Quadratic Function f ( x) ax bx c Solving Quadratic Equations by Graphing 6.1 Write each in quadratic form. Example 1 f ( x) 3( x + ) Example Graph f ( x) x + 6 x + 8 Example 3 An arrow is shot upward with an initial velocity of 64 ft /sec. The height of the arrow h(t) in terms of the time t since the arrow was released is h(t) = 64 t - 16t. How long after the arrow is released does it reach maximum height? What is the height?
Example 4 Solve m( p) = p - 6 p + 9 by graphing. real solutions 1 real solution 0 solutions Pg 338, 8 4
Solving Quadratic Equations by Factoring 6. Example 1 Example m - 3 m - 8 = 0 p + 10 p + 1 = 0 Example 3 Example 4 t = 36 4 x + 17 x = 15 Example 5 Example 6 y - 5 y - 1 = 0 a - 10 a + 5 = 0 Pg 344, 5-33 odd
Solving Quadratic Equations by Completing the Square (Opposite of Perfect Square Trinomial) 6.3 Example 1 Find the value of c that would make x + 16 x + c a perfect square. Solve by completing the square. Example Example 3 x + 6 x = 16 a + 11 a + 4 = 0 Example 4 Example 5 x - 11 x + 1 = 0 3 x + 5 x - 10 = 0 Pg 350, 7-35 odd
The Quadratic Formula and the Discriminant 6.4 Solve by Completing the Square a x + b x + c = 0 Quadratic Formula - b x = b - 4ac a Example 1 Example t - 10 t = 4 y + 4 y = 30 Example 3 Example 4 x - 3 x - = 0 x - 7 x + 1 = 0 Example 5 x - 3 x + 7 = 0
Discriminant b - 4 ac *Describes the nature of the roots* Value Perfect Square Roots Graph - 4ac b > 0 b - 4ac > 0 b - 4ac < 0 b - 4ac = 0 Find the value of the discriminant and describe the nature of the roots. Example 6 Example 7 4 x = -5 + 0x 3 x + = 5x Pg 357, 8-36 even
x = 4 x = -7 Sum and Product of Roots 6.5 Writing Equations (Quadratic) b 1. s 1 + s = a. s 1 s = c a Write a quadratic equation that has the following roots. Example 1 Example 3 1 and 5 and 1 8 3 Example 3 Example 4 7 i and 7 + i 5 + i 3 and 5 - i 3 Solve each equation and check using the sum and product of the roots. Example 5 3 x - 10 x + 3 = 0 Pg 36, 5-39
Analyzing Graphs of Quadratic Functions 6.6 Quadratic Equations y = a( x - h) k Vertex = (h, k) Axis of Symmetry: x = h Example 1 Name the vertex and the axis of symmetry for the graph of f(x) = ( x +11) 8. How is the graph of this function different from the graph of f(x) = x? How does a in y = a( x - h) k affect a parabola? a. f( x) = ( x - 4) 3 b. f( x) = -( x - 4) 3 c. f( x) = 3( x - 4) 3 1 d. f( x) = - ( x - 4) 3 4
Example Graph f(x) = 4( x + 3). Name the vertex, axis of symmetry, and direction of the opening. Example 3 Write y = x + 48 x + 10 in y = a( x - h) k form. Example 4 Write the equation of the parabola that passes through the points at (9, -3), (6, 3), and (4, 7). Pg 37, 1-45 odds
Graphing Quadratic Inequalities 6.7 Example 1 Example Graph y < - x - 6 x + Graph y x + 3 x - 4 Example 3 Solve 0 < x - x - 8 by Graphing. Example 4 Solve x + 9 x + 14 < 0 by Factoring. Pg 381, 7-49 odds
Standard Deviation (SD or each value in a set of data differs from the mean. Standard Deviation 6.8 X ) a measure of variation, or spread, which measures how much SD or = X (x - X ) + (x - X ) +...(x - X ) n 1 n x x x 1 n = 1st term = nd term = nth term X = mean n = number of data Team 1994 Average Team 1994 Average Arizona 7.69 LA Rams 9.13 Atlanta 7.00 Miami 9.65 Buffalo 33.73 Minnesota 9.79 Chicago 3.3 New England 34.34 Cincinnati 8.43 New Orleans 6.71 Cleveland 7.7 NY Giants 35.59 Dallas 3.85 NY Jets 5.00 Denver 3.34 Philadelphia 40.00 Detroit 30.04 Pittsburgh 30.99 Green Bay 6.13 San Diego 33.86 Houston 31.88 San Francisco 39.75 Indianapolis 6.48 Seattle 8.00 Kansas City 9.16 Tampa Bay 9.57 LA Raiders 31.3 Washington 35.70 Graphing Calculator 1. Clear List STAT, 4, nd L, enter 1. STAT, EDIT, enter all values in 3. STAT, CALC, 1-Var Stats, enter twice. 4. To clear again: STAT, 4, nd STAT, enter twice. L 1 Pg 389, 5-0 skip 9
Number of Students The Normal Distribution 6.9 Frequency distribution shows how data are spread out over the range of values. Histogram a bar graph that displays a frequency distribution. Favorite Subjects 0 18 16 14 1 10 8 6 4 0 19 19 14 10 6 5 Math Science History English Art Other Subjects Normal Distribution (bell curve) a symmetric curve which indicates that the frequencies are concentrated around the center portion of the distribution. Example 1 The useful lives of 10,000 batteries are normally distributed. The mean useful life is 0 hours, and the SD is 4 hours. a. Sketch a normal curve. b. How many batteries will last between 16 and 4 hours? c. How many batteries will last less than 1 hours? d. What is the probability that a battery will last between 16 and 8 hours?
Number of Students Skewed Distribution a distribution curve that is not symmetric. High left, tail right: + skewed High right, tail left: - skewed Favorite Subjects 0 18 16 14 1 10 8 6 4 0 19 19 14 10 6 5 Math Other Science History English Art Subjects Positively Skewed Pg 395, 5-17