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.