A REVIEW OF THE PRECALCULUS

By: Will Puckett

For those who don’t already know…What is Calculus?Definition of CALCULUS a : a method of computation or

calculation in a special notation (as of logic or symbolic logic)

b : the mathematical methods comprising differential and integral calculus —often used with “the”

Parent Functions and their Graphs

http://learn.uci.edu/oo/getOCWPage.php?course=OC0111113&lesson=004&topic=13&page=1

http://www.wkbradford.com/posters/geomforms.html

These formulas can be used to find the volume of a three dimensional solid.

http://www.wkbradford.com/posters/geomforms.html

These formulas can be used to find the surface area of a three dimensional solid, which is equal to the sum of the areas of all sides of the figure added together

The Quadratic Formula

http://blogs.discovermagazine.com/loom/2008/05/04/quadratic-vertebrae/

Discriminant and its Implications

The discriminant of a function is shown as

If the discriminant is…○ <0, the function has no real solutions○ =0, the function has one real solution○ >0, the function has two real solutions

Try it out Determine the number of solutions of the

equation y= x^2 + 7x + 33, and solve using the quadratic formula to find those solutions.

Exponents When dividing two powers with the same base,

subtract the exponents(a^b)/(a^c)=a^(b-c)

When multiplying two powers with the same base, add the exponents(a^b)(a^c)=a^(b+c)

Any number raised to the power of zero equals 1A^0=1

A negative exponent is equal to the multiplicative inverse of the functionA^-b=1/a^b

Solve the following expressions

1. (x^8) / (x^3) =2. (x^2 +2) (x^2 – 2) =3. 468x^0 =4. 2x^(-3) =

Symmetry of a graph A graph is symmetrical with respect to

the x-axis if, whenever (x, y) is on the graph, (x,-y) is also on the graph

A graph is symmetrical with respect to the y-axis if, whenever (x, y) is on the graph, (-x,y) is also on the graph

A graph is symmetrical with respect to the origin if, whenever (x, y) is on the graph, (-x, -y) is also on the graph

Tests for symmetry1. The graph of an equation is symmetric with

respect to the y-axis if replacing x with –x yields and equivalent equation

2. The graph of an equation is symmetric with respect to the x-axis if replacing y with –y yields an equivalent equation

3. The graph of an equation is symmetric with respect to the origin if replacing x with –x and y with –y yields an equivalent equation

Check the following equations for symmetry wrt both axes and the origin

1. x – y^2 = 02. Xy = 4

3. Y = x^4 – x^2 + 3

1. X-axis2. Origin3. Y-axis

Even and Odd Functions A function is even if

for every x in the domain, -x is also in the domain, and f(-x)=f(x)

A function can be even if and only if it is symmetrical to the y-axis Example of an even

function:○ Y=x^2

A function is odd if for every x in the domain, -x is also in the domain, and f(-x)=-f(x)

A function can be odd if and only if it is symmetrical to the origin. Example of an odd

function:○ Y=x^3

Asymptotes of graphs Horizontal asymptotes

If the power of the denominator is…○ >power of numerator: y=0 is a horizontal asymptote○ =power of numerator: y=ratio of the coefficients is a horizontal

asymptote Vertical asymptotes

Vertical asymptotes are found by finding the zeroes of the denominator

Oblique asymptotesIf the power of the numerator is larger than the power of the

denominator, you must use the long division method in order to find the asymptote of the graph

Graph the functionF (x) = 2(x^2 – 9)

(x^2 – 4)

Relative Extrema Relative extrema are also commonly known

as local extrema, or relative maximums and minimums. A relative minimum is the lowest point on the y-

axis that a function reaches between two points of inflection when concave up

A relative maximum is the highest point on the y-axis that a function reaches between two points of inflection when concave down

A polynomial of degree “n” can have a maximum of “n-1” relative extrema

Determine whether A, B, C, and D are relative maximums or relative Minimums

http://image.wistatutor.com/content/feed/tvcs/relative20maximum20help20graph20of20function.JPG

A- Relative MinimumB- Relative MaximumC- Relative MinimumD- Relative Maximum

Transformations of GraphsThere are four different types of

transformations that can change the appearance of a graph.

Rigid transformationsTranslationReflection

Non-rigid transformationsStretchShrink

Translation A transformation in which the graph of a

geometric figure is shifted up, down, or diagonally from its original location without any change in size or orientation

Y=x^2Y=(x^2)+2

The graph was shifted up two units on the y-axis

Graphs produced using mathgv

Reflection A transformation in which the graph of a

function is reflected about an axis of reflection, such as the x-axis or a line such as y=2, creating a symmetrical figure with its original graph.

The graph was flipped, or reflected, about the x-axis

Y=x^2 Y=-(x^2)

Graphs produced using mathgv

Stretch or Compress A transformation in which the graph of a

function is either compressed or stretched horizontally, changing the shape of the graph.

Y=abs(x) Y=3abs(x) Y=1/3abs(x)

The two red curves represent the transformations in which the graph was stretchedOr compressed. When multiplied by three, it was compressed towards the y-axis.When divided by three, it was stretched away from the y-axis.

Graphs produced using mathgv

Complete the following Transformations:

Shift the graph of Y=2x + 3 to the right two and down three

Reflect the graph of y=x^2 + 2 about the x-axis

http://www.dsusd.k12.ca.us/users/bobho/Alg/parabola.htm

Types of Conic Sections Parabola- the set of all points (x,y) that are

equidistant from a fixed line (directrix) and a fixed point (focus) not on the line

Ellipses- set of all points (x,y) the sum of whose distances from two distinct fixed points (foci) is constant

Hyperbola- A hyperbola is the set of all points (x,y) the difference of whose distances from two distinct fixed points (foci) is a positive constant

Parabola Standard form of equation

with vertex at (h, k) (x - h)^2 = 4p(y - k), p cannot

equal 0○ Vertical axis, directrix: y = k - p

(y - k)^2 = 4p(x - h), p cannot equal 0○ Horizontal axis, directrix: x = h

– p The focus lies on the axis p

units from the vertex. If the vertex is at the origin… X^2 = 4py vertical axis Y^2 = 4px horizontal axis

http://people.richland.edu/james/lecture/m116/conics/translate.html

Ellipse Standard form of equation

with center (h, k) and major and minor axes of lengths 2a and 2b, where 0 < b < a (x – h)^2 + (y – k)^2 = 1

a^2 b^2 (x – h)^2 + (y – k)^2 = 1

b^2 a^2 The Foci lie on major axis, c

units from the center, with c^2 = a^2 + b^2.

http://www.tutorvista.com/math/solving-major-axis-of-an-ellipse

Hyperbola Standard form of equation

with center at (h,k) (x – h)^2 - (y – k)^2 = 1

a^2 b^2 (x – h)^2 - (y – k)^2 = 1

b^2 a^2 Vertices are a units from

the center, and the foci are c units from the center. C^2 = a^2 + b^2

http://people.richland.edu/james/lecture/m116/conics/translate.html

Circle Round shape with

all points equidistant r units from center at (h, k) where r is the radius.(x – h)^2 + (y – k)^2 = r^2

http://www.mathsisfun.com/algebra/circle-equations.html

Properties of Logarithms

http://www.apl.jhu.edu/Classes/Notes/Felikson/courses/605202/lectures/L2/L2.html

Properties of Natural Log

http://www.tutorvista.com/math/natural-logarithm-exponential

Properties of the Exponential Function

Domain: All Real numbers

Range: y>0 Always increasing

Lne^x = xe^(lnx) = xA^x = e^(xlna)

Inverse of the natural logarithmic function

http://www.craigsmaths.com/number/graphs-of-exponential-functions/

Exponential Growth and Decay

A = Ce^kt A: amount at a given

timeC: Initial amountK: rate of growth or

decay (growth when positive, decay when negative)

T: time

http://www.tutornext.com/help/exponential-growth-function

The population P of a city isP = 140,500e^(kt)

Where t = 0 represents the year 2000. IN 1960, the population was 100,250. Find the value of k, and use this result to predict the population in the year 2020.

K=.0084; P=166,203

Trigonometry

http://www.tutorvista.com/math/trigonometric-functions-chart

Graphs of Trig Functions

http://www.xpmath.com/careers/topicsresult.php?subjectID=4&topicID=14

F (x) = asin(2π/b) (x – c) + dabsA = amplitudeabsB = periodabsC = horizontal shiftabsD = vertical shift

Amplitude of a Graph Describes how high or low the graph of the

function goes on the y-axis. Changing the amplitude transforms the graph by stretching or compressing it vertically

The graph shows the difference in amplitude between f (x)= sin(x) and f (x) = 3sin(x). Notice the vertical stretch made by multiplying the function by three.

Graph produced by mathgv

Period of a Graph Describes the distance or time it takes for the

graph of the function to repeat itself, or distance from crest to crest. Changing the period transforms the graph by stretching or compressing it horizontally

The graph shows the difference in period between f (x) = sin(x) and F (x) = sin(x/2).Notice the horizontal stretch, and how the period of the modified function in red is double that of the original

Graph created by mathgv

The Unit Circle

http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215.htm

1979 AB 1Given the function f defined by f (x)

= 2x^3 – 3x^2 – 12x + 20a) Find the zeros of fb) Write an equation of the line normal to

the graph of f at x = 0c) Find the x- and y- coordinates of all

absolute maximum and minimum points on the graph of f. Justify your answers

Will Puckett2011