ALGEBRA 1 NOTES. Quarter 3. Name: Block

Transcription

1 ALGEBRA 1 NOTES Quarter 3 Name: Block

2 Table of Contents Unit 8 Exponent Rules Exponent Rules for Multiplication page 4 Negative and Zero Exponents page 8 Exponent Rules Involving Quotients page 10 Unit 9 Polynomials An Introduction to Polynomials page 16 Adding and Subtracting Polynomials page 17 Multiplying and Dividing Polynomials page 19 Special Cases for Multiplying Binomials page 22 Unit 10 Factoring Factoring Binomials page 27 Diamonds Activity page 30 Factoring Trinomials (leading coefficient = 1) page 31 Factoring Trinomials (leading coefficient > 1) page 34 Unit 11 Graphing Quadratics and Best Fit Lines/Curves Graphing Quadratic Functions page 41 Best Fit Lines and Curves page 46 Graphing Calculator Instructions page 51 1

3 Warm-Ups 2

4 Warm-Ups 3

5 Exponent Rules for Multiplication Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Product of Powers Property Power of a Power Property Power of a Product Property Definition To multiply powers having the same base, add the exponent. To find a power of a power, multiply exponents. To find a power of a product, find the power of each factor and multiply. Algebra Example Simplify the expression completely

6 7. 8. Now, let s put it all together!

7 Try it! Simplify each expression Find the area of the triangle. Hint: 6. The side length of a cube can be represented by the expression. If the side length is doubled, write an expression to represent the new volume of the cube. 6

8 Applying the Rules of Exponents to Scientific Notation Standard Form: 5,200,000,000,000 Scientific Notation: Numbers written in scientific notation are made up of three parts: Write each number in scientific notation Write each number in standard notation Directions: Simplify the following problems by applying the rules of exponents Try it!

9 Negative and Zero Exponents Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Zero Exponents Negative Exponents Definition A number to the zero power is always 1. is the reciprocal of. Algebra Example Simplify the expression completely

10 Try it! Simplify completely

11 Exponent Rules Involving Quotients Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Quotient of Powers Property Power of a Quotient Property Definition Let be a nonzero real number, and let and be positive integers such that Let and be real numbers with, and let be a positive integer. Algebra Example Simplify the expression completely

12 Putting all the rules together!

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14 Law of Exponents Summary Answers are in simplest form when 1. Coefficients have been simplified 2. All like terms (same variable) have been combined 3. All exponents are positive 4. No zero exponents Is the following example in simplest form? Why or why not? 13

15 Unit 8 Review 1. Which two expressions have a product of? 2. Which expression is equivalent to 3. Which expression is NOT equivalent to the following expression? A. B. C. D. A. B. C. D. 4. The earth is miles from the sun and Jupiter is miles from the sun. Expressed in scientific notation, what is the ratio of distance from Earth to the sun compared to the distance from Jupiter to the sun? A. B. C. D. 5. Completely simplify the following: 14

16 Unit 8 Scratch Paper 15

17 An Introduction to Polynomials What is a polynomial? A polynomial is an expression consisting of variables and coefficients, otherwise known as terms. Example of a polynomial: Look at the polynomial above and answer the following questions: a. How many terms are in the polynomial? b. What is the value of the leading coefficient? c. What is the highest degree of the polynomial? Classifying Polynomials by Degree: Classifying Polynomials by Terms: Degree Name # of Terms or more Name 5 or more Try it! Classify each polynomial by degree AND term. 1. 6x 3 9x 2. 5x 2 19x x x x x 5 2x 4 + x 3 16

18 Adding and Subtracting Polynomials Learning Target: SOL A.2b: The student will add, subtract, multiply and divide polynomials. When you add or subtract, you are really combining like terms so you DO NOT CHANGE THE EXPONENT! ADDING POLYNOMIALS To add polynomials, simply add like terms! EX: SUBTRACTING POLYNOMIALS To subtract polynomials, remember to distribute the negative, or use: KEEP CHANGE CHANGE EX: Find the sum or difference

19 Try it! Find the sum or difference Find the perimeter of each shape

20 Multiplying and Dividing Polynomials Learning Target: SOL A.2b: The student will add, subtract, multiply and divide polynomials. It s time to use the Distributive Property! Remember, when you MULTIPLY with variables, you ADD the exponents! MONOMIAL times a TRINOMIAL BINOMIAL times a BINOMIAL Using a table FOIL (distributing each term) F O I L 19

21 BINOMIAL times a TRINOMIAL Using a table Distributing each term DIVIDING a polynomial by a monomial Remember, when you DIVIDE powers, you SUBTRACT the exponents! Tara is putting a pool in her backyard. The length of the pool is four more than twice the width. The depth of the pool is half of the width. Write an expression in simplest form that represents the a. Perimeter of the pool Need to know formulas: b. Area of the pool c. Volume of the pool d. If the width is 8 ft, how much water can fit in the pool? 20

22 DIRECTIONS: Simplify the following expressions

23 Special Cases for Multiplying Binomials Directions: Use a table or the horizontal method to multiply the following binomials. What patterns do you notice? Directions: Use a table or the horizontal method to multiply the following binomials. What patterns do you notice? The Special Cases -- The Square of a Binomial Patterns The Sum Pattern Example: The Difference Pattern Example: Difference of Squares Example: 22

24 Directions: Find each product using the patterns for special cases Maggy bought a large storage box to store all of her winter clothes. The length of the box is three more than twice the width. The depth of the box is three times the width. Write an expression in simplest form that represents the a. Perimeter of the box s lid: b. Area of the box s lid: c. Volume of the box: d. If the width is 2 ft, how much space is available inside the box for all of her clothes? 23

25 1. Which of the following expressions is equal to A. B. C. D. Unit 9 Review 2. The difference of two binomials is If one of the binomials is which could possibly be the other binomial? A. B. C. D. 3. The figure below is made up of a square and a rectangle. What is the perimeter of the figure? 4. Which polynomial is equivalent to Assume the division is not equal to zero. A. B. C. D. 5. You are designing a marble planter for a city park. You want the length of the planter to be six times the height and the width to be three times the height. The sides should be one foot thick. Which function expresses the volume of the planter in terms of its height? A. B. C. D. 24

26 Unit 9 Scratch Paper 25

27 What is a perfect square? Perfect Squares Warm Up Fill in the chart below. Simplify. Number (n) Number Squared (n 2 )

28 Factoring Binomials Learning Target: SOL A.2c: The student will factor completely first- and second-degree binomials and trinomials in one or two variables. Steps for factoring a binomial 1. ALWAYS look for a first! 2. Look for a of. It must contain all the following: a. Perfect square coefficients. b. Subtraction. c. Even exponents. 3. Look for a difference of squares. 4. If none of the above are true, the binomial is. A GCF can be in the form of a number, a variable, or both! Factor out the Greatest Common Factor (GCF) Factor the Difference of Two Squares

29 Factor out the GCF and THEN the difference of two squares Special Case Factor Completely!

30 Try-its continued

31 Diamonds Warm-Up Fill in the blanks using the rules demonstrated in the first two diamonds

32 Factoring Trinomials (When the leading coefficient is 1) Learning Target: SOL A.2c: The student will factor completely first- and seconddegree binomials and trinomials in one or two variables. Remember: ALWAYS LOOK FOR A GCF FIRST! How can the diamond activity help us factor trinomials when the leading coefficient is 1? Helpful Hints If the last term is negative, one of the factors will have a subtraction sign and one of the factors will have an addition sign. Look at the middle term to determine which one is which. If the last term is positive, look at the middle term. If the middle term is positive, both factors will be addition. If the middle term is negative, both factors will be subtraction. Factor completely

33 5. Find and correct the mistake. Discuss with the person next to you! Factoring to Find Quotients. Directions: Simplify completely Find the quotient of and Factor completely

34 Simplify completely is the factored form of which polynomial? A. B. C. D. 33

35 Factoring Trinomials (When the leading coefficient is greater than 1) Learning Target: SOL A.2c: The student will factor completely first- and seconddegree binomials and trinomials in one or two variables. METHOD TO USE: Factor by Grouping Given a polynomial in the form ax 2 + bx + c, use factor by grouping when the leading coefficient a is not 1! Use the following table as a reference: Steps 1. Multiply a and c and place the product in your diamond. Example: 2. Place b in your diamond. 3. Complete the diamond (Find two factors of a c that add to get b). 4. Rewrite the original problem, replacing b with the sum of the new factors. 5. Group the terms to form pairs the first two terms and the last two terms. 6. Factor each pair by finding a GCF. 7. Factor out the common binomial. 8. Remember, you can check your answer! 34

36 Factor completely. ALWAYS check for a GCF first

37 Simplify completely The area of a rectangle is Find the dimensions of the rectangle. Then, find the actual dimensions if inches. 36

38 Factor completely

39 Simplify completely The area of a rectangle is. a. Write two expressions that represent the dimensions of the rectangle. b. Find the actual dimensions if x = 3 centimeters. 10. The volume of a rectangular prism is. If the height of the prism is x units, write an expression for the length and width of the prism. 38

40 Unit 10 Review 1. When factored completely, identify the factors of the polynomial 2. Which polynomial can NOT be factored over a set of real numbers? A. B. C. D. 4. What is the quotient of and Assume the denominator does not equal zero. A. 3. When factored completely, equals -- A. B. C. D. The polynomial cannot be factored. 5. If the dimensions of a rectangular soccer field can be represented as, find two expressions that represent the length and the width of the field. B. C. D. Length Width 39

41 Unit 10 Scratch Paper 40

42 Graphing Quadratic Functions Learning Target: SOL A.4c The student will solve quadratic equations in two variables including solving graphically. The graph of a quadratic function is a. Standard Form: Finding the Vertex The vertex is the highest or lowest point on a parabola. x-coordinate: y-coordinate: substitute x back into the equation and solve for y. Equation of Axis of Symmetry: The line that passes through the vertex and divides the parabola into two symmetric parts. Equation: Directions: Identify the vertex, axis of symmetry and roots of each parabola Vertex: Vertex: Equation of Axis of Symmetry: Equation of Axis of Symmetry: Roots: Roots: 41

43 Find the vertex and axis of symmetry for each quadratic function Trends in Quadratic Graphs Reflect (Flip) Graph Width of Graph If is positive, the parabola will open and the vertex is the point. Ex. If Ex. If Ex. = 1, the graph has a width. > 1, the graph is. If is negative, the parabola will open and the vertex is the point. Ex. If 0 < Ex. < 1, the graph is. 42

44 Try it! Match each equation on the left to its correct graph on the right. 1. A B C D Directions: Graph each quadratic equation. Answer the questions provided. Does the parabola open up or down? Is the vertex a minimum point or a maximum point? Vertex a= b= c= Table of Values x y Equation of Axis of Symmetry Roots 43

45 Does the parabola open up or down? Is the vertex a minimum point or a maximum point? Compared to the parent function (y=x 2 ), is it skinnier, wider or the same width? Vertex a= b= c= Table of Values x y Equation of Axis of Symmetry Roots Use your calculator to graph the quadratic. (Calculator instructions are on page 51). Does the parabola open up or down? Is the vertex a minimum point or a maximum point? Compared to the parent function (y=x 2 ), is it skinnier, wider or the same width? Vertex a= b= c= Table of Values x y Equation of Axis of Symmetry Roots 44

46 Does the parabola open up or down? Is the vertex a minimum point or a maximum point? Compared to the parent function (y=x 2 ), is it skinnier, wider or the same width? Vertex a= b= c= Table of Values x y Equation of Axis of Symmetry Roots Directions: Use the calculator to graph the following quadratic equation. Does the parabola open up or down? Is the vertex a minimum point or a maximum point? Vertex a= b= c= Table of Values x y Equation of Axis of Symmetry Roots 45

47 Best Fit Lines & Best Fit Curves Learning Target: SOL A.11: The student will determine the equation of the curve of best fit in order to make predictions. What is Linear Regression? A linear regression line is used to represent data that continually increase (positive correlation) or continually decreases (negative correlation) at a fairly constant rate. Given a Graph (Scatter Plot) 1. Draw a best fit line for the scatter plot. 2. Pick 2 points on the line and find slope. 3. Estimate the y-intercept. 4. Then write the equation of the line in slope-intercept form. Write the equation of the best fit line How to use the calculator to find the best fit line 1. Using the calculator, open a new spreadsheet page. Name each list and enter the x values and the y values. 2. Return to a calculator page and press Menu, 6, 1, Select the appropriate x list and y list from the drop down menu. 4. Write the equation of the line in slope-intercept (y = mx + b) form. 3. Consider the data in the table on the right. a. Write the equation of the line of best fit. x y b. Determine the approximate value of 46

48 What is Quadratic Regression? Quadratic regression is used to represent data that will increase, reach a peak, and then decrease or vice versa. Example 1 Larry made a scatter plot showing the apparent height of a football at one-second intervals during the time period the ball was in the air. Which is most likely the equation for the curve of best fit for the relationship? A. y = - 0.4x B. y = 9.0x C. y = 5.3x 2 0.9x D. y = - 0.9x x How to use the calculator to find the best fit curve 1. Using the calculator, return to the spreadsheet page. Name each list and enter the values found in the table. 2. Return to a calculator page and press Menu, 6, 1, 6 (Quadratic Regression). 3. Select the appropriate x list and y list from the drop down menu. 4. Write the equation of the parabola in standard quadratic form (y = ax 2 + bx + c.) Example 2 a. Look at the table. What do you notice about the y-values? Does this make the equation linear or quadratic? x y b. Use the graphing calculator to calculate a curve of best fit. Round to the nearest 100 th if necessary. 47

49 Example 3: The table below shows the cost, y, for a specified number of phone lines. Number of lines, x Total cost, y $10.40 $12.25 $14.10 $15.95 $17.80 $19.65 a. How do you know this data can be represented using a linear regression line? b. Use your graphing calculator to find a line of best fit. c. Determine the approximate total cost for 15 lines. d. Determine the number of lines if the cost is $6.70. Example 4: Consider the table below. Depth (in) Weight (lbs) a. How do you know this data can be represented using a quadratic regression line? b. Use your graphing calculator to find the curve of best fit. Round to the nearest 100 th if necessary. c. Determine the approximate weight at a depth of 8 inches. 48

50 Determine if the given data represents a linear function or a quadratic function. Then use your graphing calculator to find the line or curve of best fit. Round to the nearest 100 th if necessary. 1. x y Time (sec) Height (ft) The table below shows the body temperature B (in degrees Celsius) of a desert spiny lizard at various air temperatures A (in degrees Celsius). Air temperature, A ( C) Body temperature, B ( C) a. Is this data best modeled by a linear or a quadratic function? b. Write an equation for the best fit line/curve. c. Use your equation to predict the body temperature of the spiny lizard when the air temperature is 35 C. 4. The table below shows the number of students who attended Mrs. Knipper s math club in the first seven weeks of school. Week Number of Students a. Is this data best modeled by a linear or a quadratic function? b. Write an equation for the best fit line/curve. c. Use your equation to predict the number of students who attended Math Club during the 10 th week of school. 49

51 Unit 11 Review 1. A delivery service company maintains several vehicles. The table summarizes the cost for auto insurance related to the number of vehicles insured. Using the equation of a line of best fit for the data, which is the closest estimate of the total cost of insuring eight vehicles? A. $5,050 B. $5,200 C. $5,500 D. $5, Given the set of data, estimate what the value of y will be when x = 14. Round your answer the nearest tenth. 3. Select each set of data that can be represented by the linear curve of best fit A. B. C. 4. Match each equation of the curve of best fit to the correct set of data. Write the letter of the equation beneath the table. A. B. C. 50

52 Graphing Calculator Instructions Always begin by clearing the memory on your calculator! Graphing an Equation Steps: 1. Add a graphing page. This will open up a blank graph. At the bottom of the screen you will see f1(x)=. This is where you will type in your equation. To create a second graph: 1. Press the tab key. The bottom of the screen should now read f2(x)= Note: Lines must be in slope-intercept form before using the calculator to graph! Finding the Vertex Steps 1. Graph the equation. Decide if the vertex is the minimum or maximum point. 2. Press Menu 3. Select 6: Analyze Graph 4. Depending on the graph, select 2: minimum or 3: maximum 5. A pointing hand will appear. Use the cursor or the arrow keys to move the hand. You will need to select a lower bound by pressing enter. Then, select an upper bound by pressing enter. Displaying a Table of Values Steps 1. Graph the equation. 2. Press Ctrl T to display the table. Finding the Roots Using a Graph Steps 1. Open a graphing page and enter the equation. 2. Press Menu 3. Select 6: Analyze Graph 4. Select 1: Zero 5. A pointing hand will appear. Use the cursor or the arrow keys to move the hand. You will need to select a lower bound by pressing enter. Then, select an upper bound by pressing enter. To delete ALL of your graphs, press: Menu 1: Actions 5: Delete All Select Yes 51

53 Unit 11 Scratch Paper 52