PreCalc Chapter 2

Ch 2 Graphs and Functions

Ch 2.1 Rectangular Coordinate Systems

Definition of Points, Axes, Origin, and Coordinates ;
Plotting Points (4,3) ;
Plotting Points , A(-3,2), B(-1,4) , C(-2,-4), D(0,-2), E(3,0) ;
Finding Coordinates of the Given Points from the Graph

Identifying the Quadrant of Each Point on the Coordinate Plane

Determining if the Points (-3,6), (2,-1), (5,3), (4,-1) are on the Line y=-x+3

Using the Distance Formula to Find the Distance between (-4,1) and (2,-3)

Using the Distance Formula to Find the Distance between A(7,5) and B(12,-5);
Using the Midpoint Formula to Find the Midpoint of A(7, 5) and B(12, -5) ;
Finding the Equation that is on the Perpendicular Bisector of A(-4, -3) and B(5, -4) ;
Given A(5,-7), finding the Coordinates of B such that C(2,3) is the Midpoint of Segment AB

Given A(-5,7), finding the Coordinates of B such that A(-2,2) is the Midpoint of Segment AB

Ch 2.2 Graphs of Equations

Graphing  y=3x+2;
Graphing  y=1/2 x³;
Finding the Center and Radius of the Circle (x-4)² + (y+2)² =4 ;
Finding the Center and Radius of the Circle 9x²+9y²+12x-6y+4=0 ;
Finding the Equations of Upper Half, Lower Half, Left Half and Right Half of the Circle (x+3)² + y² = 64

Standard Form of an Equation of a Circle (x-h)² + (y-k)² =r²;
Graphing the Circle (x+3)² + (y+2)² = 16 ;
Graphing the Circle x²+ (y-1)² =4 ;
General Form of a Circle x²+y²+ax+ay+c=0 ;
Writing 2x²+2y²-12x+8y-24=0 in Standard Form and Graphing it

Reading the Equation of the circle (x-2)² + (y+3)² = 16/9

Writing the Equation of a Circle From a Graph

Finding the Center and Radius of the Circle x²+y²-4x+6y+12=8

Finding the Center and Radius of the Circle 4x²+8x+4y²-36y-59=0

Graphing Equation 7x+2y=10 using the TI-89 Calculator

Ch 2.3 Lines

Finding the Slope of a Line using (5,-3)Points  and (-1,7)

Finding the Equation of a line using points (-2,-7) and (3,-5) and Expressing in Slope–Intercept Form

Writing the Equation of a Line with Slope =2/3, and Point =(6,-3)

Writing the Equation of a Line given that f(3)=-2 and f(-12)=4

Parallel and Perpendicular Lines ;
Determining if the Two Lines y=2/3x-4 and 3y-2x=3 are Parallel, Perpendicular or Neither ;
Determining if the Two Lines 2x+5y=9 and 10x-4y=12 are Parallel, Perpendicular or Neither

Finding Slope of a Line Going Through Two Points ;
Finding the Slope of a Line Going Through Points (4,-1) and (-6,-3) ;
Finding the Equation of the Line Perpendicular to 3x+2y=7 and Going Through (4,5) ;
Finding the Slope and y-intercept of 7x=-4y-8

Determining the Equation of the Line that is Parallel to x-3y=9 and Passing Through (6,2) ;
Determining the Equation of the Line that is Perpendicular to 2x+3y=-6 and Passing Through (3,4) ;
Determining the Equation of the Line that is Perpendicular to y=4 and Passing Through (-3,-2)

Ch 2.4 Definition of Function

Finding f(3),f(1/2) , f(2.73) for Function f(x)=20.18x²-33.43x+45.77 using the TI-83/84 Calculator

Finding f(2),f(1/3) , f(3.9) for Function f(x)=19.5x²-34.7x+40.6 using the TI-85/86 Calculator

Evaluating the Difference Quotient f(x+h)-f(x)/h given the Function f(x)=x²-1

Finding the Domain and Range of the Function f(x)= -(x+4)²+4 from the Graph

Given f(x)=x², finding f(-6),f(3), f(Δ), f(▯), f(a), f(-a), -f(a), f(a)+f(h), f(Δ+ ▯ ),f(x+h)  and the Domain of  f(x)=x²;
Given g(x)= x²/x+1, finding g(-3),g(▯) ,g(1/a) ,1/g(a) ,g(√a) , √g(a), g(Δ+▯), g(x+h),g(x+h)-g(x) /h  and the Domain of g(x)= x²/x+1

Finding the Domain of  f(x)=√8-3x;
Finding the Domain of  g(x)=√4x-3 / x²-4;
Using Vertical Line Test ;
Finding Zeros of a Function ,
Finding the Domain and Range of  f(x)=√16-x²;
Evaluating the Difference Quotient f(x+h)-f(x) / h given the Function g(x)=3x²-2x+7

Ch 2.5 Graphs of Functions

Discussing Even Functions and Odd Functions ;
Determining if f(x)=x² is Odd, Even, or Neither ;
Determining if f(x)=x³ is Odd, Even, or Neither

Determining if f(x)=x4-x²-3 is Odd, Even, or Neither ;
Determining if f(x)=½x³-2x is Odd, Even, or Neither

Function Transformations: Horizontal and Vertical Stretches and Compressions ;
Graphing g(x)=½x² and h(x)=2x² from f(x)=x² ;
Graphing g(x)=(½x)² and h(x)=(2x)² from f(x)=x² ;
Graphing f(x)=3|x| from g(x)=|x| ;
Graphing  f(x)=√2x from g(x)=√x

Function Transformations: Horizontal and Vertical Translations ;
Comparing   f(x)=x² and f(x-1)=(x-1)² ;
Comparing f(x)=x² and f(x)-2=x²-2 ;
Graphing f(x)=|x+3|+2 from g(x)=|x| ;
Graphing f(x)=(x-2)²-4 from g(x)=x²

Function Transformations: Reflections Across the x-axis and y-axis ;
Graphing  g(x)=-√x and h(x)=√-x from f(x)=√x ;
Graphing f(x)= -|x| from g(x)=|x| ;
Graphing f(x)=³√-x from g(x)=³√x

Functions Transformations: A Summary

Graphing f(x)=-½|x-1|-2 from f(x)=|x| ;
Graphing f(x)=√(2x-4) +1 from f(x)=√x

Graphing f(x)=2(x+4)²-3 from f(x)=x² ;
Graphing f(x)= 2/x-3 +1 from f(x)=1/x

Discussing Even Functions and Odd Functions ;
Determining f(x)=2x5-7x3+4x Even Function, Odd Function or Neither ;
Given f(x)=x², describing how  f(x)=x²+2,  f(x)=x²-2, f(x)=(x+2)², f(x)=(x-2)²,  f(x)=2x², f(x)=½x² , f(x)=(2x)² , and f(x)=(½x)² Differ ;
Graphing  f(x) where x-3 if x≤-2,  -x²  if -2<x<1,  and -x+4 if x≥1 ;
How would the Graph of f(x) Change for f(x+2) , f(x)-3, 2f(x), and f(½x) ?
Finding the New Point given (3,-1) with y=2f(x)+4 ;
Describing the Graph of f(x)=2|x-3|+1 from f(x)=|x|

Interpreting and Graphing Piecewise Functions

Graphing Piecewise Function f(x) where 2x-1 if x<1,  and -x+2 if  x≥1;
Graphing Piecewise Function  f(x) where -x²-1 if x<0,  and x²+1 if x≥0 ;
Graphing Piecewise Function f(x) where -x² if x<-1,  3  if x=-1,  and x+2 if x>-1

Finding the Domain and Range of a Piecewise Function from the Graph

Graphing a Piecewise Function f(x) where x+1 if x<-3 ,  5  if -3≤x≤0,  and x² if 0<x

Zeros and Vertex in Standard Form, Vertex Form, and Factored Form ;
Expressing f(x)=-4x²+16x-13 in f(x)=a(x-h)²+k Form ;
Finding the Zeros for f(x)=-4x²+16x-13 using Quadratic Formula -b±√b²-4ac / 2a or in f(x)=a(x-h)²+k
Form, Determines the if the Function has a Maximum or Minimum, and the Value of Maximum/Minimum ;
Finding the Equation of a Parabola given the Vertex (4, -7) and x–intercept of -4

Finding the Vertex from y=(x-3)²-1 ;
Finding the Vertex from y=(x+3)²+5 ;
Graphing y=(x-3)²-4, and Identifying the Vertex, Axis of symmetry, and the Intercepts ;
Graphing y=-2(x+1)²+8, and Identifying the Vertex, Axis of symmetry, and at least Two Additional Points

Finding the Vertex of Quadratic Functiony=-2(x+1)²+4 , Equation of Axis of Symmetry, and
y-intercept and Graphing the Function

Finding the Vertex of a Quadratic Function f(x)=x²+2x-4 by Completing the Square, Finding the Equation of Axis of Symmetry, y–intercept and Graphing the Function

Finding the Vertex of a Quadratic Function f(x)=-2x²-12x-19 by Completing the Square

Finding the Vertex of a Quadratic Function f(x)=-2x²-12x-19 using Vertex Formula (-b/2a, f(-b/2a))

Finding the Vertex of a Quadratic Function f(x)=x²-4x-5 using Vertex Formula (-b/2a, f(-b/2a)), Equation of Axis of Symmetry, x-intercepts and Graphing the Function ;
Finding the Vertex of a Quadratic Function f(x)=-2x²+10x-7 using Vertex Formula  (-b/2a, f(-b/2a))
, Equation of Axis of Symmetry, and Graphing the Function

Finding the Zeros of a Quadratic Function f(x)=3x²-x-1 using Quadratic Formula -b±√b²-4ac/2a

Finding the Equation of a Quadratic Function from a Graph

Determining the Values of b and c of the Quadratic Function f(x)=-2x²+bx+c with Vertex (3,-17)

Finding the Quadratic Function that passes through(-4,0) , (3,0), and  (0,6) using f(x)=a(x-r1)(x-r2)

Finding the Quadratic Function that has Zeros -3 and 4 using f(x)=a(x-r1)(x-r2)

Quadratic Function Application Using Formulas – Rocket Launch

Ch 2.7 Operations on Functions

Given f(x)=x/x-2 and g(x)=3x/x+4, finding (f+g)(x), (f-g)(x),(fg)(x) ,(f/g)(x) ;
Given f(x)=x²+3x and g(x)=7x-1, finding f(2), f(Δ),f( ▯),f(g(x)), g(3), g(Δ), g(▯), g(f(x)), f(g(2)),g(f(5))

Given f(x)=2x-1, g(x)=x³-5 and h(x)=5-x², finding the Composite Function Values (f°g)(3), (g°f)(3), and (h°g)(-1) ;
Given f(x)=4x+1 and g(x)=x²-x+5, finding the Composite Function (f°g)(x) and (gºf)(x)

Adding the Functions of f(x)=2x²-3x and g(x)=7x+3

Subtracting the Functions of f(x)=4x²+2x and g(x)=3x-5

Dividing the Functions of f(x)=2x+10 and g(x)=x²+2x-15

Adding and Multiplying Functions of and f(x)=4x²-3x+1 and g(x)=-2x³+5x

Determining the Composite Function Values (f°g)(-2), (gºf)(3), (fºf)(0), (gºg)(-4) using a Table

Determining the Composite Function Values (f°g)(1), (g°f)(9), (fºf)(0), (g°g)(8) using Graphs

Finding Composite Function (f°g)(x) of f(x)=√x-1 and g(x)=4x²

Finding the Composite Function (g°f)(x) of f(x)=4x²-3x+1 and g(x)=-2x³+5x

Finding the Composite Functions (f°g)(x) and (g°f)(x) of f(x)=1/x+5 and g(x)=3/x -5

Finding the Domain of the Composite Function (fºg)(x) of f(x)=(x+2)² and g(x)=3x-2

Finding the Domain of the Composite Function (fºg)(x) of f(x)=x²+1 and g(x)=√x

Finding the Domain of the Composite Function (fºg)(x) of f(x)=1/x-4 and g(x)=1/x

Finding the Domain and Range of the Composite Function (fºg)(x) of f(x)=√x-1 and g(x)=x²+1