Chapter 9 Quadratic Functions and Equations 1
9 1Quadratic Graphs and their properties U shaped graph such as the one at the right is called a parabola. A parabola can open upward or downward. A parabola that opens upward has a minimum or lowest point. A parabola that opens downward has a maximum or highest point. The vertex of a parabola is its minimum or maximum point. All parabolas have a line or axis of symmetry. What is the vertex of the graph below? Is it a minimum or maximum? The graph opens downward, so you are looking for the highest point. The vertex is ( 3, 2) and it is a maximum. 2
Graphing y= ax 2 The farther away (a) is from zero, the skinnier the graph. This is only true when its in y= ax 2 or y= ax 2 + c form. Graphing y= ax 2 + c Same is true above, but the (c) value moves the parabola up and down. (+c) moves up, ( c) moves down 3
Graphing y= ax 2 versus Graphing y= ax 2 +a graph open up a graphs open down 4
Any function in the form y = ax 2 + bx + c where a 0 is called a quadratic function. The graph of a quadratic function is a parabola. Problem What is the graph of This is a quadratic function where a =, b = 0 and c = 4. The graph will be a parabola. Use a table to find some points on the graph. Then use what you know about parabolas to complete the graph. 5
You try! 1. Graph y = x 2 4 2. Graph y = 3x 2 + 2 Steps 1. Does it open up or down? 2. Shift up or down? 3. What are the points? 6
An acorn drops from a tree branch 70 ft above the ground. The t function h= 16t 2 + 70 gives the height h of the acorn in feet after t seconds. What is the graph of this function? At about what time does the acorn hit the ground? Why would you not calculate the heights for any negative values of t? 7
9 2 Quadratic Equations To find the axis of symmetry ( x coordinate) about the x axis, use the formula... x= b/2a This formula, comes from the quadratic function y = ax 2 + bx + c where a 0 Ex. What is the axis of symmetry of y= x 2 6x + 4? x= 3 if you know the (x), can you find out (y)? y= 5 8
Recall that the general equation for a quadratic function is y = ax 2 + bx + c. Using this general equation, the equation for the axis of symmetry is Since the vertex lies on the axis of symmetry, the x coordinate of the vertex is What are the equation of the axis of symmetry and the coordinates of the vertex of the graph of y = 3x 2 + 6x 4? The equation of the axis of symmetry is x = 1 and the coordinates of the vertex of the graph are ( 1, 7). 9
9 3 Solving Quadratic Equations An equation in the form ax 2 + bx + c = 0 where a 0 is called a quadratic equation. Its related quadratic function is y = ax 2 + bx + c. If you graph the related quadratic function, the solutions of the quadratic equation are x values where the graph crosses the x axis. The related function of x 2 + 4 = 0 is y = x 2 + 4. The graph of y = x 2 + 4 is shown below. The related function of x 2 2x + 1 = 0 is y = x 2 2x + 1. The graph of y = x 2 2x + 1 is shown below. The related function of x 2 x + 2 = 0 is y = x 2 x + 2. The graph of y = x 2 x + 2 is shown below. The graph crosses the x axis where x = 2 and x = 2. The equation x 2 + 4 = 0 has two solutions, 2 and 2. The graph touches the x axis where x = 1. The equation x 2 2x + 1 = 0 has one solution, 1. The graph does not touch the x axis. The equation x 2 x + 2 = 0 has no real number solutions. 10
Make sure when solving equations, functions and inequalties that you do the order of operations backwards. It isn't absolutely necessary, but it makes it a lot easier. What are the solutions of 81x 2 = 49? What are the solutions of x 2 + 9 = 0? 11
9 4 Factoring to Solve Quadratic Equations If the product of two or more numbers is 0, then one of the factors must be 0. You can use this fact to solve quadratic equations. this is called the zero product property What are the solutions of the equation (4a + 12)(5a 20) = 0? Since the product is 0, either (4a + 12) or (5a 20) must equal 0. 4a + 12 = 0 or 5a 20 = 0 4a + 12 12 = 0 12 or 5a 20 + 20 = 0 + 20 4a = 12 or 5a = 20 a = 3 or a = 4 12
Review from chapter 8 on factoring Steps to factor a polynomial with an "a" value of 1!!! 1) Can you pull out anything the polynomial or the two binomials share in common? 2) Are their any special cases? ex. (a 2 +2ab+b 2 ), (a 2 2ab+b 2 ), (a 2 b 2 ) 3) multiply the coefficient (a) (which is 1) by constant (c) including their signs to get a product. 4) take the product and figure out two numbers that multiply to give you this product, but add/subtract to give you (b). 5) These will be your last terms of your binomial, with their respective signs. first terms of both will be your variable 6) Use product zero rule if they want you to solve for the variables. y = ax 2 + bx + c 13
Review from chapter 8 on factoring Steps to factor a polynomial with an "a" value other than 1!!! 1) Can you pull out anything the polynomial or the two binomials share in common? 2) Are their any special cases? ex. (a 2 +2ab+b 2 ), (a 2 2ab+b 2 ), (a 2 b 2 ) 3) multiply the coefficient (a) by constant (c) including their signs to get a product. 4) take the product and figure out two numbers that multiply to give you this product, but add/subtract to give you (b). 5) Whatever you get for #4, change (bx) to this in order to make 2 separate binomials. 6) Does each binomial have anything you can pull out in order to create a binomial that is exactly the same as the other. 7) Factor out the two similar binomials, and take whatever is left over and make a separate binomial. 8) Use product zero rule if they want you to solve for the variables. y = ax 2 + bx + c 14
You try! y 2 + 3y + 2 = 0 6t 2 + 13t + 6 = 0 15
Factoring to make a quadratic equation If you can rewrite a quadratic equation as a product of factors that equals zero, you can solve the equation. To solve equations in this manner, you must use all your factoring skills. What are the solutions of the equation x 2 x = 20? 16
x 2 You try! 5x = 14 2h 2 9h = 5 17
9 5 Completing the perfect square trinomial You have learned to square binomials. Notice how the coefficient of the a term is related to the constant value in every perfect square trinomial. (a + 1) 2 = (a + 1)(a + 1) = a 2 + 2a + 1 = 1 (a 1) 2 = (a 1)(a 1) = a 2 2a + 1 = 1 (a 2) 2 = (a 2)(a 2) = a 2 4a + 4 = 4 (a + 3) 2 = (a + 3)(a + 3) = a 2 + 6a + 9 = 9 formula to figure out the perfect squared trinomial (c) c = (b/2) 2 What is the value of c such that x 2 14x + c is a perfect square trinomial? The coefficient of the x term is 14. Using the pattern, c = or 49. So, x 2 14x + 49 is a perfect square trinomial. 18
a 2 + 8a + c You try! d 2 + 12d + c 19
You can use completing the square to solve quadratic equations. Problem What are the solutions of the equation x 2 + 2x 48 = 0? 20
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9 6 The Quadratic Formula and the Discriminant Old Faithful!!!!! If a quadratic equation is written in the form ax 2 + bx + c = 0, the solutions can be found using the following formula. This formula is called the quadratic formula. 22
What are the solutions of x 2 + 7x = 60? Use the quadratic formula. First rewrite the equation in the form ax 2 + bx + c = 0. 23
x 2 + 32x + 175 = 0 You try! 4x 2 + 24x = 35 24
Using the discriminant to figure out the number of solutions.. discriminant = - the expression underneath the radical sign b 2 4ac > 0 b 2 4ac b 2 4ac = 0 b 2 4ac < 0 2 solutions 1 solution no solutions 25
3x 2 5x = 6 You try! number of solutions 26