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Graphing Equations in Rectangular Coordinate System

Explore how to graph equations, identify intercepts, and interpret information from graphs in the rectangular coordinate system. Practice solving linear equations and rational equations. Learn to model situations using linear equations.

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Graphing Equations in Rectangular Coordinate System

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  1. Chapter 1 Equations and Inequalities

  2. Aim #1.1: How do we graph and interpret information? • Analytic Geometry this new branch of geometry founded by Rene Descartes brings Algebra and Geometry together. • Key Terms • Rectangular Coordinate System or Cartesian Coordinate System • X-axis, • Y-axis • Quadrants

  3. Graphs of Equations Solution of an equation that satisfies the equation.

  4. Using a Graphing Calculator • Using your calculator- • Y = 4 – x2 • Understanding the Viewing Rectangle • [-10, 10, 1] Minimum X-value, Maximum X- value and X scale

  5. Intercepts • X-intercept- is the point on the graph where it intersects with the X- axis. Ex. (X, 0) • Y-intercepts – is the point on the graph where it intersects with the Y-axis. Ex, (0, Y) • Look at Text- Identifying Intercepts

  6. Interpreting Information from Graphs • Line Graph used to illustrate trends or data over time. • Look at Text

  7. Summary: Answer in complete sentences. • What is the rectangular coordinate system? • Explain why (5, -2) and (-2, 5) do not represent the same point. • What does [-20, 2, 1] by [ -4, 5, 0.5] viewing rectangle mean? • Determine whether the following is true or false. Explain. • If the product of a point’s coordinates is positive, the point must be in quadrant I.

  8. Warm-up: 8/23/2011 • Take out your homework and unit plan.- • Copy the questions and Answer. • Explain how to graph an equation in the rectangular coordinate system. • Explain how to graph the point (7, -8) on the rectangular coordinate system. • Sketch a function that models the following situation. • As the blizzard got worse, the snow fell harder and harder. (Label the x-axis Time and Y-axis snowfall)

  9. Aim #1.2: How do we solve equations? • What is a linear equation? • A linear equation has one variable and can be written in the form of y= mx +b, where a and b are real numbers and a ≠ 0. • How do we solve? 4x + 12= 0

  10. Practice: • 4x + 5 = 29 • 6x – 3 = 63

  11. How do we solve other types linear equations? • Solve 2 (x – 3) – 17 = 13 • 4( 2x + 1) = 29 + 3 (2x – 5)

  12. Practice: • 5x – (2x – 10 ) = 35 • 3x + 5 = 2x + 13 • 13 x + 14 = 12 x – 5

  13. How do we solve an equation w/ a fraction? • Steps: • Multiply the entire equation by a multiple of the denominator. For this example, it would be … • This would get rid of all denominators. • Then combine like terms and isolate the variable.

  14. Guided Practice:

  15. It is an equation containing one or more rational expressions. Solving: Hint: Multiply the entire equation by the LCD or multiple of x, 5, and 2x. What is a Rational Equation?

  16. Solving a Rational Equation

  17. Solving a Rational Equation

  18. Categorizing the Different Equations Key Terms • Identity Equation is an equation that is true for all values of x. Ex.: x + 3= x + 2 + 1 • Conditional Equationis true for a particular value of x. Ex. 2x + 5 = 10 x = 2.5 • Inconsistent Equation is an equation that is not true for any value of x. Ex. x = x + 7

  19. Summary: Answer in complete sentences. • What is a linear equation in one variable? Give an example. What are some other types of linear equations? 2. Does the following make sense? Explain your reasoning. • Although I can solve 3x + 1/5 = ¼ by first subtracting 1/5 on both sides, I find it easier to multiply by 20, the least common denominator, on both sides. 3. Is the equation (2x- 3)2 = 25 equivalent to 2x – 3 = 5? Explain.

  20. Warm-up: 8/25/2011 • Take out your homework and unit plan.- • Copy the questions and Answer. • Solve for X. • 7x – 5 = 72 • 6x – 3 = 60 • 13x + 14 = 12 x – 5

  21. Warm-up: 8/25/2011 • Take out your homework and unit plan. • Copy the questions and Answer. • Solve for X.

  22. Aim #1.3:How do we use linear equations to model situations? • Steps to solving Word Problems 1. Read the problem. Twice. 2. Define x = 3. Write your equation. 4. Solve. 5. Check your solution. Does it make sense?

  23. Using the 5-step strategy and find the number. • When two times a number is decreased by 3, the result is 11. What is the number? • Let x= number • Equation: 2x – 3 = 11 Solve for x.

  24. When a number is decreased by 30% of itself, the result is 20. What is the number? • Let x = the number • Equation x • __ = ___ • Solve and Check your answer.

  25. 2L + 2W = P Solve for l Your goal is to isolate L. Subtract 2W on both sides. Then divide by 2 on both sides to isolate L. What’s your final answer? Note: 2W is like one term because you are multiplying W by 2. Solving a Formula for a Variable

  26. Check for Understanding • Solve the formula for W. • 2L + 2W = P

  27. Solving a Formula for a Variable that Occurs Twice • Solve for P. • A = P + P r t • Factor out the P from P + P rt. • Then divide by the expression in parenthesis to isolate P.

  28. Check for Understanding • Solve the formula for C. • P = C + MC

  29. Summary: Answer in complete sentences. • Explain how to solve for P below and then solve. • T= D + pm • What does it mean to solve for a formula? • Write an original word problem that can solved using a linear equation. Then write out all the steps for the solution.

  30. Take out your homework and Unit Plan. What is the difference between an identity, conditional and inconsistent equation? Solve for r. Agenda Go over hw. Quiz Review Warm-up: 8/26/2011

  31. Take out your homework and Unit Plan. Solve. Reminder- Quiz Friday- P.6B through- 1.5 Note: We are skipping 1.4 at this time. Warm-up: 8-29-2011

  32. Warm-up: 8/30/2011 • Take out your homework and Unit Plan. • Solve for x. • Reminder Quiz Friday P.6B -1.5

  33. Warm-up: 8-31 or 9-1 • Take out your homework and unit plan. • First, write the value (s) that make the denominator (s) zero. Then solve the equation. • What type of equation is this?

  34. Aim #1.5: How do we solve quadratic equations? • Definition of a Quadratic Function: • A quadratic equation is an equation that can be written in the general form of ax2 + bx + c = 0, where a, b, and c are real numbers and a≠0. i.e. Quadratics are second degree polynomial functions.

  35. Notice in the examples to the left that there is no b term. Steps: 1. Isolate the x2 term. 2. The find the square root of both sides. 3. Simplify. Solving Equations by the Square Root Property

  36. Check for Understanding: • Solve by the Square Root Property.

  37. Zero-Product Principle • If the product of two algebraic expressions is zero, then at least one of the factors must equal to zero. • If AB= 0 then A= 0 or B= 0 • Example: x2 + 7x + 10 = 0 • Factor. (X + 5)(x+2) = 0 • Set each factor = 0 X+ 5 = 0 or x+2 = 0 • Solve. x = -5 or x = -2

  38. Is there a GCF, that can factored out? Subtract 4 on both sides and set equation = 0. Now factor. Solving Quadratic Equations by Factoring

  39. Solving Quadratics by Completing the Square • Completing the Square is a strategy to solve quadratics when: • The Trinomial can not be factored • Zero Product property can not be used Completing the square allows us to convert the equation so that it can be solved using the square root property.

  40. Completing the Square: • If x + bx is a binomial, then by adding , which is the square of half the coefficient of x,a perfect square trinomial will result. That is,

  41. Completing the Square • What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial. • x2+ 8 x • Solution:

  42. Completing the Square • What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

  43. Completing the Square • What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.

  44. Steps: 1. Subtract 4 on both sides. 2. Take the b term and divide by 2 and square it. 3. Now add it to both sides of the equation. 4. Now you can express the left hand side as a square. 5. Apply the square root property and solve for x. Solving Quadratics using Completing the Square

  45. Solving Quadratics using Completing the Square • Guided Practice:

  46. Steps: Divide the entire equation by 9, so that a = 1. Add -4/9 to both sides. Complete the square. Then solve for x. Solving Quadratics using Completing the Square

  47. Solving Quadratics using Completing the Square • Guided Practice:

  48. What is the Quadratic Formula? Quadratic formula: If ax2 + bx + c = 0 and a≠0

  49. Solve x2 + 6 = 5x Steps: Subtract 5x on both sides, so the equation = 0. Identify the values for a, b, and c. a= 1, b = -5, c = 6 3. Then substitute into the Using the Quadratic Formula

  50. What is the discriminant? Property of the Discriminant For the equation ax2 + bx + c= 0, where a ǂ 0, you can use the value of the discriminant to determine that number of solutions. If b2 – 4ac > 0, there are two solutions. If b2 – 4ac = 0, there is one solution. If b2 – 4ac <0, there are no solutions

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