AP Calculus AB Unit 1 Project
Mind Map (Covers 1.1-1.6)
By: Kevin Virk Grade 12
Southpointe Academy
1.2 Functions and Relations
1.1
Lines
Increments
How much a point moves and can be represented in ∆x=x₂-x₁ and ∆y=y₂-y₁
Increments are useful for finding slope
Slope
Is associated with the coefficient "m" and is used to know a line's incline. The formula for a slope is ʸ²⁻ʸ¹⁄ₓ₂₋ₓ₁ or as it is commonly referred to as, "rise over run"
Perpendicular Lines
Two lines can be considered as perpendicular if the slopes of both lines are the negative reciprocal of each other (m1 x m2 = -1)
In this example there are the lines y=3x+2 and y=-1/3x+10/3 which are perpendicular to each other
Fun Fact
There is one solution to the system of equations that the lines represent as there is only one intersection
Slope of 0
Horizontal Line
Undefined Slope
Vertical Line
Parallel Lines
Two lines can be considered as parallel if the slopes of both lines are identical
(m1=m2)
In this example there are the lines y=3x+8 and y=3x+4 which are parallel to each other
Fun Fact
There are no solutions for the system of equations that the lines represent as they will never intersect
Coincident Lines
When there are lines that have the same slope and y-intercept thus are on top of each other but appear to look like one line
(m1=m2) (b1=b2)
In this example both lines are y=9/2x+8
Fun Fact
There are infinite solutions for the system of equations the lines represent as they with "intersect" infinitely
Applications
It can help with Physics for calculating velocity from a displacement vs time graph. There is also a use in Economics as lines such as Budget Lines can be used and there can be a understanding leading to better economic analysis
Even and Odd Functions Symmetry
Domain and Ranges
Domain: Symbolized by D, it is a set of guidelines/restrictions for the x axis
Range: Symbolized by R, it is a set of guidelines/restrictions for the y axis
Piecewise Functions
A function that will change depending on the domain that is listed
Functions
A rule that will help to create a relationship between and independent and dependent variable
Composite Functions
A function within a function. The innermost function will be solved first. Example: f(g(x))
In the example below g(x) would be solved and then substituted to solve for f(x)
Ways to represent
Function Machine
Mapping
Table of Values
Graphs
Equation
How to Recognize
There are simple steps to know if something is a function
Graphically: Vertical Line Test
If there is a curve and you place a vertical line anywhere in the graph and that vertical line does not intersect more than one time, the graph is a function
In this example, the circle on the left is a relation while on the curve on the right is a function
What would this mean: Relation vs Function
A relation would be considered as any set of ordered pairs due to which any function can be a relation. However not any relation is a function as a function is a relation in which there is a unique y present for every value of x
You can see this by mapping the relations and seeing if the ordered pairs have unique x values. However the y can have many x values
X cant have more than one y value or else it fails the Vertical Line Test
Function Machine
Table of Values
A chart which helps to show a set of ordered pairs and can be later turned into a graph
Equations
Mapping
Graphing
Graphically
If the function is plotted as a graph then you can check the following to see whether it is even or odd
Even Functions: Reflected on the y axis
Odd Functions: Symmetrical on Origin
Algebraically
To completely verify whether something is a even or odd function algebra can be used. Graphically verifying is good but algebraically is gives you a definite answer
Even Functions: f(-x)=f(x)
Odd Functions: f(-x)=-f(x)
End Behaviour
Even functions will go from Quadrant I -> II or Quadrant III -> IV
Odd Functions will go from Quadrant I -> III or Quadrant II -> IV
Absolute Value Functions
A V shaped function written as f(x)=|x| in which each point that is plotted is its absolute value.
How to draw
You will look at the domain and make a table of values for each domain and combine to make one table of values. Then you will graph it accordingly and verify by substitution
1.3
Exponential Functions
Exponential Growth
Growth which multiplies and its growth increases as time goes on.
Its equation is: y=aBˣ in which B>1 and "a" is the initial amount
Exponential Decay
When a quantity decays (opposite of growth) by a multiple and its eventually it will reach a limit during which the decay will be virtually nothing. That point is its asymptote
Its equation is: y=Bˣ in which B<1
Real Life Applications
Exponential Functions can be used to find:
Compound Interest : A=A₀(1+ⁱ⁄ₙ)ⁿᵗ
Continuous Compound Interest : A=A₀(e)ᶦᵗ
Bacteria's Growth : A=A₀(B)ᵗ⁄ₚ
Half Life : A=A₀(B)ᵗ⁄ₚ
The number e
Definition: It is a mathematical constant with the value of 2.718... and is seen as the base for a natural logarithm (ln)
How is it found:Using the formula of (1+1/n)ⁿ is a way to find the value of e
Asymptotes
A point in a graph that a function will never touch but will get near to as it is a restriction in the graph. The curve will approach infinity as it comes near the asymptote
Domain: How to find
Look at your function and isolate for x and the constant. Then you will further isolate for x and create an inequality statement which is correctly representing the function and/or graph 
Range: How to find
Look at your function and substitute x with 0. Then you will solve for y and create an inequality statement to represent the function and/or graph
Notation
You can use Interval Notation to represent the domain and range or solutions that are on a number line or inequality solutions.
How to write
Open Intervals:(x,y) and an open circle on the graph is represented to show that this point is not included in the Domain and Range, so a < or > sign will be represented by those brackets.
Note that if a graph goes till infinity () brackets will be used
Closed Intervals:[x,y] and a filled in circle on the graph is represented to show that this point is included in the Domain and Range, so a ≤ or ≥ sign will be represented by those brackets
Exponential Growth is different than linear growth as exponential growth is a quantity being multiplied while linear growth is a quantity repeatedly being added to by the same number
How to identify
Graphically: If there seems to be an increase or decrease of a quantity as the x value changes. This is simple to spot
The B Value, as previously stated can also be seen to see whether a there is growth or decay
A Table of Values can also be utilized to recognize if there has been growth or decay as the numbers will increase/decrease
An illustration of Exponential Growth and Decay
1.4
Parametric Equations
Relations
A set of ordered pairs of (x,y) and each variable is a function to a third variable of "t" which is called a parameter
Circles
General Equation: (x-h)²+(y-k)²=R²
x is the x coordinate
y is the y coordinate
(h,k) is the center of the circle
R is the radius of the circle
Ellipses
Similar to a circle this has a more oval like shape as it can be stretched horizontally or vertically. Its General Equation is:
[(x-h)²/a²] + [(y-k)²/b²] = 1
x is the x coordinate
y is the y coordinate
(h,k) is the center
a is the distance from the center to the major axis' end
b is the distance from the center to the minor axis' end
Lines and Other Curves (Segments)
A parametric equation does not always have to be a function as it does not pass the vertical line test at times. However a curve can be made in certain instances by making x and y specific functions. A line segment can also be made by having certain functions for x and y.
These can be graphed using a graphing calculator to show the parametric equation. Which an example of is x(t)=t and y(t)=3t
However for Parametric Equations a circle can be represented by having (sin(t),cos(t)) as your set of ordered pairs with a Domain of: 0≤t≤2π.
Note: The ordered pair can also be (cos(t),sin(t)) and the coefficient must be the same for both values in the ordered pair as the coefficient will be your radius
The coefficient in the values in the ordered pair is 2
However for Parametric Equations an ellipse can be represented by having (sin(t),cos(t)) as your set of ordered pairs with a Domain of: 0≤t≤2π.
Note: The ordered pair can also be (cos(t),sin(t)) and the coefficient must be different for both values in the ordered pair
1.5
Functions and Logarithms
One to One Functions
A function that has no two ordered pairs different first coordinates but same second coordinates. To know if a function is one-to-one, the Horizontal Line Test can be conducted
Inverses
This is when the x and y values in a function are replaced with the other variable. For example in y=3x+3 the inverse would be x=3y+3 which would make it y=1/3x-1. If f(x) is your original function f⁻¹(x) will be your inverse of the function
Finding Inverses
Only one-to-one functions can have an inverse
You can also find inverses by using a table of values. You can swap the x and y values in the original function. For example (3,5) -> (5,3) in the inverse. This can be graphed to graphically portray an inverse of a function.
Logarithmic Functions
Known as the inverse of an exponential function, the logarithmic function is represented by y=logₐx in which a>0 and isn't 1.
This image shows an exponential function (blue) and logarithmic function (red)
Properties of Logarithms
There are properties of logarithms used to help evaluating expressions
logₐ(a)=1
logₐ(a)ˣ=x
aˡᵒᵍₐ⁽ˣ⁾=x
Product Rule:log(xy)=logₐ(x)+logₐ(y)
Quotient Rule:log(x/y)=logₐ(x)-logₐ(y)
Power Rule:logₓ(A)ᴮ = B⋅logₓ(A)
Change of Base: logₐ(b)=(logₙ(a))/(logₙ(b))
Real World Applications
Measure the Decibels in Sound: 10log₁₀(I/I₀)
Measure the Intensity/Magnitude of an Earthquake: Ms-Mw=log₁₀(I/I₀)
pH or pOH which helps to know how acidic or basic a substance is: pH = −log ([H+]) or pOH=-log([OH-])
Horizontal Line Test
If there is a function and you place a horizontal line anywhere in the graph and that line does not intersect more than one time, the graph is a one-to-one function
Function A passes the test and is one-to-one
Function B does not and is just a function
1.6
Trigonometric Functions
Radian Measure
The radian measure for ∠ACB is the length of the arc that is cut from the unit circle. That is arc length and has the formula of a=R*θ.
a is arc length
R is the radius
θ is the angle in radians
Graphs of Trigonometric Functions
As seen below in the example:
Sin(x) can be graphed with the formula AsinB(x-C)+D
Cos(x) can be graphed with the formula AcosB(x-C)+D
Tan(x) can be graphed with the formula AtanB(x-C)+D
Periodicity
Whenever an angle and another angle 2π more are in standard position, the lines will coincide so they will have the same value for cos(x) and sin(x). Something similar for tan(x) applies but it is an angle and another one π more.
Inverse Functions
None of the 6 trigonometric functions are one-to-one so they do not have inverses. However, by restricting the domain you can get the inverses
Transformations for sin(x) and cos(x)
The following can be graphed
y=Asin(B(x-C))+D
y= Acos(B(x-C))+D
A=Amplitude / Vertical Stretch / Reflection to x-axis
B = Horizontal Stretch / Reflection to y-axis
C = Phase Shift
D = Vertical Shift / Midline
Period: 2π/B
Even and Odd Trigonometric Functions
cos(x) and sec(x) are even trigonometric functions because they are symmetrical to the y axis and f(-x)=f(x)
sin(x) and csc(x) are odd trigonometric functions because they are symmetrical to the origin and f(-x)=-f(x)
tan(x) and cot(x) are odd trigonometric functions because they are symmetrical to the origin and f(-x)=-f(x)
Degrees and Radians
Every 180° on the unit circle is π.
So to get Degrees it is: Degrees = Radians 180°/π
To get Radians it is: Radians = Degrees π/180°
Trigonometric Ratios
sinθ=opp/hyp=y/R
cosθ=adj/hyp=x/R
tanθ=opp/adj=y/x
cscθ=R/y
secθ=R/x
cotθ=x/y
Forms of Linear Equations
Slope Intercept Form: y=mx+b
y is the y coordinate
x is the x coordinate
m is the slope
b is the y-intercept (The y-coordinate of the y-intercept is (0,b)
This is the formula that is generally used as the slope and the y-intercept can be easily seen
General Linear Equation: Ax+By=C
This form is useful to easily find x and y intercepts and useful for solving systems of two linear equations
Point-Slope Equation: (y-y1)=m(x-x1)
m is the slope
y1 is the y coordinate of point one
x1 is the x coordinate of point one
y is the y coordinate of second point
x is the x coordinate of second point
Inverse of sin(x) and sec(x)
Inverse of cos(x) and csc(x)
Inverse of tan(x) and cot(x)
Works Cited
Byju’s. Coincident Lines. cdn1.byjus.com/wp-content/uploads/2020/09/coincident-lines-example-solution.png.
CollegeBoard. Logo. 2022, https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQAHVxrBuSMZuyiOGi2jkYib7sE2AFBzuVzcqi5JQwnH1XZab85XtK3Hfz8YxX49qaR-L8&usqp=CAU
“Composite Functions.” Online Math Learning, www.onlinemathlearning.com/image-files/composite-functions.png.
“Exponential and Logarithmic Graph.” MathBits Notebook, mathbitsnotebook.com/Algebra2/Exponential/loginverse.jpg.
“Exponential Function Graphs.” ShutterShock, image.shutterstock.com/image-vector/graph-exponential-function-growth-decay-260nw-2102473285.jpg.
Function Machine. s3.amazonaws.com/illustrativemathematics/images/000/000/782/medium/Task_1_8c7a6a9a2e1421586c40f125bd783de3.jpg?1335065782.
“Horizontal Line Test.” Calculus How To, www.calculushowto.com/wp-content/uploads/2019/06/horizontal-line-test.png.
“Parametric Equations.” Story of Mathematics, www.storyofmathematics.com/wp-content/uploads/2021/02/graphing-a-circle-using-its-parametric-equations.png.
“Piecewise Functions Precalculus OnRamps.” YouTube, i.ytimg.com/vi/kSMR58mUPcU/maxresdefault.jpg.
Sine Cosine Tangent Waves. images.saymedia-content.com/.image/t_share/MTczOTQ5NDI5NzMxMzcwODc1/trigonometry-graphs-of-sin-x-cos-x-and-tan-x.jpg.
"Southpointe Notes."Rupprecht, https://classroom.google.com/u/0/c/NDk1ODQ3ODU5OTU3/m/NDk1ODQ3ODYwMDc5/details
Varsity Tutors. Absolute Value Function. www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/absolute-value-functions/abs-graph.gif.
---. Parallel and Perpendicular Lines. www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/parallel-lines/parallel_lines_1.gif.
---. Parallel and Perpendicular Lines. www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/perpendicular-lines-and-slopes/perpendicular-lines1.gif.
Vertical Line Test. mathworld.wolfram.com/images/eps-svg/VerticalLineTest_1000.svg.
An example of a line segment would be x(t)=t+2 y(t)=4t+1 and it would make a straight line. An example of a curved segment would be x(t)=t^2+2 y(t)=4t+1
Parametrizing a line segment
If you have two points which make a line segment, you can parametrize that to get the parametric equations. To do this, you would find your two points (x1,y1) and (x2,y2) and then assign one as T=0 and another as T=1. Then using the Formula of x(t)=aT+B and y(t)=cT+D, you can substitute T=0 to get the D value. Afterwards you can substitute for T=1 in those equations and get x2=aT+B and y2=cT+D and solve for a and c to get your final equations in the Formula that was first stated. You do not have to use T=0 and T=1 however that is the most simple method.
In this example, points of (2,1) and (3,5) were parameterized and using the method explained we get x(t)=t+2 and y(t)=4t+1