Chapter Four Notes N P U2C4

Transcription

1 Chapter Four Notes N P U2C4 Name Period Section 4.3: Quadratic Functions and Their Properties Recall from Chapter Three as well as advanced algebra that a quadratic function (or square function, as it was called in Chapter Three) is a polynomial function of degree 2 whose graph is that of a parabola. As you learned in Chapter Three, graphs of any polynomial function can be transformed by adding/subtracting some number from x, adding/subtracting some number from the entire function, and negating the x or the overall function. To simplify transformations, the vertex form of a quadratic function is often used to express a quadratic function because one can easily identify the transformations of a basic quadratic function f(x) = x 2. Vertex form of a quadratic function: f(x) = a(x h) 2 + k For this form, (h, k) is the vertex (maximum if a is negative or minimum if a is positive) for the quadratic function. Axis of the parabola is x = h. The parabola is symmetric about this imaginary line. The parabola opens upward if a > 0 and opens downward if a < 0. Example 1: Find the vertex and axis of the graph of the function f(x) = 3(x 1) Then determine if the function has an absolute minimum or maximum. Find all x-intercepts and y- intercepts. Specify the intervals of x where f is increasing and decreasing. Determine the domain and range of the function. Then graph the function. page 1 N P U2C4

2 As stated before, though, the standard form of a polynomial is f(x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0. For a quadratic function, the standard form of a quadratic function is f(x) = ax 2 + bx + c where a is not zero. To find the vertex in this form, let us consider the quadratic formula from advanced algebra and b b 2 4ac which you used in Example 1. For some function ax 2 + bx + c = 0, x. However, more 2a than one answer comes from the quadratic formula if b 2 4ac 0. Since a vertex does not have a mirror point like the remainder of points on a parabola, the x coordinate for the vertex, known as h, must only b b equal, meaning h =. To find k, simply substitute the value of h in for x and find the k 2a 2a associated with it. Example 2: Find the vertex and axis of the graph of the function f(x) = 8x + 4x Then determine if the function has an absolute minimum or maximum. Find all x-intercepts and y- intercepts. Specify the intervals of x where f is increasing and decreasing. Determine the domain and range of the function. Then graph the function. page 2 N P U2C4

3 Finding the vertex could have been accomplished another way though using a method you have previously used called completing the square where you get the x 2 and x terms on a side by themselves and add b 2 2 to both sides. Example 3: Change the function f(x) = x 2 4x + 6 into vertex form and find the vertex and axis of the graph of the function. Then determine if the function has an absolute minimum or maximum. Find all x-intercepts and y-intercepts. Specify the intervals of x where f is increasing and decreasing. Determine the domain and range of the function. Then graph the function. page 3 N P U2C4

4 As you may have noticed, two of the functions graphed had two x-intercepts, one with rational numbers and one with irrational numbers, so a square root must have been involved. You saw that on the last example when no real x-intercepts existed. The radical in the quadratic formula determines the number and kinds of x-intercepts that will result. The expression in the radical (b 2 4ac) is called the discriminant. As follows is a summary how you can know the possible answers resulting from solving the discriminant. The Discriminant and Its Properties Discriminant Value Number of x-intercepts Type of x-intercept(s) b 2 4ac > 0 b 2 4ac < 0 b 2 4ac = 0 Writing equations for parabolas also works if you are provided a vertex and a point on the parabola. Example 4: Write an equation in standard form for a parabola with vertex ( 1, 3) and a point on the parabola (1, 5). HOMEWORK: Page 304 of your online textbook, questions 21, 22, 36, 40, 42, 43, 48, 50, 52 Section 4.4: Build Quadratic Models from Verbal Descriptions and from Data Quadratic functions also have applications in finances, particularly in finding maximum revenues and amounts, architecture, motion, and a variety of other areas. Example 1: The Sweet Drip Beverage Company sells cans of soda pop in machines. It finds that sales average 26,000 cans per month when the cans sell for 50 cents each. For each nickel increase in the price, the sales per month drop by 1000 cans. (a) Determine a function R(x) that models the total revenue realized by Sweet Drip, where x is the number of $0.05 increases in the price of the can. (b) Find a graph of R(x) that clearly shows a maximum for R(x). (c) How much should Sweet Drip charge per can to realize the maximum revenue? What is the maximum revenue? page 4 N P U2C4

5 (a) Determine a function R(x) that models the total revenue realized by Sweet Drip, where x is the number of $0.05 increases in the price of the can. (b) Find a graph of R(x) that clearly shows a maximum for R(x). (c) How much should Sweet Drip charge per can to realize the maximum revenue? What is the maximum revenue? page 5 N P U2C4

6 Vertical free-fall is also derived from the use of a quadratic function. Galileo learned that the height s or an object projected with vertical velocity v 0 was true to the form of the equation s(t) = 0.5gt 2 + v 0 t + s 0 where v 0 is the initial vertical velocity and s 0 is the initial height. One can also find the final velocity of the object v(t) = gt + v 0 where t is time in seconds. g is the acceleration due to gravity, which is appropriately 32 ft/sec 2 or 9.8 m/sec 2 (depending on which system of measurements you are using). Example 2: The Sandusky Little League uses a baseball throwing machine to help train 10-year-old players to catch high pop-ups. It throws the baseball straight up with an initial velocity of 48 ft/sec from a height of 3.5 ft. (a) Find an equation that models the height of the ball t seconds after it is thrown. (b) What is the maximum height the baseball will reach? How many seconds will it take to reach that height? (a) Find an equation that models the height of the ball t seconds after it is thrown. (b) What is the maximum height the baseball will reach? How many seconds will it take to reach that height? Example 3: 500 feet of fencing is available to enclose a rectangular lot along side of highway 77. A separate company will supply the fencing for the side along the highway, so only three sides are needed, as shown below. What is the maximum area that can be enclosed, and what would be the length and width of the lot? page 6 N P U2C4

7 Example 3: A parabolic suspension bridge is 200 meters long. The towers are 40 meters high, and the lowest point of the cable is 10 meters above the roadway. Find the vertical distance from the roadway to the cable at 50 meters from the center. First, it may be helpful to draw a diagram. HOMEWORK: Page of your online textbook, questions 4, 6, 8, 10, Section 4.5: Inequalities Involving Quadratic Functions In Section 4-3, we graphed quadratic equations. You can use that method to solve for quadratic inequalities. Example 1: Solve the inequality 0 > x page 7 N P U2C4

8 Example 2: Solve the inequality x 2 + 6x 5. Example 3: Solve the inequality 4x 2 1 4x 1. page 8 N P U2C4

9 Example 4: Using the graphs of f(x) and g(x) shown below, write the solution in interval notation for when (a) f(x) > g(x) and (b) f(x) g(x). (a) f(x) > g(x) (b) f(x) g(x) HOMEWORK: Page 319 of your online textbook, questions 3 6, 9 21 (multiples of 3) page 9 N P U2C4