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March 18
 

March 18

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    March 18 March 18 Presentation Transcript

    • March18th Today: Warm-Up Test Review Khan Academy Results/Schedule Begin Unit on Quadratic Equations
    • Khan Academy: Saturday/Sunday -- 1409 minutes = 23.48 Hours Topics for March 24th:Graphing Parabolas in Standard Form Solving Quadratics by Factoring 1
    • Number Sense: Space & Volume
    • Number Sense: Space & Volume
    • Number Sense: Space & Volume
    • Number Sense: Space & Volume
    • Number Sense: Space & Volume3D Sphere
    • Number Sense: Space & Volume
    • Test Review: Top 4 missed questions from Fridays test: v.1 4th; (44% correct) #8. 32x2 = 50 3rd; (42% correct) #10. x3 - 121x = 0 2nd; (40% correct) #4. -3x3 - 12x2 = 01st; (37%) #3. The product of (9 - 4t)(9 + 4t) results in:
    • Quadratic Equations: Todays Objectives:1. Understand the characteristics of Quadratic Equations, (What they are, and what they arent).2. Recognize the Graph of a Quadratic Equation3. Describe the Differences between Quadratic & Linear Equations4. Solve Quadratic Equations by factoring5. Listen Carefully, take notes, ask questions when needed.
    • Quadratic Equations: 1. What is a Quadratic Equation? From the Latin quad, as in quaduplets, quadrilaterals, and quarters...Quad means 4. A square has four sides. A variable in aquadratic equation can have an exponent of 2, but no higher.An exponent of 2 is a number squared.... The following are all examples of quadratic equations:x2 = 25, 4y2 + 2y - 1 = 0, y2 + 6y = 0, x2 + 2x - 4 = 0 The standard form of a quadratic is written as: ax2 + bx + c = 0, where only a cannot = 0
    • Quadratic Equations: We have been solving quadratic equations recently without actually calling them Quadratics. Lets review. Solve: x2 - 13x = 0 x( x - 13) = 0 x = 0, or x = 13 One more example. Solve: y = x2 - 4x - 5. To find the x-intercepts, we set the equation to x2 - 4x - 5 = 0 ( x - 5)( x + 1) = 0 x = 5 or x = -1Which brings us to: What do Quadratic Equations look likeand how are they different from linear equations?
    • Linear Equations: Y = 2x + 0 is a linear equation.Linear Equations are straightlines and cross the x and y axisonly one time. For eachy, there is only one x. The greatest degree of anyexponent in a linear equationis 1. The relationshipbetween x and y is constant;the slope stays the same.
    • Linear vs. Quadratic Equations A. The graphs of quadratics are not straight lines, they are always in the shape of a Parabola. B. Parabolas can cross an axis more than once.C. Unlike linear equations, each value of Y in a quadraticequation has more than one value of x. Because Y is 0 at theX-intercept, when we set the equation = to 0, we get thevalues of the x-intercepts. D. The slope of a quadratic is not constant. The slope- intercept formula will not work with parabolas.
    • Parabolas: ...In Sports
    • Parabolas: ...In Archeticture
    • Parabolas: ...In Nature
    • Parabolas: ...EverywhereFinally, the most important Parabola of all
    • Objective 4: Solving Quadratic Equations by Factoring There are 2 ways to factor Quadratic Equations and we have done both already. Lets review: Method 1: Set the equation = to 0 and solve: Example A. x2 + 6x + 9x2 + 6x + 9 = 0; (x + 3) (x + 3) = 0, x = -3.This is a perfect squaretrinomial, and the parabola only crosses the x axis at -3 andwould be in this shape:
    • Objective 4: Solving Quadratic Equations by Factoring Example B. x2 + 16x + 48 = 0(x + 12) (x + 4) = 0; x = -12, x = -4. This parabola is to the left of the Y axis Method 2: Solve x2 = 64. Remember the standard form? ax2 + bx + c = 0, where only a cannot = 0 In this case, b is 0, and c is 64. We can solve by taking the square root of both sides. The result is x = + 8; x = 8, and x = -8
    • Factoring Quadratic EquationsFrom the warm-up exercises, we have seen the variousways to factor quadratic equations. The solutions, orroots, tell us where the graph crosses the x axis.Given this information, we can begin to plot the graph.However, there is still more information we need tocomplete the graph.
    • Graphing Parabolas & Parabola TerminologyRemember, all Quadratic Equations are in the form ofa Parabola. Parabolas are in one of these forms: To solve and graph a quadratic equation, we need to know where the graph crosses the x and y axis:
    • Graphing Parabolas & Parabola Terminology Important points on a Parabola:1.Axis of Symmetry:The axis of symmetry is the verticleor horizontal line which runs through the exact centerof the parabola.
    • Graphing Parabolas & Parabola TerminologyImportant points on a Parabola:2. Vertex: The vertex is the highest point (themaximum), or the lowest point (the minimum) on aparabola. Notice that the axis of symmetry always runs through the vertex.
    • Finding the Axis of Symmetry and Vertex 1. Finding the Axis of Symmetry: The formula is: x = - b/2a Plug in and solve for y = x2 + 12x + 32We get - 12/2; = -6. The center of the parabola crosses the xaxis at -6. Since the axis of symmetry always runs throughthe vertex, the x coordinate for the vertex is -6 also.
    • Finding the Axis of Symmetry and VertexThere is one more point left to find and that is they-coordinate of the vertex. To find this, plug in thevalue of the x-coordinate back into the equation andfind y. y = -12 + 12(4) + 32. Y = 1 + 48 + 32. Y = 81. The bottom of the parabola is at -1 on the x axis, andway up at 81 on the y axis.
    • Warm- Up Exercises The slope is 2, which is positive and the Y- intercept is -2 Therefore, the correct graph is A
    • Warm- Up Exercises The Y-intercept is:0 The slope is: 2 The equation of the line is: Y = 2x + 0Write the equation for the line above
    • Warm- Up Exercises3. Write the inequality for the graph below The Y-intercept is: 2 The slope is: -3 The line is solid, not dotted. The equation is: Y < -3x + 2
    • Class Work:
    • 90% of 90 girls and 80% of 110 boys haveshown up in the concert hall on time.How many children are late?
    • ParabolasA parabola with -x2