Algebra 1 Notes Quarter

Transcription

1 Algebra 1 Notes Quarter Name: ~ 1 ~

2 Table of Contents Unit 9 Exponent Rules Exponent Rules for Multiplication page 6 Negative and Zero Exponents page 10 Exponent Rules Involving Quotients page 12 Unit 10 Polynomials Naming Polynomials page 15 Adding and Subtracting Polynomials page 17 Multiplying and Dividing Polynomials page 19 Unit 11 Factoring Factoring Binomials page 25 Factoring Trinomials, Part 1 page 27 Factoring Trinomials, Part 2 page 29 Unit 12 Graphing Quadratics and Best Fit Lines/Curves Graphing Quadratic Functions page 32 Best Fit Lines and Curves page 37 Graphing Calculator Instructions page 42 ~ 2 ~

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5 Diamonds Fill in the blanks using the rules demonstrated in the first two diamonds ~ 5 ~ 30

6 Exponent Rules for Multiplication Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Property Definition Example Product of Powers Power of a Power Power of a Product Simplify the expression completely. 1. x² x³ x 5 2. y -3 y 7 y [(-3)³]² 4. (-2w) 2 5. (6x 4 y 2 ) 3 6. [(a + 1) 3 ] 4 Now, put them all together! 7. (4x²y)³ x 5 8. (3x 2 y 5 z 3 ) 2 5x 4 y 6 z ~ 6 ~

7 9. (-3xy²)³ (-2x²y)² 10. (2x 6 y 4 ) 2 + (3x 3 y 2 ) (x 5 ) 2 + (2x 3 ) 7 Scientific Notation: Applying the Rules of Exponents to Scientific Notation Write each number in scientific notation , Write each number in standard notation x x 10-5 Simplify the following problems by applying the rules of exponents. 5. (1.4 x 10 4 )(7.6 x 10 3 ) 6. (1.5 x 10-3 ) 2 ~ 7 ~

8 Try it! Simplify each expression. 1. 2x 3 y 4-3x 5 y 8 2. (4x 6 y 10 ) x 11 y -4 (2x 3 y 7 ) 4 4. (3x 5 y 8 ) 2 (-4x 3 y 9 ) 2 5. (5.6 x 10-2 )(4 x 10 4 ) 6. (3.4 x 10 4 ) 3 ~ 8 ~

9 Warm - Up Fill in the following table. Power Decimal Whole # or Power Fraction Decimal Whole # or Fraction What patterns do you notice? ~ 9 ~

10 Negative and Zero Exponents Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Rule Definition Example Zero Exponents Negative Exponents Simplify the expression completely x -2 y (5x) (4n -2 ) _ 6. 45x 3 y -8 m -3 z -5 ~ 10 ~

11 7. 8x -2 y _ 4x -10 y ( 4 ) xy (xy 2 ) -3 x -2 y -5 Try it! 1. (x 2 y 4 z 5 ) y 5 x a a -4 b 2 b -2 3c 5 d -3 ~ 11 ~

12 Exponent Rules Involving Quotients Learning Target: SOL A.2a: The student will apply the laws of exponents to perform operations on expressions. Property Definition Example Quotient of Powers Power of a Quotient Simplify the expression completely. 1. x _ y 3 x 3 y x 5 y 7 z x 6 yz 8 Putting it all together! ~ 12 ~

13 x a -3 b 5 c x a 5 b 5 c x -3 y 2 (2xy 3 ) (-2x 4 y -5 z 6 ) 2 (3x -3 y -6 z 7 ) -2 x 5 y -1 xy Try it! ( ) ( ) m 4 m r 3 s 4 t -9-12r -12 st x 10 9 m 7 3r 6 s 2 t 2r -8 s -5 t x 10 4 ~ 13 ~

14 Unit 9 Scratch Paper ~ 14 ~

15 What is a Polynomial? Vocabulary Definition Example(s) Monomial A number, a variable, or a product of the two. Polynomial A monomial or the sum or difference of many monomials. Degree of a Monomial The sum of the exponents of the variables in the monomial. Term In a polynomial, each individual monomial is called a term. Degree of a Polynomial The greatest degree of the polynomial s terms. Standard Form of a Polynomial A polynomial is written in standard form when the terms are listed in descending order by degree (highest degree to lowest degree). ~ 15 ~

16 Classifying Polynomials by Degree: Degree Example Name or more Classifying Polynomials by Terms: # of Terms Example Name 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 ~

17 Adding and Subtracting Polynomials Learning Target: SOL A.2b: The student will add, subtract, multiply and divide polynomials. You can add and subtract polynomials using two different formats: vertical and horizontal. When you add or subtract, you are really combining like terms so you DO NOT CHANGE THE EXPONENT! 1. (4x 2 + 6x 9) + (5x 2 7x + 2) 2. (9x 3 4x 2 + 2x 6) + (4x 3 3x + 1) 3. (2x 2 + x 5) + (6x 2 3x + 7) 4. (6x 3 + 2x 2 5x + 9) (10x 3 5x 2 + 3x 8) 5. (7x 2 5x) (2x 2 + 6x 1) 6. (-4x 3 + 9x 2 8) (5x 2 + 7x 3) ~ 17 ~

18 Try it! 1. (9w 4) + (w + 5) 2. (6x 2 3x + 1) + (x 2 + x 1) 3. (5x 3 6x 2 + 4x 1) (2x 3 + 8x 2 9x 5) 4. (8b 2 + 6) (3b 2 + 6b + 1) 5. (x 2 + xy + 2y 2 ) + (6x 2 y 2 ) 6. (4x 2 + y 2 ) (-3xy + y 2 ) ~ 18 ~

19 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 exponents! A monomial times a polynomial 1. 3x²(4x² 6x + 7) 2. -5xy(8x³y² + 3x²y 10x) 3. 4x(8x² + 9x 1) + 3x(7x² 6x + 2) 4. 8x(5x² + 3x 3) 3x(9x² + x 2) A binomial times a binomial Using a Table Using FOIL (Horizontal Method) 5. (5x 2)(6x + 1) 6. (x y)(x 5y) F O I L ~ 19 ~

20 A binomial times a trinomial Using a Table Horizontal Method 7. (3x + 2)(4x² 2x 7) 8. (x 4)(x² + 6x + 3) Dividing a polynomial by a monomial Remember, when you DIVIDE powers, you SUBTRACT the exponents! 9. 18x 3 y x 2 y 5 36xy x x 3 8x 2-3xy 2 8x 2 Application 11. 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 Formulas to Know 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 ~

21 Special Cases for Multiplying Binomials Directions: Use a table or the horizontal method to multiply the following binomials. (x + 4)(x + 4) (y 2) 2 (3z + 13)(3z + 13) What patterns do you notice? Directions: Use a table or the horizontal method to multiply the following binomials. (x + 9)(x - 9) (x 15)(x + 15) (6x 3 + 7)(6x 3-7) What patterns do you notice? The Special Cases The Square of a Binomial (Difference of Squares) The Sum and Difference Patter Directions: Find each product using mental math. (2x + 11)(2x + 11) (9z + 1)(9z - 1) (4y x 2 )(4y x 2 ) (6z - 14)(6z + 14) (a + 5b) 2 (3x 5-4y) 2 ~ 21 ~

22 You Try It! DIRECTIONS: Simplify the following expressions using any method. 1. 3x(5x² - 9x + 2) 2. 3a(4a² + 7a 8) 5a(7a² - 3a + 4) 3. (7b 4)(2b + 8) 4. (3x 6)(x 2) 5. (4x + 5)² 6. (x 8)(x + 8) 7. (8x - 3)(4x 3 2x 2 + 6x 7) y y 5 24y 2 6y 2 ~ 22 ~

23 Unit 10 Scratch Paper ~ 23 ~

24 Perfect Squares Warm Up Fill in the chart below. Simplify. Number (n) Number Squared (n 2 ) 1. (5x 12)(5x + 12) 2. (3x + 14y)(3x 14y) 3. (11x 2 6y)(11x 2 + 6y) 4. (15x + 7)(15x 7) 5. (8xy 1)(8xy + 1) What is a Perfect Square? ~ 24 ~

25 Factoring Binomials Learning Target: SOL A.2c: The student will factor completely firstand second-degree binomials and trinomials in one or two variables. 1. Steps for Factoring Binomials 2. a. b. c Factor out the Greatest Common Factor (GCF) 1. 3x + 6y 2. ab + ac 3. 12x 3 y 4 6x 2 y 7 Factor the Difference of Two Squares 4. 4x x 2 y x 4 16y 4 ~ 25 ~

26 Factor out the GCF and THEN the difference of two squares 7. 18x x 6 3y 2 Prime 9. x y Try it: Factor Completely! x² x² 144y² x² + 20y² 4. x² y² 5. 5x 2 y 3 15x 3 y 4 6. y² x² x 4 y x² x² x² x² x + 42y 14. 4x x x 2 + 7x ~ 26 ~

27 Factoring Trinomials, Part 1 Learning Target: SOL A.2c: The student will factor completely firstand second-degree binomials and trinomials in one or two variables. ALWAYS LOOK FOR A GCF 1 st! Do you remember the diamonds activity? Factor completely. 1. x² + 7x x² 14x x² 6x x³ 14x² 20x Simplify completely. 5. Find the quotient of x 2 + 9x 36 and x 3. 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. ~ 27 ~

28 Try it! 1. x² 10x x² + x x² + 5x x² + 80x x² 21x x² 12x x² + 6x 5 8. x 3 2x 2 48x Simplify completely. 9. x² x x² 9x 52 (x 8) (x + 4) ~ 28 ~

29 Factoring Trinomials, Part 2 Learning Target: SOL A.2c: The student will factor completely firstand second-degree binomials and trinomials in one or two variables. ALWAYS look for a GCF first! 1. 2x x x 2 x x 2 24x x 3 30x 2 + 9x Simplify completely x x x x 15 (3x + 5) (4x - 1) 7. The area of a rectangle is 6x x 36. Find the dimensions of the rectangle. Then, find the actual dimensions if x = 2 inches. ~ 29 ~

30 Try it! 1. 4x 2 + 8x x 3 13x 2 10x 3. 4x x x 2 25x x x x 2 13x 12 Simplify completely. 7. 6x x x 2-5x - 10 (x + 5) (3x - 2) 9. The area of a rectangle is 15x x 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 10x x 2 30x. If the height of the prism is x units, write an expression for the length and width of the prism. ~ 30 ~

31 Unit 11 Scratch Paper ~ 31 ~

32 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 PARABOLA. Standard Form: Finding the Vertex: Axis of Symmetry Equation: 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 is negative, the parabola will open and the vertex is the point. Ex. If = 1, the graph has a width. Ex. If > 1, the graph is. Ex. If 0 < < 1, the graph is. Ex. ~ 32 ~

33 Graph each quadratic equation. Answer the questions provided. 1. Does the parabola open up or down? Is the vertex the minimum point or the maximum point? a = b = c = Vertex: Axis of Symmetry: Table of Values: Roots: 2. Does the parabola open up or down? Is the vertex the minimum point or the maximum point? Compared to the parent function (y = x 2 ), is it skinnier, wider or the same width? a = b = c = Vertex: Axis of Symmetry: Table of Values: Roots: ~ 33 ~

34 Use your calculator to graph the quadratic. (Calculator instructions are on page 42). 3. Is the vertex the minimum point or the maximum point? Vertex: Axis of Symmetry: Table of Values: Roots: Try it! Match each equation on the left to its correct graph on the right A B C D ~ 34 ~

35 Try it, Continued Directions: Graph each quadratic. Answer the questions provided. 5. a = b = c = a. Graph opens: up or down b. Vertex is a: min or max c. Coordinates of vertex: (, ) d. Axis of symmetry: e. Table of values: x y f. Roots: Directions: Use the calculator to graph each quadratic. 6. a = b = c = a. Graph opens: up or down b. Vertex is a: min or max c. Coordinates of vertex: (, ) d. Axis of symmetry: e. Table of values: x y f. Roots: ~ 35 ~

36 Scatter Plot Warm Up Plot the data in each table as a scatter plot on the coordinate grid provided. 1. Gold s Gym followed 3 members for a week. Each person recorded the calories they burned during their treadmill workout. The results are in the table below. (Hint: Count by 5 s on the x-axis and by 20 s on the y-axis!) Minutes on Treadmill Calories Burned Calories Burned Minutes on Treadmill 2. A football was kicked from an initial height of 2.5 feet. The height was recorded at different intervals and recorded in the chart below. (Hint: Count by 0.3 on the x-axis and by 5 s on the y-axis!) Seconds Distance in the air from the ground (ft) Distance from the Ground (ft) ~ 36 ~ Seconds in the Air

37 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 increases (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 Calculator Steps 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 below. Write the equation of the line of best fit. Then, determine the approximate value of f(25). x y ~ 37 ~

38 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 Calculator Steps 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. ~ 38 ~

39 Practice: 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 liner 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. Practice: 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. ~ 39 ~

40 Try It! 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? Write an equation for the best fit line/curve. b. 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? Write an equation for the best fit line/curve. b. Use your equation to predict the number of students who attended Math Club during the 10 th week of school. ~ 40 ~

41 1. Consider the equation 2 8. a. Find the vertex: b. Identify the axis of symmetry: Review Day Warm Up c. Create a table of values and graph the equation. x y_ d. What are the roots of this equation? 2. Identify the following using the graph shown. Coordinate of the Vertex: Axis of Symmetry: Roots: 3. 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. Then find f(8). x y ~ 41 ~

42 Graph an Equation Graphing Calculator Instructions Always begin by clearing the memory on your calculator! 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. Find the Roots Using a Graph s 5: Delete All Select Yes 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. ~ 42 ~

43 Unit 12 Scratch Paper ~ 43 ~