/negative direction as the x-values (on the graph) continue in the positive or negative direction
As an example:
f(x)=x
The limit as x goes to (positive) infinity---> the function will continue in the positive direction (x=(positive) infinity)
The limit as x goes to negative infinity---> the function will continue in the negative direction (x=negative infinity)
As an example:
f(x)=x
The limit as x goes to (positive) infinity---> the function will continue in the positive direction (x=(positive) infinity)
The limit as x goes to negative infinity---> the function will continue in the negative direction (x=negative infinity)
An Example of How End Behavior is Written:
More about End Behavior:
Transformations
y--->-f(x) Vertical flip
y--->f(-x) Horizontal flip
y--->2f(x) Vertical shrink by a factor of 2
y--->1/2f(x) Vertical stretch by a factor of 1/2
y--->f(1/2x) Horizontal stretch by a factor of 1/2
y--->f(2x) Horizontal shrink by a factor of 2
y--->f(x)+3 the graph moves up
y--->f(x)-3 the graph moves down
y--->f(x-3) the graph moves to the right 3
y--->f(x+3) the graph moves to the left 3
y--->f(-x) Horizontal flip
y--->2f(x) Vertical shrink by a factor of 2
y--->1/2f(x) Vertical stretch by a factor of 1/2
y--->f(1/2x) Horizontal stretch by a factor of 1/2
y--->f(2x) Horizontal shrink by a factor of 2
y--->f(x)+3 the graph moves up
y--->f(x)-3 the graph moves down
y--->f(x-3) the graph moves to the right 3
y--->f(x+3) the graph moves to the left 3
Types of Functions
Regressions
To Find Regressions with a calculator:
Step 1: Press 2nd and Stat Plot--- turn Plot 1 on, make sure the type of plot is scatter, the x-list is L1, and the y-list is L2
Step 2: Press stat and go to edit--- Enter all the x values (from the table you have) into L1 and all the y values (from the table you have) into L2
Step 3: Exit out of that and make sure your window for your calculator is in the correct position
Step 4: Press stat and go to calf--- Press the type of regression you desire (ex. Linear, Exponential, Cubic, etc.)
Step 5: Enter in: L1,L2,Y1
(to get Y1: Press VARS, go to Y-Vars, go to function, and press Y1---this means the regression will be stored in Y=)
Step 6: Press enter and graph---the regression will appear
To See the Regression equation--- Press Y=
To Determine Which Regression's linear association stronger:
Finish step 5 and then look at each regressions R-squared Value
-If the R-squared value is closer to 1 or -1 then the regression is perfectly linear and the linear association is strong
Step 1: Press 2nd and Stat Plot--- turn Plot 1 on, make sure the type of plot is scatter, the x-list is L1, and the y-list is L2
Step 2: Press stat and go to edit--- Enter all the x values (from the table you have) into L1 and all the y values (from the table you have) into L2
Step 3: Exit out of that and make sure your window for your calculator is in the correct position
Step 4: Press stat and go to calf--- Press the type of regression you desire (ex. Linear, Exponential, Cubic, etc.)
Step 5: Enter in: L1,L2,Y1
(to get Y1: Press VARS, go to Y-Vars, go to function, and press Y1---this means the regression will be stored in Y=)
Step 6: Press enter and graph---the regression will appear
To See the Regression equation--- Press Y=
To Determine Which Regression's linear association stronger:
Finish step 5 and then look at each regressions R-squared Value
-If the R-squared value is closer to 1 or -1 then the regression is perfectly linear and the linear association is strong
Questions:
#1:What is the domain of f(x)=sqt(x+5)?
#2:Sketch a cubic function and state the end behavior of a cubic function.
#3:Graph this function: -2(x-2)+5=F(x) and state the transformations of this function.
ANSWERS:
Question #1
#2:Sketch a cubic function and state the end behavior of a cubic function.
#3:Graph this function: -2(x-2)+5=F(x) and state the transformations of this function.
ANSWERS:
Question #1
Step 1: Substitute 0 in for y and solve for x
Step 2: square both sides of the equation
Step 3: To get x by itself---subtract 5 by both sides
Step 4: x=-5
ANSWER: Domain: X ≥ -5
Question #2
Step 2: square both sides of the equation
Step 3: To get x by itself---subtract 5 by both sides
Step 4: x=-5
ANSWER: Domain: X ≥ -5
Question #2
Step 1: Sketch a cubic function
Step 2: Look at end behavior:
As x goes to positive infinity-->f(x)=infinity
As x goes to negative infinity-->f(x)=negative infinity
Question #3
Step 2: Look at end behavior:
As x goes to positive infinity-->f(x)=infinity
As x goes to negative infinity-->f(x)=negative infinity
Question #3
Remember Transformations of a function!!!
Common Core:
Save the Rhinos!
Black Rhinos are slowly disappearing from Earth. Construct a linear function that would help show the decrease of Black Rhinos on Earth in the past 5 years. How many Rhinos will be left in the next 3 years?
In the last 5 years:
Inital ---50 Rhinos
5 years ago (1)---42 Rhinos
4 years ago (2)---34 Rhinos
3 years ago (3)---26 Rhinos
2 year ago (4)---18 Rhinos
1 year ago (5)---10 Rhinos
Answers:
Black Rhinos are slowly disappearing from Earth. Construct a linear function that would help show the decrease of Black Rhinos on Earth in the past 5 years. How many Rhinos will be left in the next 3 years?
In the last 5 years:
Inital ---50 Rhinos
5 years ago (1)---42 Rhinos
4 years ago (2)---34 Rhinos
3 years ago (3)---26 Rhinos
2 year ago (4)---18 Rhinos
1 year ago (5)---10 Rhinos
Answers:
Step 1: Look at the rate each year
Step 2: Make a equation that would show the decrease of Black Rhinos on Earth
Step 3: Check the regression of the line with the calculator (R^2=1)---There is a strong association-- linear
Step 4: It will be 8 years since the initial amount of Black Rhinos
---Plug 8 in for the value of x and solve
Step 5: Rhinos will be extinct in 3 years
----OR----
Make a Linear Regression:
Enter in the points
Step 2: Make a equation that would show the decrease of Black Rhinos on Earth
Step 3: Check the regression of the line with the calculator (R^2=1)---There is a strong association-- linear
Step 4: It will be 8 years since the initial amount of Black Rhinos
---Plug 8 in for the value of x and solve
Step 5: Rhinos will be extinct in 3 years
----OR----
Make a Linear Regression:
Enter in the points
Make a Scatter Plot
Make a Linear Regression (need help? -- go to the Regression Notes in Functions)
Linear Regression:
Linear Regression Graphed
To find out how many Black Rhinos will be left in 3 years
Step 1: Press 2nd--->Calc
Step 2: Enter in 8 to x= because it will be 8 years since the initial amount
Step 3: Answer (Rhinos will be extinct)
Step 1: Press 2nd--->Calc
Step 2: Enter in 8 to x= because it will be 8 years since the initial amount
Step 3: Answer (Rhinos will be extinct)
Answer: -8x+50=y or 50-8x=y
In 3 years, Black Rhinos will be extinct!!
In 3 years, Black Rhinos will be extinct!!