Precalculus 3.1a notes - Exponential functions

3.1a Exponential Functions and logarithmic functions are examples of transcendental functions.

The exponential function f with base a is denoted by

f(x) = a^x

where a > 0, a not equal to 1, and x is any real number.

Examples: a⁰ = 1 any number to the zero power is one, why?

4²/4² = 4^2-2 = 4⁰= 1

Graphs of Exponential Functions - the domain, like those of polynomial functions, is the set of all real numbers.

f(x) = a^x , a > 1

g(x) = a^-x , a > 1

Domain: all reals;

Range: y > 0 ;

y-intercept (0,1);

Asymptote y = 0

both graphs are continuous

Plot each graph and see the differences:

f(x) = a^x , the graph is increasing

g(x) = a^-x, the graph is decreasing

Transformations of Graphs of Exponential Functions:
a. f(x) = a^x-h + k

Basic graph h = ________ k = ________ a = __________

b. f(x) = 2^x-2 This is the graph shifted ___ units to the ________.

c. f(x) = 2^x+3 This is the graph shifted ___ units to the________.

d. f(x) = 2^x + 5, This is the graph shifted _____ units _____.

e. f(x) = 2^-x This is the graph reflected in the _____.

f. f(x) = -2^x This is the graph reflected in the _____.

Homework #25 pg. 225; #1-33 (odd)

Precalculus 3.1b notes- The Natural Base e

The natural base is e = 2.71828…

The function f(x) = e^x is the natural exponential function.

Domain: all the reals

Range: y>0

y-intercept: (0,1)

Compound Interest Formulas:

After t years, the balance A in an account with principal P and annual interest rate r (expressed as a decimal) is given by the following formulas:

1. For n compoundings per year: A = P((1 + r/n)^nt) .

Example: You want to invest $500 for 3 years in a bank. They offer you the following choices, which do you take?

1. 3% interest compounded 4 times a year

2. 4% interest compounded 2 times a year

3. 5% interest compounded 1 time a year

2. For continuous compounding: A = Pe^rt.

Example: You have a credit card with $2000 on it. Your interest is 15.4%. How much interest will you have paid on it by the end of the year?… by the end of 2 years?

Homework #26 pg. 225; #35, 39, 43, 47, 51, 53, 59 - 69 odd, 73, 76

Geometry - 7.1 + 7.2 guided notes

Geometry Unit 7 sec 7.1 and 7.2

Rigid Motion in a Plane/Reflections

Guided Notes:

I) Rigid Motion in a Plane sec 7.1

A) Transformation: Movement of an original figure (preimage) onto a new figure ( image).

Example:

In a figure of two triangles, ABC and A'B'C', the preimage is triangle ABC and the image is triangle A'B'C' (stated triangle A prime, B prime, C prime)

Triangle ABC will fit exactly on top of the other by doing a transformation of the figure.

Vertex A maps to Vertex A', vertex B maps to Vertex B' and vertex C maps to vertex C'

B) Isometry: A transformation that preserves lengths, angle measures, parallel lines, and distance between points. They are called rigid transformations.

II) Reflections sec 7.2

C) Line of Reflection: the line that a figure is reflected over - if we have a triangle reflect over a line, the image can be fitted exactly over the preimage by folding the paper on the line of reflection

Now let's look at specific line reflections

1. Reflection in x-axis: (x,y) maps to (x,-y)

Example: (4 , 3) becomes (4 , -3)

2. Reflection in y-axis: (x,y) maps to (-x,y)

Example: (4 , 3) becomes (-4 , 3)

3. Reflection in y = x: (x,y) maps to (y,x )

Example: (4 , 3) becomes (3 , 4)

4. Reflection in y = -x: (x, y) maps to (-y, -x)

D) Line of Symmetry: A figure allows a copy of a figure to be mapped onto itself.
examples:

In nature, art, and in industry, we find many forms that contain a line of reflection.

1. Butterfly

2. Leaf

3. Architecture

4. Automobiles

E. Axis of Symmetry - when the figure is its own image under a reflection in a line.

Example: An Isosceles triangle - draw a line from the vertex of the angle that is not a base angle perpendicular to the base side (non-congruent side). This line is the axis of symmetry because it divides the triangle into 2 congruent triangles.

Triangle ABC is an isosceles triangle with sides AB = BC and angle A = angle C. Draw a perpendicular line from vertex A through AC and where they intersect label this point D. Line segment BD is the axis of symmetry and the reflection line so triangle ABD = triangle CBD.

Example 2: Can you think of any letters of the alphabet that have line symmetry?

A, B, C, D, E, H, I, K, M, O, T, U, V, W, X, Y

How about any words?
MOM, BIKE, HIKED, CHECK, BOB, DEED, RADAR,

(sometimes the lines are horizontal (MOM) and other times they are vertical (BIKED)

Precalc 2.7 Graphs of Rational Functions

2.7 Graphs of Rational Functions

A. Guidelines for Graphing Rational Functions

Let f(x) = N(x)/D(x), where N(x) and D(x) are polynomials with no common factors.

1. Find and plot the y-intercept (if any) by evaluating f(0).

2. Set the numerator equal to zero and solve the equation N(x) = 0. The real solutions represent the x-intercepts of the graph. Plot these intercepts.

3. Set the denominator equal to zero and solve the equation D(x)=0. The real solutions represent the vertical asymptotes. Sketch these asymptotes using dashed vertical lines.

4. Find and sketch the horizontal asymptotes of the graph using a dashed horizontal line.

5. Plot at least one point between and one point beyond each x-intercept and vertical asymptote.

6. Use smooth curves to complete the graph between and beyond the vertical asymptotes.

7. Test for Symmetry (origin, x-axis, y-axis)

Example:

g(x) = (x² + 1)/x

1. g(0) = (0 + 1)/0 = undefined so no y-intercept

2. x² + 1 = 0

x² = -1

x = plus and minus the square root of negative one

Therefore no real x-intercepts

3. D(x) = 0 so x=0 vertical asymptote

4. n = 2 and m = 1 so n>m, the graph has no horizontal asymptote.

5. points to plot (-3, -3.3), (-2, -2.5), (-1, -2), (0, error), (1,2), (2,2.5), (3,3.3)

6. Sketch the graph using steps #1 - 5

7. Test for symmetry (origin, x-axis, y-axis) - none

B. Slant Asymptote

If the degree of the numerator of a rational function is EXACTLY ONE MORE than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. (n = m + 1)

From last example:

g(x) = (x² +1)/x

n = 2 and m = 1 so slant asymptote

divide them out using long division and you get “x + 1/x”

Dropping the remainder, you get the slant asymptote y=x.

Add this asymptote line to the graph.

Example 2:

f(x) = (2x² - 5x + 5)/ (x² - 2)

1. f(0) = -2.5 so (0, -2.5) is a y-intercept

2. 2x² - 5x + 5 = 0

using the quadratic equation you have 5/4 + i √15 and 5/4 - i √15

So the roots are imaginary so there are no x-intercepts

3. D(x) = 0

x² - 2 = 0

x² = 2

x = √2 and x = - √2

4. n = 2 and m = 2 so n = m so (2/1) = 2 = y

So the horizontal asymptote is y = 2.

5. Finding points (-2, 11.5), (-1, -12), (0, -2.5), (1, -2), (2, 1.5)

6. Using these points and asymptotes, graph the polynomial function.

Application:

Page Design: you have a rectangular page with a width of x units and a height of y units. It has a margin of 1″ on both sides of the x length and 2″ on the y length.

The inner page then has a width of x - 1 - 1 or x - 2

and a height of y - 2 - 2 or y - 4. The page contains 30 square inches of print.

Therefore it has an area of A = xy.

The smaller inner rectangle has an area of 30 square inches of print.

30 = (x - 2)(y - 4)

30/(x-2) = y - 4

30/(x - 2) + 4 = y

(30 + 4(x-2)) / (x-2) = y

(30 + 4x - 8)/ (x - 2) = y

(4x + 22)/ (x - 2) = y

Recall: A = xy so plugging y in

A = x ((4x + 22)/(x-2))

What is the domain? Since the margins on the left and right are each 1 inch so x>2.

Sketching this:

You see that the Area is minimum when x = 5.87

Homework #23; pg. 204; #1,2,31-39 odd, 47-53 odd, 70, 75

Geometry 7.3 Rotations - guided notes

Geometry Unit 5, sec 7.3

Rotations - guided notes

I) Rotations - sec. 7.3

A. Definition: A rotation is a transformation in which a figure is turned about a fixed point.
This point is called the center of rotation.

1. Rotation 90 degrees (counterclockwise): (x,y) maps to (-y,x).

Example: (4 , 1) becomes (-1 , 4)

2. Rotation 180 degrees (counterclockwise): (x,y) maps to (-x,-y).

Example:(4 , 1) becomes (-4 , -1)

3. Rotation 270 degrees (counterclockwise) or -90 degrees (clockwise): (x,y) maps to (y,-x).

Example: (4 , 1) becomes (1 , -4)

B. Rotational symmetry: a figure has rotational symmetry if the figure can be mapped onto itself by a rotation of 180 degrees or less.

Example: A circle, Square, Rhombus, Equilateral Triangle, and more. Any regular polygon with center angle 180 degrees or less.

An example of a figure without rotational symmetry: trapezoid

Geometry 8.3+8.4 Similar polygons guided notes

A. Similar polygons - If all corresponding angles are congruent and all corresponding sides are proportional, then the polygons are similar.

Example: If quadrilateral ABCD and quadrilateral DFGH have the following relationship:

angle A = angle D, angle B = angle F, angle C = angle G, angle D = angle H, and

(AB)/DF = BC/FG = CD/GH = AD/DH,

then we know quadrilateral ABCD ~ quadrilateral EFGH.

B. Statement of Proportionality: Set up ratios using corresponding sides. These ratios are all proportional.

Example: Pentagon ABCDE ~ Pentagon FGHIJ

Because the pentagons are similar, we know angle A = angle F, angle B = angle G, angle C = angle H, angle D = angle I, angle E = angle J and

AB/FG = BC/GH = CD/HI = DE/IJ = AE/FJ

C. Using scale factors: Set up a ratio using a pair of corresponding sides. Reduce, if possible.

Example: If you want to enlarge a picture that is 3” in width by 5” in length and have the corresponding width of the enlarged picture be 10”, what is the new length?

3/5 = 10/x

3x = 50

x = 50/3

If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Geometry 7.4 - Translations and Vectors

I) Translations sec. 7.4

A) Definition: A translation is a transformation that shifts every two points the same distance in the same direction. It is an isometry.

example: Line segment PQ maps to line segment P’Q’.

PQ = P’Q’ , PP’ is parallel to QQ’, PQ is parallel to P’Q’

PP’ = QQ’
Therefore, it creates a parallelogram.

B) Theorem 7.5: If 2 lines are parallel, then a reflection in a line followed by a reflection in another line results in a translation.

EX. line k and line m and parallel and a distance "d" apart.

When you reflect line segment PQ over line k and then over line m, you have P”Q”.

PP” = 2d, PQ mapped to P”Q” is a translation.

PP” is perpendicular to line k, and PP” is perpendicular to line m.

A vector is a quantity that has both direction and magnitude, or size, and is represented by an arrow drawn between two points.

The initial point, or starting point, of the vector is P and the terminal point, or ending point, is Q. The vector name is PQ with the notation of a ray except it does not have the bottom part of the arrow.

The horizontal component of vector PQ is the horizontal shift from point P to point Q and the vertical component of vector PQ is the vertical shift from point P to point Q.

Example, given point P (3,2) and point Q(7,8).

The horizontal shift is from3 to 7 or 4 units.

The vertical shift is from 2 to 8 or 6 units.

the component form is written like the following:

Therefore, for this example, <4,6>.

Translations in a Coordinate Plane:
Sketch a parallelogram with the following vertices:

R(-4, -1), S(-2, 0), T(-1, 3), and U(-3, 2)
Then sketch the image of the parallelogram after translation (x, y) ⇒ (x + 4, y - 2)

R(-4 , -1) ⇒ R’ ( -4 + 4, -1 - 2) = R’ (0 , -3)
S (-2, 0) ⇒ S’ (-2 + 4, 0 - 2) = S’ (2, -2)
T (-1 , 3) ⇒ T’ (-1 + 4, 3 - 2) = T’ (3 , 1)
U (-3 , 2) ⇒ U’ (-3 + 4, 2 - 2) = U’ (1, 0)

The Component Form of this parallelogram would be
<4,-2>.

Draw the vectors between parallelogram RSTU to R’S’T’U’. You have now made a 3-D figure.
To find the magnitude, use the distance formula:
vector's magnitude = √(x₁-x₂)²+(y₁-y₂)²

Since the difference of the x’s is the horizontal component
and
the difference of the y’s is the vertical exponent we have:

vector's magnitude = √ (horizontal component² + vertical component²)

So this would be
√(4² + (-2)²) = √(16 + 4) = √20.

Geometry 8.1+8.2 Ratio and Proportion

Geometry unit 8, sec 8.1+8.2

Ratio and Proportion

Guided notes:

I) Vocabulary:

A) Ratio: A comparison of same quantities (or units), usually expressed in fractional form.

Ex. a/b a:b or you can use "a to b" where b is not = 0.

B) Proportion: Two ratios set equal to each other

1. Means: bottom left, top right, or 2 inside terms.

2. Extremes: top left, bottom right, or 2 outside terms.

3. The means and the extremes can be multiplied and the results are still the same.

4. Cross products are equal!

In a proportion, the product of the means is equal to the product of the extremes.

Example:

a/b = c/d

ad = bc

1) 1/2 = 2/4

(1)(4) = (2)(2)

4 = 4

2) 9/12 = 3/4

(9)(4) = (12)(3)

36 = 36

Geometric Mean: If two means are equal, then they are the geometric mean, or the mean proportional.

For 2 positive numbers “a” and “b” the positive number x, such that

a/x = x/b

ab = x(x)

ab = x²

1) what is the geometric mean of 3 and 9?

3/x = x/9

27 = x²

x = the square root of 27

2) what is the geometric means of 9 and 4?

9/x = x/4

36 = x²

x = 6

Precalculus Guided Notes 3.2

Precalculus 3.2 Logarithmic Functions and their Graphs

Recall the exponential function f (x) = a^x, a > 0, a is not equal to 1.

Every function of this form passes the Horizontal Line Test and therefore must have an inverse. This inverse function is called the logarithmic function with base a.

Definition of Logarithmic Function:

For x > 0 and 0 is less than a which is not equal to 1, then

y = log_ax if and only if x = a^y.

The function f (x) = log _a x is called the logarithmic function with base a.

Examples 1: log ₃ 81 =

so we think: 3^x = 81
3⁴ = 27 so x = 4

Example 2: log ₄ 1024 = 5

Example 3: log _a 1 = 0

Properties of Logarithms

1. log _a1 = 0 because a⁰ = 1.

2. log _aa = 1 because a¹ = a.

3. log _aa^x = x and a^(log_ax) = x. 4 by inverse properties

4. If log_a x = log_ay, then x = y by One to One Property

Example: Graph of y = log_ax , a > 1

Domain: (0, ∞ )

Range: (- ∞, ∞ )

Intercept: ( 1, 0)

Increasing

y-axis is a vertical asymptote

(log a x ) → - ∞ as x → 0+

Continuous

Reflection of graph of y = a^x in the line y = x.

Transformations of Graphs of Logarithmic Functions:

a. f(x) = log _a (x - h) + k Basic graph

h = ________ k =________ a = __________.

b. f (x) = log _a (x - 2) This is the graph f (x) = log_a x

shifted 2 units to the right.

c. f (x) = log _a (x +3) This is the graph

f (x) = log _a x shifted 3 units to the left.

d. f (x) = log _a x +5 This is the graph

f (x) = log _a x shifted 5 units up.

e. f (x) = log _a (-x) This is the graph

f (x) = log _ax reflected in the y-axis.

f. . f (x) = -log _a x This is the graph

f (x) = log _a x reflected in the x-axis.


The Natural Logarithmic Function

The function defined by

f (x) = log _e x = ln x, x > 0

is called the natural logarithmic function.

Properties of Natural Logarithms:

1. ln 1 = 0 because e⁰ = 1

2. ln e = 1 because e¹ = e.

3. ln e^x = x and e ^{ln x} = x by inverse properties.

4. If ln x = ln y, then x = y by One - to - one properties

Example:

1. Find the domain, the intervals in which the function is increasing or decreasing,
and approximate any relative maximums or minimums:

f (x) = x/(ln x)

a. Graph the function:

1. Domain = ___________

2. Increasing = ____________

3. Decreasing = ____________

4. Maximum or minimums ________________

Geometry 8.5 Proving Triangles are Similar

Geometry 8.5 guided notes:

If two angles of one triangle are congruent to the 2 other angles of a second triangle, then the triangles are similar.

I) Proving Triangles are similar

A) Side-Side-Side (SSS) Similarity Theorem:

If the corresponding pairs of sides of two triangles are proportional, Then the triangles are similar.

Example:

B) Side-Angle-Side (SAS) Similarity Theorem

If an angle of one triangle is equal to an other triangle, and the sides forming these angles are in proportion, then the triangles are similar.

Example:

Geometry Unit 8, sec 8.6+8.7 guided notes

Geometry Unit 8, sec 8.6 & 8.7

Proportions and Similar Triangles & Dilations - Guided notes

I). Proportions and Similar Triangles: Sec 8.6

If a line parallel to one side of a triangle intersects the other 2 sides, then it divides the 2 sides proportionally, and the two triangles are similar.

Example: Triangle ABC with line segment DE parallel to side AC, with point D on AB and point E on BC. Since DE is parallel to AC, then (DB)/(DA) = (BE)/(EC) and (BA)/(BD)=(BC)/(BE)=(AC)/(DE)

If BD = 7, AD = 3, BE = 8, ED = 6, EC = x, and AC = y, solve for x and y.

(DB)/(DA) = (BE)/(EC)

7/3 = 8/x

7x = 24

x = 24/7

(BA)/(BD) = (BC)/(BE) = (AC)/(DE)

10/7 = (8 + 24/7)/(8) = y/6

10/7 = y/6

7y = 60

y = 60/7

B) Theorem 8.7: If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other 2 sides.

Example: Given diagram - Triangle ABC with ray CD from vertex C bisecting angle ACB

If ray CD bisects angle ACB, then (AC)/(AD) = (CB)/(DB)

If AC = 6, AD = 3, DB = x and CB = 8 then

6/3 = 8/x

6x = 24

x = 4

II) Dilations: Sec 8.7

C) Dilations: Changes the size of a figure, so it is NOT an isometry. It has center, C, and a scale factor, k.

1. Reduction- the scale factor is between 0 and 1

(this means that 0 is less than the scale factor which is less than 1)

2. Enlargement - the scale factor is k>1.

*NOTE: To find scale factor, divide Image by Preimage

(CP’)/ (CP) = k = scale factor

Example:

Given center point C and preimage Triangle PQR that maps to image Triangle P’Q'R’, CP = 6 and CP’ = 3, then the scale factor is 3/6 or 1/2.

Given center point C and preimage Triangle STU that maps to image Triangle S’T'U’, CS' = 10 and CS = 5, then the scale factor is 10/5 = 2/1. Remember that scale factors have to be in a ratio!

D) Dilation in coordinate geometry:

(x,y) maps to (kx, ky) where k is the scale factor.

Example: If you have a point P (3,2) and the scale factor is 6, then (x, y) maps to (kx, ky) so…

P (3,2) maps to P' (3 x 6, 2 x 6) = (18,12).

Geometry 12.1 notes

12.1 Exploring solids

Polyhedron - is a solid that is bounded by polygons called faces that enclose a single region of space.

An edge of a polyhedron is a line segment formed by the intersection of two faces.

A vertex of a polyhedron is a point where three of more edges meet.

Plural = polyhedra

Example:

Regular Polyhedron - all faces are congruent and all the faces are regular polygons (polygons that have all angles congruent and all sides congruent).

Convex Polyhedron - any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.

Cross-section - intersection of the plane and the solid. pg. 720

Platonic Solids -

Regular Tetrahedron (4 faces, 4 vertices, 6 edges) - associated with the element of fire.

Cube (6 faces, 8 vertices, 12 edges) -

Regular Octahedron (8 faces, 6 vertices, 12 edges) - associated with the element of air.

Regular Dodecahedron (12 faces)

Regular Icosahedron (20 faces, 12 vertices, 30 edges) - associated with the element of water. A soccer ball’s pattern is a truncated icosahedron.

Euler’s Theorem:

The number of faces (F), vertices (V), and edges (E), of a polyhedron are related by the following formula:

F + V = E + 2

Example: a polyhedron has 20 faces; all triangle

Since a triangle has 3 sides and each of the triangles are going to share an edge, you have

20 + V = ((3)(20))/2 + 2

20 + V = (60)/2 + 2

20 + V = 30 + 2

20 + V = 32

V = 12

Homework: #40 pg. 723 #10-30 even, 32-35, 54, 55, 60-62, 69

Geometry 12.2 -notes

Geometry 12.2 notes

Prism - is a polyhedron with two congruent faces called bases that lie in parallel planes.

The other faces are called lateral faces - are parallelograms formed by connecting the corresponding vertices of the bases.

The segment connecting these vertices are lateral edges.

Example:

Right Rectangular Prism Oblique Triangular Prism

Surface Area of a Right Prism:

SA = 2 (Area of the Base) + (perimeter of the base) (height between the bases)

SA = 2A + Ph

Right Prism - height is the perpendicular distance between bases and each lateral edge is perpendicular to both bases.

Example:

Rectangular Prism with length = 16 cm, width = 4 cm, and height = 9 cm.

SA = 2 B + Ph

SA = 2 (16)(4) + (16 + 4 + 16 + 4)(9)

SA = 488 square cm.

Recall the area of an equilateral triangle is

A = the square root of 3 times the side squared divided by 4

Cylinder - is a solid with congruent circular bases that lie in parallel planes.

The altitude or height of the cylinder is the perpendicular distance between the bases.

Example:

Given a Cylinder with radius of 5 ft. and the height of 12 ft., what is the surface area? Use 3.14 for pi.

SA = 2 B + Ph

The base is a circle so Area of a circle is = pi (5)(5) = 78.5 square feet

The perimeter of a circle is the circumference so C = pi (10) = 31.4

SA = 2B + Ph

SA = 2(78.5) + (31.4)(12) = 157 + 376.8 = 533.8 square feet.

The Surface Area of a cylinder is

2 (pi) (r^2) + 2( pi) (r)( h)

Precalculus 3.5 notes

3.5 Exponential and Logarithmic Models

5 most common:

1). Exponential growth model: y = ae^(bx), b>0.

2). Exponential decay model: y = ae^(-bx), b>0.

3). Gaussian model: y = ae^{(-((x-b)2)/c)}

4). Logistic growth model: y = a/(1+be^(-rx))

5). Logarithmic models:

a. y = a + b ln x; b. y = a + b log x

Graphs:

Examples:

Exponential Growth: Given P = 240360e^(0.012t)

t = 0 represents 2000, when will the population reach 275,000?

275000 = 240360 e^(0.012t)

1.144117158 = e^(0.012t)

ln 1.144117158 = 0.012 t

11.21944152 = t

Therefore, 2000 + 11 = 2011.

Example 2:

Country: Croatia, in 1997 had 5.0 million people and is estimated to have 4.8 in 2020, how many will there be in 2030?

Two points: (0, 5) and (23, 4.8)

Find the exponential growth model using y = ae^(bt)

5 = ae^(b*0)

5 = ae⁽⁰⁾

5 = a

4.8 = ae^{(b * 23)}

4.8 = 5e^(23b)

.96 = e^(23b)

ln .96 = 23b

-.0017748693 = b

Croatia’s equation is y = 5e^{(-.0017748693t)}

So in 2030, the population is 4.7 million

The “b” is the growth rate.

Since “b” is negative in Croatia, the population is decreasing.

Gaussian Models y = ae^-((x-b)2)/c

The model is commonly used in probability and statistics to represent populations that are NORMALLY DISTRIBUTED. For STANDARD normal distributions, the model takes the form:

y = (1/(standard deviation times the square root of 2 pi) )times (e^{((-x^2)/(2 standard deviation squared))})

The graph of a Gaussian model is called a bell-shaped curve.

Example:

The SAT scores in 1997 followed:

y = 0.0036e^{((-(2-511)^2)/(25088))} where 200 is less than or equal to x which is less than or equal to 800.

The maximum point is (511.11112, 0.0036)

A logistic growth curve: is a rapid growth followed by a declining rate of growth OR a sigmoidal curve.

y = a / ( 1 + be^(-rx))

Example: Spread of a virus (flu)

y = 5000/(1 + 4999e^(-.8r))

a. How many students are infected after 5 days?

54 people

b. If classes are cancelled when 40% or more are infected of a campus of 5000 students, how long will it take? 10.14 days.

Magnitudes of Earthquakes

R = log (I/ I_o)

Magnitude R of an earthquake of Intensity of I where I₀ usually equals 1.

Example: Find the magnitude of R of an earthquake of Intensity I, let I₀ = 1.

a). Taiwan in 1999, I = 39811000

R = log 39811000 = 7.600003087 = 7.6

Example 2: What is the Intensity if R = 8.6?

8.6 = log I

10^(8.6) = I

I = 398107170.6

Homework: #30 pg. 266; #1-29 odd, 39

Geometry 12.3 notes

Geometry 12.3 notes

Pyramid - is a polyhedron in which the base is a polygon and the lateral faces are triangles with a common vertex.

Altitude - of a pyramid is the perpendicular distance between the base and the vertex.

Base edge - is the intersection of the base and any lateral face.

Lateral face - has slant height “l” - the altitude of any lateral face.

Regular Pyramid - has a regular polygon for a base and it’s altitude meets the base at the center.

Regular Polygon - recall all sides are congruent and all angles are congruent.

How do you find the surface area of a pyramid?

1. Find the area of the base and the area of all the triangles that make up the lateral faces.

2. If the pyramid has a regular base, since all the triangles would have the same measurements, you can use the following formula:

Surface Area of a Regular Pyramid: B + (1/2 )Pl

B = the area of the base

P = Perimeter of the base

l = height of the lateral face

Circular Cone, or cone, has a circular base and a vertex that is not in the same plane as the base.

Altitude - the perpendicular distance between the vertex and the base.

Slant Height - l - the altitude from the vertex of the cone to the base of the cone on the side of the cone.

Surface Area of a Right Cone:

SA = (pi) r ² + (pi) r l

Examples:

You have a regular pyramid with the altitude of the pyramid = 321 feet. The base edge is 300 feet. What is the slant height?

You know the altitude meets at the center of the pyramid so the altitude of the pyrmid, the slant height of one of the lateral faces and 1/2 the length of the base form a right triangle.

using a² + b² = c²

(321)² + (150)² = l ²

125541 = l ²

c = 354.32 = slant height.

Example: What is the surface area of the pyrmid?

SA = B + (1/2) P l

SA = (300)(300) + (4)(300)(354.32)

SA = 515184 square feet

Example:

Given a right cone with an altitude of the cone = 4cm and the diameter = 6 cm, what is the surface area?

Again, the altitude, the radius and the slant height form a right triangle:

using a² + b² = c²

(4)² + (3)² = l ²

25 = l ²

l = 5cm = slant height.

Therefore the surface area = (pi) (3)(3) + (pi)(3)(5) = 9(pi) + 15(pi) = 24(pi) = about 75.40 square cm.

HW: pg. 738 #18 - 36 even, 50, 52, 53

Geometry 12.4 notes

Geometry 12.4 Volume of Prisms and Cylinders

Volume of a solid is the number of cubic units contained in its interior.

Volume Congruence Postulate - if 2 polyhedra are congruent, then they have the same volume.

Volume Addition Postulate - the volume of a solid is the sum of the volumes of all its nonoverlapping parts.

Cavalieri’s Principle - if 2 solids have the same height and the same cross-sectional area at every level, then they have the same volume.

Volume of a Right Prism = Bh

B = Area of the base

h = height of the solid

Volume of a Cylinder = Bh = (pi) r²h

Example:

You have a right prism with a length of 5 cm, a width of 4 cm, and a height of 8 cm. What is it’s volume?

V = Bh = (5)(4)(8) = 160 cubic cm

Example:

You have a right cylinder with a diameter of 8 feet and a height of 13 feet. What is it’s volume?

V = Bh the radius will be 1/2 of 8 = 4 feet so

V = (pi)(4²)(13) = 208(pi) cubic feet or about 653.45 cubic feet.

Homework: #43 pg. 746; #10-19, 21-24, 28-32 even, 36-42 even, 45-49, 51-55, 57-59

Geometry 12.5 notes

Geometry 12. 5 Volume of Pyramids and Cones

Volume of a Right Pyramid = (1/3)Bh

B = Area of the base

h = height of the solid

Volume of a Right Cone = (1/3) Bh = (1/3) (pi) r²h

Examples:

You have a regular pyramid with a square base that has a side of 3 cm and a height of 6 cm. What is it’ volume?

V = (1/3)Bh = (1/3)(3)(3)(6) = 18 cubic cm.

Example:

You have a right cone solid with a radius of 6.7 m and a height of 10.2 m. What is it’s volume?

V = (1/3) Bh = (1/3) (pi) (6.7²)(10.2) = 152.63(pi) cubic meters or about 479.4887203 cubic meters.

Geometry 12.6 notes

Geometry 12.6 Surface Area and Volume of Spheres

Sphere - is the locus of points in space that are a given distance from a point.

Chord of a sphere - is a segment whose endpoints are on the Sphere.

Diameter of a sphere - is a chord that contains the center.

Radius of a sphere - is a segment that goes from the center of the sphere to an endpoint on the sphere. It is 1/2 the diameter in measure.

Surface Area of a sphere = 4 (pi) r²

Volume of a sphere - (4/3) (pi) r³

Great Circle of a sphere - if a plane intersects a sphere, the intersection is either a single point or a circle.

If the plane contains the center of the sphere, then the intersection is a Great Circle of the Sphere.

Every great circle of a sphere separates a sphere into 2 congruent halves called hemispheres

Homework: worksheet 12.6 B

Geometry 12.7 notes

Geometry 12.7 Similar Solids

Similar solids - two solids with equal ratios of corresponding linear measures, such as heights or radii.

Common ratio is called the scale factor.

Example:

If a rectangular prism has sides of 3, 2, and 2, and a second one has lengths of 5, 4, and 4, are these two solids similar?

Set up ratios:

3/6 = 2/4 = 2/4

1/2 = 1/2 = 1/2 , so yes they are.

If 2 similar solids have a scale factor of a:b, then the corresponding areas have a ratio of a²:b² and corresponding volumes have a ratio of a³:b³.

Example:

If one sphere has a volume = 8pi and a second one has a volume of 125pi, what is the ratio of their areas?

((8pi)/(125pi))^1/3 = 2/5

(2/5)² = 4/25

Example:

The scale factor of the model car is 1:16. If the model car is 5.5 in. What is the height of the car?

1/16 = 5.5/x
x = 88 inches

Homework: Worksheet 12.7 B

Precalculus 3.2 Logarithmic Functions and their Graphs

Precalculus 3.2 Logarithmic Functions and their Graphs

Recall the exponential function f (x) = a^x, a > 0, a


1 . Every function of
this form passes the Horizontal Line Test and therefore must have an inverse. This
inverse function is called the logarithmic function with base a.

Definition of Logarithmic Function:

For x > 0 and 0 < a


1,

y = log _ax if and only if x = a^y.

The function

f (x) = log _a x

is called the logarithmic function with base a.

Examples:

log ₃ 27 = ____________ log ₄ 64= _________ log _a 1 = ___________

Properties of Logarithms

1. log _a 1 = ________ because a ⁰ = 1

2. log _a a = ________ because a ¹ = a.

3. log _a a ^x = _________because a^(log _a x ) = x

4. If log _a x = log _a y, then x = y.

Example:

Graph of y = log _a x , a > 1

Domain: (0, ∞ )

Range: (- ∞, ∞ )

Intercept: ( 1, 0)

Increasing

y-axis is a vertical asymptote

(log _a x ) → - ∞ as x → 0⁺

Continuous

Reflection of graph of y = a^x in the line y = x.

Transformations of Graphs of Logarithmic Functions:

a. f(x) = log _a (x - h) + k Basic graph - h = ________ k = ________ a = __________

b. f (x) = log _a (x - 2) This is the graph f (x) = log _a x shifted ___ units to the
________.

c. f (x) = log _a (x +3) This is the graph f (x) = log _a x shifted ___ units to the
________.

d. f (x) = log _a x +5This is the graph f (x) = log _a x shifted ___

units________.

e. f (x) = log _a (-x) This is the graph f (x) = log _a x reflected in the ______.

f. . f (x) = -log _a x This is the graph f (x) = log _a x reflected in the ______.

Examples:

The Natural Logarithmic Function

The function defined by

f (x) = log _e x = ln x, x > 0

is called the natural logarithmic function.

Properties of Natural Logarithms:

1. ln 1 = 0 because e⁰ = 1

2. ln e = 1 because e¹ = e.

3. ln e^x = x and e ^{ln x} = x.

4. If ln x = ln y, then x = y.

Example:

1. Find the domain, the intervals in which the function is increasing or decreasing,
and approximate any relative maximums or minimums:

f (x) = x/(ln x)

a. Graph the function:

1. Domain = ___________

2. Increasing = ____________

3. Decreasing = ____________

4. Maximum or minimums ___________________

4.1 Radian and Degree Measure

4.1 Radian and Degree Measure
Trigonometry means measurement of triangles.

When placed on the coordinate system, the initial side is placed on the x-axis. This is standard position of the angle. Positive angles are generated by counterclockwise rotation, and negative angles by clockwise rotation. An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side. The endpoint of the ray is the vertex of the angle. In standard position, the vertex of the angle is on the origin or where the x-axis and y-axis intersect (point (0, 0)).

If two angles have the same initial and terminal sides, one clockwise angle and one counterclockwise angle, these angles are coterminal.
Examples:
What is coterminal to:

1. 75 degrees ________

2. 125 degrees ________________

3. π/3 radians ___________

4. 3π/4 radians ______________

Radian Measure - one way to measure angles.
One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.
Recall: the radian measure of an angle of one full revolution is 2π, therefore

½ revolution = π radians

1/4 revolution = π/2 radians

1/6 revolution = π/3 radians

1/8 revolution = π/4 radians

1/12 revolution = π/6 radians

Two positive angles a and b are complementary (complements of each other) if their sum is π/2 or 90 degrees.
Two positive angles a and b are supplementary (supplements of each other) if their sum is π or 180 degrees.

Examples - What is complementary and supplementary to:
1. π/3 Complementary _______________

Supplementary ______________

2. 62 degrees - Complementary __________

Supplementary ______________

Degree Measure:

A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex. A full rotation is 360 degrees. Therefore:

½ rotation = 180 degrees

1/4 rotation = 90 degrees

1/6 rotation = 60 degrees

1/8 rotation = 45 degrees

1/12 rotation = 30 degrees

With this in mind:

½ rotation = 180 degrees and ½ rotation = π radians, we can conclude:

½ rotation = 180 degrees = π radians

1/4 rotation = 90 degrees = π/2 radians

1/6 rotation = 60 degrees = π/3 radians

1/8 rotation = 45 degrees = π/4 radians

1/12 rotation = 30 degrees = π/6 radians

1 degree = π/180 radians and 1 radian = 180degrees/π

Quadrants:

I. 0 degrees to 90 degrees OR 0 radians to π/2 radians

II. 90 degrees to 180 degrees OR π/2 radians to π radians

III. 180 degrees to 270 degrees OR π radians to 3π/2 radians

IV. 270 degrees to 360 degrees OR 3π/2 radians to 2π radians

Example: Which quadrant does the angle lie?

1. 72 degrees _________

2. 158 degrees __________

3. -260 degrees ______

To convert degrees to radians, multiply degrees by π radians/ 180 degrees.
To convert radians to degrees, multiply radians by 180 degrees/π radians.

Examples:

1. 83.7 degrees = _______radians

2. -7π/6 radians = ________degrees

3. 395 degrees = _______radians

4. 28π/15 radians = ______degrees

To find the arc length of made by a central angle with a given radius:
S = θ r
Examples:
Given radius 5 cm and central angle of 2π/3, what is the arc length ________
Given radius 8 feet and central angle 35 degrees, what is the arc length _______
Linear and Angular Speed:

Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is:
Linear speed = (arc length)/time = s/t.
Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, the angular speed of the particle is:
Angular speed = (central angle)/time = θ/t.
Example:

An earth satellite in circular orbit 1250 kilometers high makes one complete revolution every 90 minutes. What is its linear speed? Use 6400 kilometers for the radius of the earth.

HW 32, pg. 291; #7-21 odd, 33, 34, 35-61 odd, 67-73 odd, 75-85 odd, 95

4.2 Trigonometric Functions: The Unit Circle

4.2 Trigonometric Functions: The Unit Circle

Unit Circle (x² + y²)= 1

As the real number line is wrapped around the unit circle, each real number t corresponds to a point (x,y) on the circle. For example, the real number 0 corresponds to the point (1,0). Because the unit circle has a circumference of 2π, the real number 2π also corresponds to the point (1,0).
Examples:

1.) π/2 corresponds to (x,y) = (0, 1)

2.) π corresponds to (x,y) = (-1, 0)

3.) 3π/2 corresponds to (x,y) = (0, -1)

In general, each real number t corresponds to a central angle θ (in standard positition) whose radian measure is t.

Trigonometric Functions:
Definitions of Trigonometric Functions:
Let t be a real number and let (x,y) be the point on the unit circle corresponding to t.

sine t is written as sin t = y

cosecant t is written as csc t = 1/y

cosine t is written as cos t = x

secant t is written as sec t = 1/x

tangent t is written as tan t = y/x

cotangent t is written as cot t = x/y

Degrees	0°	30°	45°	60°	90°
Radians	0 radians	π/6	π/4	π/3	π/2
sin θ
cos θ
tan θ
csc θ
sec θ
cot θ

Withing the unit circle, you can see that :

A). the value of x is betweeen -1 and 1

B). the value of y is between -1 and 1

Definition of a Periodic Function
A function f is periodic if there exists a positive real number c such that
f (t + c) = f (t)
for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

Example:
26π/12 = 2π + 2π/12 , you have sin (26π/12 ) = sin (2π + 2π/12 ) = sin 2π/12 = ½

Even and Odd Trigonometric Functions
The cosine and secant functions are even.

cos ( -t) = cos ( t )

sec ( -t ) = sec ( t )

The sine, cosecant, tangent and cotangent are odd.

sin( -t ) = -sin t

csc ( -t ) = - csc t

tan ( -t ) = -tan t

cot ( -t ) = -cot t

Examples:

sin ( -30°) = -½ -sin (30°) = -1/2

same answer

cos ( -60°) = ½

cos (60°) = ½

same answer

tan ( -45°) = -1

tan (45°) = -1

same answer

sin ( -330°) = ½

sin (330°) = -( -½) = ½

same answer

Evaluating Trigonometric Functions with a Calculator
Make sure you set your calculator to the proper mode of measurement (degrees or radians). Also remember to enclose all fractional angle measures in parentheses.
Examples:
cos (5π/3) = 1/2

Precalculus 4.4 notes

4.4 Trigonometric Functions of any Angle
Definitions of Trigonometric Functions of Any Angle:
Let < θ be any angle in standard position with (x, y) a point on the terminal side of < θ

and r² = x² + y²

so therefore:

r = the square root of (x² + y²) - and cannot equal zero.

sin θ = y/r

csc θ = r/y , y cannot equal 0.

cos θ = x/r

sec θ = r/x , x cannot equal 0.

tan θ = y/x , x cannot equal 0.

cot θ = x/y , y cannot equal 0.

Example: the given point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle.

1. (5, -12) When you plot this it is in the IV quadrant. Using Pythagorean theorem,

(5)² + (-12)² = r²

25 + 144 = r²

169 = r²

13 = r

so you see the hypotenuse is 13. Therefore the values of the six trigonometric functions are:

sin θ = -12/13

csc θ = 13/-12

cos θ = 5/13

sec θ = 13/5

tan θ = -12/5

cot θ = -5/12

Reference Angles:
Let θ be an angle in standard position. Its reference angle is the acute angle θ’ formed by the terminal side of θ and the horizontal axis.

Quadrant II

θ’ = π - θ (radians)

θ’ = 180° - θ (degrees)

Quadrant III

θ’ = θ - π (radians)

θ’ = θ - 180° (degrees)

Quadrant IV

θ’ = 2π - θ (radians)

θ’ = 360° - θ (degrees) Example: Find the reference angle:

1. θ = 57° θ’ = 57°

2. θ = 125°

θ’ = 180° - 125° = 55°

3. θ = 295°

θ’ = 360° - 295° = 65°

4. θ = 2π/3

θ’ = π - 2π/3 = π/3

5. θ = 5π/6

θ’ = π- 5π/6 = π/6

6. θ = 11π/6

θ’ = 2π - 11π/6 = π/6

Evaluating Trigonometric Functions of any angle:
To find the value of a trigonometric function of any angle θ:
1. Determine the function value for the associated reference angle θ’.

2. Depending on the quadrant in which θ lies, prefix the appropriate sign to the function value.

By using reference angles and the special angles discussed in the previous section, you can greatly extend the scope of exact trigonometric values.

Example:
Evaluate the trigonometric function exactly:
1. sin -20π/3 = sin -2π /3 = sin 4π/3 =(since 4π/3 is in the III quadrant) = -sin(ref <>

2. cos( -300°) = cos 300 = (ref <) = cos 60° = ½

3. tan (- 495°) = tan (-135) = (ref <) = tan 45° = 1

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