Chapter 2: Polynomial and Rational Functions Power Standard #7

Transcription

1 Chapter 2: Polynomial and Rational s Power Standard #7 2.1 Quadratic s Lets glance at the finals. Learning Objective: In this lesson you learned how to sketch and analyze graphs of quadratic functions. 1

2 Definition of a Polynomial function: Let n be a nonnegative integer and let,,...,,, be real numbers with 0. A polynomial function of x with degree n is... the function.... What we really need to get out of this is that the order of polynomial functions lead with the highest exponent of the first letter in the alphabet. ( Quadratic functions have a degree polynomial of 2, or a Second Degree polynomial. The graph of a quadratic is a U shape, which is called a parabola., 0. A couple of characteristics: If the leading coefficient is + the function opens upward with a minimum point. If the leading coefficient is then the function opens downward and has a maximum point. Domain:, Range:,,,, depending on which direction it opens to. 2

3 Standard Form of a Quadratic: f() = h +, 0 Where: The graph of f is a parabola with a vertical axis of symmetry, x = h. What does that mean? The axis of symmetry is the h = The vertex is at (h, k). If a > 0, the parabola opens upward, and if a < 0, the parabola opens downward. The process to find this form is to completing the square. So we need to solve + + = 0 Completing the Square review = + + 1: make f(x)=0 and subtract the constant. 1: 0 = (2 + 8) + 5 2: Take of GCF of the numbers 3: Take b, divide it by 2, then square it. 4: make the perfect square with the value. Add to the left side. 5: Add/subtract the left side back and put f(x) back. 2: 5 = ( + 4) 3: = 2 = 4: 5 + = 2( ) 3 = : 3 = = + 3

4 Finding x & y intercepts Remember that x-intercepts exist on the x axis, so this is where 0. We plug 0 in for y and solve. You may need to factor, use the square root method, or complete the square to find your x-intercepts. There may not be any real solutions. To find the y-intercept, simply set 0 and solve for y. If you have a constant in an equation, it is usually the y-intercept. Example find the vertex, axis of symmetry & x & y intercepts of the parabola. Graph these and a few points. 6 8 CTS: Vertex: AOS: y-intercept: x-intercept: 4

5 Your turn: OYO Sketch the graph of 2 8 and identify the vertex, axis, and y & x-intercepts of the parabola. Graph a few points. Vertex: AOS: y-intercept: x-intercept: Your turn: Answer! Sketch the graph of 2 8 and identify the vertex, axis, and x-intercepts of the parabola. (- 1, - 9); AOS: x = - 1; (- 4, 0) and (2, 0) 0, 8 5

6 OYO Again without graphing 4 2 Find: AOS: Vertex: y-int: x-int: Example of finding a. Find the standard form of the equation of the parabola that has vertex at (1, -2) and passes through the point (3, 6). From the vertex we have this much of the equation: () = 1 2. To find a we substitute the point (3, 6) and solve for a. 6

7 Graph 4 5. what is the minimum value of y? The minimum occurs at as well as the maximum. This is also at the vertex s x value in a quadratic. So Evaluating the function at for is another way to find the axis of symmetry, the x value in the vertex, and the where the minimum or maximum value is of a quadratic. : a rough one. Find the minimum value of the function without a calculator. () = At what value of x does this minimum occur? What is the minimum point! Completing the square my be too rough. 7

8 Max Height of a Projectile The path of a softball is given by the function = , Where () is the height of the baseball (in feet) and x is the horizontal distance from home plate (in feet). What is the maximum height reached by the ball? We ll need to use the equation for the axis of symmetry., Application The daily cost of manufacturing a particular product is given by () = Where x is the number of units produced each day. Find how many units should be produced to minimize cost. = = 0.1( 70) = 35 = = =

9 Application The daily cost of manufacturing a particular product is given by Where x is the number of units produced each day. Find how many units should be produced to minimize cost. = Producing 35 units per day will minimize cost. Homework P.96 #12, 21-29odd, 31, 35, 39, 41, 51, 65, 70 9