Unit 2 Test Bonus: 1) The rate at which a tablet of Vitamin C begins to dissolve depends on the surface area of the tablet. One brand of tablet 3 cm long in the shape of a cylinder with hemispheres of diameter 0.5 cm attached at both ends. A second tablet is to be manufactured in the shape of a regular cylinder with height 0.5 cm. Find the diameter of the second tablet so that its surface area is equal to that of the first tablet. (5 points) 2) On the back of the paper, draw a picture of Old McDonald’s Farm. (0-2 points)

Functions and Graphs Unit 3

Unit Essential Question How do we represent all of the different functions graphically?

Coordinate Systems Lesson 3.1

LEQ How do we use the distance formula and midpoint formula to determine perpendicular bisectors?

Distance and Midpoint Formula

Right Triangle/Perpendicular Bisectors We can show that three points can create a right triangle by using the Distance Formula and the Pythagorean Theorem. Example Page 140 # 16 A perpendicular bisector is a line that passes through a segment that creates a right angle and bisects the segment. Examples Page 140 #’s 22, 24

Homework: Pages #’s 9-23 odds

Bell Work:

1) Is the triangle created by the points A(4,2), B(8,3), and C(4,9) a right triangle?

Bell Work: Find the coordinates of point A(-a, 2a) if it is in the second quadrant has a distance of 10 units away from point P(4,2).

Graphs of Functions Lesson 3.2

Sketching Graphs There are going to be times this year where we will draw graphs of functions and equations by hand, but for now we will introduce how to use the TI-83 Plus. In order to use our calculator, all of the functions must be in the form y =.

Finding Minimums, Maximums, and Zeros Some graphs will have a minimum and maximum that needs to be found. The minimum is the smallest value in the range and the maximum is the largest value in the range. (Remember that Range refers to y-coordinates.) Zero’s refer the points where a function or graph will cross the x-axis, which means that y is equal to zero. (also the x- intercept(s)) Sometimes there may not be a maximum, minimum, or zero.

Finding Intercepts The x and y intercepts are where the lines cross the x and y access respectively. To find the x-intercepts we use y = 0. (Zero’s) To find the y-intercepts we use x = 0. (Value x = 0)

Graphing Circles:

Circles:

Bell Work: Find the equation of the perpendicular bisector of the segment created by points A(10,12) and B(-2,18). What would be the equation of a circle that has a center of C(-4,2) and a diameter of 22 units?

Class Examples with Circles: Examples Pages #’s 26, 36, 40, 48, 52, 62

Homework: Page 155 #’s 35, 37, 39, 41, 45, 47, 49, 51, 55, 61

Bell Work:

Lines and Slope Lesson 3.3

LEQ How can the slope between two points be used to find parallel and perpendicular lines?

Slope

Different Forms for Linear Equations:

Remember: All vertical lines have the equation x = a, where a is the x- intercept. Vertical lines are parallel to the y-axis and perpendicular to the x-axis. All horizontal lines have the equation y = b, where b is the y- intercept. Horizontal lines are parallel to the x-axis and perpendicular to the y-axis.

Parallel and Perpendicular Lines Lines that are parallel have slopes that are the same! Lines that are perpendicular have slopes that are the opposite reciprocal!

Homework: Pages 170 – 173 #’s 9, 11, 19, 27, 29, 31, 33, 47, 49, 51, 53,55 (skip d), 59, 61 We will be having a quiz on Friday, come ready with questions.

Bell Work:

Quiz Tomorrow Small Quiz tomorrow on distance, midpoints, circles and lines. You will need to know: How to find the distance between two points or midpoint. How to find the equation of a circle (Center and Radius) How to find slope and equations of lines in standard form or slope-intercept form.

Bell Work: Come up with three examples of real world things that have a function or purpose. (Example: A microwave has a purpose to heat leftovers.)

Functions Lesson 3.4

What is a function? A function (f) is a correspondence that assigns each element (x) in the Domain to exactly one element (y) in the Range. Domain = all possible x-values for a function. Range = all possible y-values for a function. Range can be represented as y or f(x) which means that “f is a function of x.” Any letter can be used for a function! f(x), g(x), h(x), m(x)…

Example:

Homework: Pages #’s 3, 5, 9, 15

Bell Work:

Example: Looking at the graph on the board, answer the following : A) f(-2) B) f(3) C) Domain of f D) Range of f E) For what domain is f(x) < 1? F) On what intervals is f(x) decreasing?

The Difference Quotient: Sometimes it is necessary to find the slope of a secant line through two points on curve. That is where the difference quotient comes in handy… See Board…

Bell Work:

Example:

Asymptotes

Example:

Homework: Pages 188 and 189 #’s 7, 13, 17, 19, 21, 23, 27, 29, 31, 33, 35, 39, 41

Bell Work:

Graphs of Basic Functions Lesson 3.5

Lesson Essential Question (LEQ) How do we determine the change in basic functions when performing operations inside and outside of the functions?

Even or Odd? Even Function: A function is said to be even if f(-x) = f(x) for every value of x in the domain. If it is even, then we know the function is symmetrical with respect to the y-axis. Odd Function: A function is said to be odd if f(-x) = -f(x) for every value of x in the domain. If it is odd, then we know the function is symmetrical with respect to the origin. Examples on Page 204 #’s 2, 4, 8, 10

Basic Functions:

Changes in Graphs:

Homework: Pages #’s 1, 3, 5, 7, 9, 11, 12, 13

Bell Work:

Reflecting a Function

Vertically Compressing or Stretching

Horizontally Compressing or Stretching

Homework: Page #’s 16, 25, 39 (parts i and j), 40 (parts i, j, and k), 41 – 44

Bell Work:

Piecewise Functions! Sometimes it is necessary to use multiple expressions within the same function. These are called piecewise functions. When multiple expressions are used within a given function, each expression is only defined over specific intervals. It is important to note what intervals correspond to which expressions. Examples on board…

Examples of Piecewise Functions Page 207 #’s 46, 48, 50

Homework: Page 207 #’s 45, 47, 49 (Graph them by hand! Don’t Cheat!)

Bell Work: Sketch the graph of the following piece-wise function: x – 1, x < -4 f(x) = { |x|, -4 ≤ x ≤ 1 -x² + 6, x > 1

Quiz Tomorrow: We are going to have a small quiz tomorrow on functions, you need to know: Domain and Range Determining Intervals of Increasing and Decreasing Evaluating Functions Is the function even, odd, or neither? Graphs of Functions (shifting, reflecting) Piecewise Functions No Calculators will be allowed for this QUIZ! Sorry…

Review:

Bell Work:

Quadratic Functions Lesson 3.6

Terminology The minimum or maximum of a quadratic function is the vertex of the parabola. The zeros of a quadratic function are the points where the parabola crosses the x-axis. It is possible for a quadratic function to have only one zero, or no zeros. We can use the TI83 to find all of these!!!

To find the vertex:

Vertex Form of a Quadratic Function

Finding Distance:

Homework: Pages #’s odds, odds

Bell Work: Find the equation of the quadratic function in standard form for the parabola that would pass through the x-axis at -4 and have a vertex of (-1,9). Extra Sweeeeeeeet Question: Without looking at the graph, where would the other zero be for the function described above? How do you know?

Word Problems!!! Page 221 #’s 38, 40, 44, 46

Homework: Pages #’s 39, 43, 45, 47, 49

Bell Work:

Amber runs and jumps from point A to point B 15 feet away. If the path of her jump follows the path of a parabola, and she reached a maximum height of 5 feet, then find an equation in standard form that models it.

Operations of Functions Lesson 3.7

Basic Operations of Functions

Compositions:

Homework: Pages #’s 3, 7, 11, 15, 21, 23, 25, 29, 35

Bell Work:

Class Work: Pre-Calculus in class assignment: Pages #’s 2, 4, 8,12, 22, 28, 32 This assignment will be collected at the beginning of class tomorrow!!!

Bell Work:

Lesson 3.8 Inverse Functions

First we need to know: What is a one-to-one function? This is when each x-value in the domain corresponds to its own private y-value in the range. We can prove this by showing: If f(a) = f(b) in the Range R, then a = b in the domain D. For now, we are only looking at simple inverse functions, and they must be one-to-one functions to have an inverse.

Bell Work:

How do we find an inverse?

Homework: Page 243 #’s odds

Bell Work:

Classwork/Homework Page 243 #’s 15 – 18 and 20 – 34 evens We will be reviewing all of Unit 3 next Monday and Tuesday, and Having our Unit 3 Test on Wednesday!!!

Domain and Range:

The relationship between the graphs: Since the domain and range are switched for a function and its inverse, this creates a special relationship when looking at the graphs: The function and its inverse are reflected across the oblique line y = x. Anytime a function and its inverse will intersect, it will occur on the line y = x.

Homework: Page 243 #’s 20 – 38 evens

Unit 3 Test: MidPoint/Distance/Perpendicular Bisectors Minimums/Maximums/Zeros/Intersection Points Standard Form of a Circle Linear Functions/Slope/Parallel/Perpendicular Evaluating Functions Domain/Range Sketches of Functions (Vertical/Horizontal Shifts and Reflections) Piecewise Functions Quadratic Functions (Includes Word Problems) Operations and Compositions of Functions Inverses of Functions

Test Review: Pages #’s 3, 7, 11, 13, 16, 17, 22, 23, 51, 56, 57, 59, 61, 64, 65, 66, 67, 69, 70 Not included in this group of problems: How to shift basic functions horizontally and vertically. Inverse Functions