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: ...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