Algebra 2 Trig Notes

Exponential Models

• Graphs and models exponential growth and decay.
• Writes situations that represent each type of model.
  • Linear model
  • Quadratic model
  • Exponential y = a(b)^x model

Scenarios

• Dylan shoots a rocket (vertically) from the top of a building.
• Pam rides her bike at a constant speed to her friend's house.
• Pam invested some money into an account and she rides he like at a constant speed to her friend's (compound interest).

Exponential Function

• y = a(b)^x
  • Where a is initial value (constant) and either b > 1 or 0 < b < 1
• Example: graph of y = 2^x

End Behavior

• As x approaches +\infty, f(x) \rightarrow +\infty
• As x approaches -\infty, f(x) \rightarrow 0

Domain of Function

• Example: x \in \mathbb{R} or (-\infty, \infty)

Range of Function

• Example: {y | y > 0} or (0, \infty)

Growth Factor

• b is called the growth factor for exponential growth where b > 1. Quantity increases by a constant percentage each time period.
  • The percentage increase, r, written as a decimal, is the rate of increase or growth rate.

Decay Factor

• b is called the decay factor for exponential decay where 0 < b < 1. Quantity decreases by a constant percentage each time period.
  • The percentage decrease, r, written as a decimal, is the rate of decrease or decay rate.
• Time in years.

Examples

• Ex1: The value of Aileen's vase from her grandmother will increase in value approximately 17% per year for the first few years. What's the car's value after 3 years?
  • a = 1500
  • A(t) = a(1 + r)^t
  • Decay r = 0.17
  • A(3) = 1500(1 + 0.17)^3
  • A(3) = 20,012.55
  • In 3 years, the car will depreciate in value to $20,000 (roughly).
• Ex2. Steven purchased a car this past weekend for 35,000. The car depreciates in value 1.5% per year. The current value is 1500, what's the approx value in 10 years?
  • a = 35,000
  • A(t) = a(1 - r)^t
  • Decay r = 0.015
  • A(10) = 1500 (1 + 0.015)^{10}
  • A(10) = 1740.81
  • In ten years, the value is about 1740

Exponential Functions and Transformations

• Apply transformations (stretch/compress, reflect, and translate) to graphs of exponential functions.
  • Parent function: y = (b)^x
• Y = a(b)^x
  • For vertical stretch/vertical compression/reflection over X-axis
    • Vert Stretch, (a) > 1
    • Vert compress, 0 < a < 1
    • NOTE !!! Know these for logs graphs
    • reflect over X, a < 0
• y = a(b)^{x-h} + k
  • For horizontal and vertical translation
    • (h) -><
    • (k) - ^

Logarithmic Functions

• Define logarithm base \,b of \,x = when b > 0 (b \ne 1), \log_b x = y is the same as b^y = x

Examples

• EX1. use the definition above to write the equivalent form.
  • log form exponential form
  • a. log_3 9 = 2 9 = 3^2
  • b. log_4 t = -1 4^{-1} = t
  • c. \,log_a 1 = 0 9^0=1
  • d. log_256 y = 8 256 = 2^8
  • e. log_{10} 100 = 2 100 = 10^2
• EX2. EVALUATE EACH LOGARITHM
  • a. \,log_4 64 = x 4^x = 64 -> x=3
  • b. \,log_8 0.125 = x 8^x = 0.125 -> x = -3
  • c. \,log_{32} 2 = x 32^x = 2 -> x = \frac{-4}{5}
  • d. log_2 \frac{1}{16} = x 2^x = \frac{1}{16} -> x = -4

Graph of Log Functions

• Vertical asymptote at x = 0

Inverse of a Logarithm Function

• Interchange the domain and range (aka switch x and y in function)

  • y = \log_b x, b > 1
  • \,D = (0, \infty)
  • R = (-\infty, \infty)
  • y = b^x (inverse invert f)

• Know how to find the domain + range for these, especially when we get into transformations!

Properties of Logarithms

• Define:
  • \,X = \log_b m
  • Y = \log_b n
• There are 3 properties of logarithms!
  • 1) product property
  • 2) Quotient property
  • 3) Power # property
• \,m \cdot n = b^X \cdot b^Y = mn = (b^X)b^Y
  • By log definition: M= b^x
  • \,mn = b^{x+y}
  • mn = b^x b^y
  • n = b^Y
  • \log_b mn = X+Y
• \frac{m}{n} = b^{X-Y}
  • \log_b \frac{m}{n} = X-Y
  • \logb \frac{m}{n} = \logb m - \log_b n
• \logb m^n = n \cdot \logb m

Simplify (condense) the expression to a single logarithm

• a) 2 \log4 6 - \log4 9
  • power prop
  • \log_4 \frac{36}{9}
  • quotient prop
  • \log_4 4 = 1
• b) \log2 5x + \log2 3x
  • \log_2 (5x \cdot 3x)
  • \log_2 (15x^2)
  • (5) + \log_2 x
• c) \log7 6^2 - \log7 9
  • power prop
  • \log7 (x \cdot 3x) - \log7 36y^2
  • \log_7 36y^2
  • quotient prop
  • 5log(x-5) + \frac{1}{4}log 7 ->
  • \frac{1}{5} simplify
  • log(x-5)^5 + log 7^{\frac{1}{4}}
  • \log (x-5)^5 \cdot 7^{\frac{1}{4}}
• Evaluate \,log ->
  • simplify power prop

Expand the expression to the sum/difference of logarithms

• a. \,log_2 54x^5

  • \,log2 (2 \cdot 3^3) + \log2 (x^5)
  • \log2 2 + 3 + 5\log2 x

• b) \,log \frac{\sqrt[3]{25}}{36}

• when these 2 ##log_3 5 +7 2log z(3 2 21 matchups, its either 1 or when these 2 matchup, its either 1 +log when theses since in 2 since it canebout !

Parabolas

A parabola is the set of all points (x, y) such that each point (x,y) is equidistant to the focus and directrix

• The locus lies on the axis of symmetry.

• The vertex lies halfway between the focus and the directrix.

• The directrix is perpendicular to the axis of symmetry.

• In Chapter 4, the parabolas had a vertical axis of symmetry and opened up or down. In this lesson, the parabolas can have a horizontal axis of symmetry and open right or left.

• The four cases are shown:

  • focus: (0, p)

  • directrix: y=-p

  • vertex (0,0)

  • x^2 = 4py, p>0

  • x^2 = 4py, p<0

  • Focus. (p,0) tocus: 1p.a)

  • Directrix \,x=-P

  • Vertex:(0, 0)

  • \,y^2=4px p>0

  • \,j= 4px p<0

Examples

• Ex1: Identify the focus and directrix of the parabola given by x=-\frac{1}{6}y^2 Graph the parabola.

• Ex2: Write an equation for parabola with vertex (0, 0) and directrix y=-2. Flo,2y

• Ex3: Write an equation for parabola with focus (5, 0) and vertex (0, 0).

• Ex4: Write an equation for parabola with directrix y=3 and focus (0, -3)

• Ex5: Graph and identify the focus and directrix.

• Ex6: Translate the graph in Ex5 right 2 units and down 3 units. Write the equation of the graph.
  Standard form of a parabola with vertex (h, k) and with prtical axis of symmetry is

• horizontal axis of symmetry is

Classifying Conic Sections

• If the graph of Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 is a conic, then the type of conic can be determined as follows:

  • Discriminant
  • B^2 - 4AC < 0, B = 0, and A = C -> Circle
  • B^2 - 4AC < 0 and either B \ne 0 or A \ne C -> Ellipse
  • B^2 - 4AC = 0 -> Parabola
  • B^2 - 4AC > 0 -> Hyperbola

• If B = 0, each axis of the conic is horizontal or vertical. If B \ne 0$$, the axes are neither horizontal nor vertical.

• Ex1: 2x2 +y2 - 4xー4= 0

• Ex2: 4x2 -ア2 - 16xー4y ー4＝0

• Ex3: 2x+2y2 -12x ＋4y+2=0

• Ex4: 4x2 - 9y2+32x ー144y ー548 =0 Mymeol

Hyperbolas

A hyperbola is the set of all points (x, y) such that the difference of the distances between P and the two fixed points, called the foci, is constant.

• The line through the foci intersects the hyperbola at two points, the vertices

• The line segment joining the vertices is the transverse axis (VI-Ua) and its midpoint is the center of the hyperbola.

• A hyperbola has two branches and two asymptotes

• The line segment perpendicular to the transverse axis and passing through the center is the conjugate axis

• The asymptotes contain the diagonals of a rectangle centered at the hyperbolas center.
  note:a2isthe1stdenominatorcregardiessotvalue

• The foci of a hyperbola lie on the transverse axis, c units from the centerwhere_C?=a7 Ex1:Graph thehyperbola4x2-9y2=36.

• Ex1:Graph thehyperbola4x2-9y2=36.

• Ex2: Write the equation of the hyperbola with foci at(0, -3) and (0, 3) and vertices at (0, -2) and (0, 2).

• Ex3:Writethe equationofthehyperbolawith center(0, 0), conjugate axis 20 units, and vertex at(9, 0).

• Ex4: Translate the graph in Ex1 right 2 units and down l unit, Write the equation of the graph.

• Ex5: Write the equation of the hyperbola with foci(3, -13) and (3, 9) and transverse axis 16 units.

Ellipses

An elipse is the set of all points (X, y) such that_the Sii thedaLe het Axedeont Mlle he Ao i is Coni스nt

• The line through the foci intersects the ellipse at two points, the _ CCCS
• The line segment joining the vertices is the
• The line perpendicular to the major axis at the center intersects the ellipse at two points callea the
• The line that joins these points is the __Midor aYIS

Examples

• Ex1: (Graph the ellipse 9x2 + 16y = 144. 1=a
• Ex2; Write the equation of the ellipse witha Center (0, 이), Vertex (-4, 이, Focus (2, 이.
• Ex3: Write the equation of the ellipse with Center (0, 이, Vertex (0, 7), Co-Vertex (-6, 이).
• Ex4: Translate the graph in Ex12 units left and 3 units up. Write the equation of the ellipse.
• Ex5: Write the equation of the ellipse with foc (-2,8) and (-2,4) and a major axis of 8 units.

Circles

A circle is the set of all points (x, y) such that each point (x,y) is equidistant to the focus , also called the center.

• The distance r between the center and point (x, y) on the circle is the radius
• The distance formula canbeusedtoobtainan equation ofthe circle whose centeris the origin andwhose radius is r.

Examples

• Ex1: The point(1, 4) is on a circle whose center is the origin. Write the equation of the circle.
• Ex2:Write an equation ofthe line thatis tangentto the circle x2 + y2=13at (2,3).
• Ex3: Graph the circle x2+ y?=16 that is translated 1unittothe right and 2units down.
• Ex4:Writethe equationofthe circle fromEx3.
• Ex5:Writeequation ofthe circle that has center(4, -2) and passes through (9, -3).

The Standard Deviation

The five-number summary is notthe most common numerical description of adistribution.

• Thatdistinction belongs to the combination ofthe meantomeasure conter and the standard deviation tomeusurespread
  The standard deviation measures spread by lukingat hoh far vauves are from the moun
• Oftenweare interested inthe standard deviation of apopulation.There are 2 typesofstandard deviation,population,Ux _andsumple,Sx.
• Typically,wecannotgetthedata fromthe entirepopulation, sowecalculate the sample standard deviation. Thesamplestandarddeviationistheestimate _ofthepopulationstandarddeviation based on the sample.

The Variance sỷ and Standard Deviation Sx

Thevariancesyofasetofobservations istheaverageofthesquaresofthedeviationsofthe observationsfromtheirmean.

• Insymbols,thevarianceofnobservationsX1,X2,…,X,is
• ThestandarddeviationSxisthesquarerootofthevariancesx: Sx=
• Thedeviationsx,-Xdisplay the spread ofthevawesx;apoutthemean Someofthesedeviationswillbemegutive _and some willbepuritive because someoftheobservations falloneithorfideotthemeun
  The variance istheAverageofthesquareddeurations. Sosyands,will be largeif thevaluesarewidklefwidelyspreadabtthemean and small if thevaluesareclosetothemean

Ex1: Metabolic Rate

Here are the metabolic rates of7 men who took part in a study of dieting, The units are calories osua7hper 24 hours. 1전하시e지 ie02 1792 1666 1362 1614 1460 1867 1439 게 e기n Calculate the variance and standard deviation(by hand).
*Why do we square the deviations?
*Why do we emphasize the standard deviation rather than the variance?

Normal Distribution - The 68-95-99.7 Rule

All normal distributions obey the 68-95-99.7 Rule:
*In the normal distribution with mean y and standard deviation σ:
About 68% of the data fall within 1o of I
about 95% of the data full within 2o of I
about 99.7% of the data full within 36 of I

*Figure 2.12 The 68-95-99.7 rule for Normal distributions.

Ex1:

Young women's heights
*Using the 68-95-99.7 rule
The distribution of heights of young women aged 18 to 24 is approximately Normal with
mean =64.5 inches and standard deviation or = 2.5 inches.
*Figure 2.13 shows the application of the 68-95-99.7 rule in this example.

1. What % of young women at between 59.5in and 64.5in?
2. What % are between 67in and 72 in?
3. What % are below 59.5 inches?
4. What percentile is a woman at with height 69.5in.?

Ex2:

a. IQs are normally distributed with a mean of 100 and standard deviation of 15.
*What percent of people have an IQ higher than 115?
*What percent of people have an IQ lower than 70?

b. Women's heights are normally distributed with x = 63.5" and sx = 2.5".
*What percent of women are between 61" and 66"?

*What percent of women are taller than 5'1"?

c. A test had a normal distribution. 130 students took the test and scored an average of 76 with standard deviation 7.
*How many students got an A?
*How many students passed?

Analyzing Data

*Measures of Central Tendency
Common measure for the "middle" of the data.
min QI med Q3 max
mean average of the data
median middle data value
mo de data value(s) that occors most often
*(bimodal)data set that has 2 modes (more than 2 modes is not useful info)
5-Number Summary
Consists of minimum, quartile 1, median, quartile 3, maximum

P.0 examples:
4 5 6.5 7 9
4 6 7 8 9