Note: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Sep 15 2:51 PM

Transcription

1 2.1 Linear and Quadratic Name: Functions and Modeling Objective: Students will be able to recognize and graph linear and quadratic functions, and use these functions to model situations and solve problems. Polynomial Function Let n be a nonnegative integer and let a 0, a 1, a 2,..., a n be real numbers with a n 0. The function given by f(x) = a n x n + a n-1 x n a 2 x 2 + a 1 x + a 0 is a polynomial function of degree n. The leading coefficient is a n. Note: The zero function f(x) = 0 is a polynomial function. It has no degree and no leading coefficient. Examples: Which of the following are polynomials? For those that are polys, state the degree and leading coefficient. 1.) f(x) = 4x x 2 - (3/4) 2.) h(x) = 6x x ) g(x) = 5x 2 + 4x 4 Sep 15 2:51 PM Polynomial Functions of No and Low Degree Name Form Degree Zero function Constant function Linear function Quadratic function Linear Functions slope formula: m = point-slope form: The slope m in y = mx + b is called the rate of change. Sep 15 2:58 PM 1

2 Example Write an equation for a linear function f such that f(-3) = 5 and f(6) = -2 and graph. Sep 15 3:04 PM Quadratic Functions Standard form f(x) = ax 2 + bx + c Use x = - b to get the 2a x-coordinate of the vertex. Plug x back in to get the y-coordinate. Vertex form f(x) = a(x - h) 2 + k Vertex: (h,k) In either case, the axis of symmetry (or axis for short) is a vertical line given by x = x-coordinate of the vertex. Use the quadratic formula, factoring, completing the square or square roots to get the x-intercepts of a quadratic function. When a > 0, the parabola opens up. When a < 0, the parabola opens down. Sep 15 3:17 PM 2

3 Examples 1.) Let's look at f(x) = 3x x Sep 15 3:26 PM 2.) Write an equation for the quadratic function whose vertex is (-2,-5) and (-4,-27) is a point on the graph. Sep 15 3:28 PM 3

4 Vertical Free-Fall Motion The height s and vertical velocity v of an object in free fall are given by s(t) = -16t 2 + v 0 t + s 0 and v(t) = -32t + v 0, where t is time (seconds), v 0 is the initial vertical velocity (ft/sec) of the object and s 0 is the initial height (ft). Example: The Sandusky Little League uses a baseball throwing machine to help train young players to catch high popups. It throws the baseball straight up with an initial velocity of 48 ft/sec from a height of 3.5 feet. a.) Find the equation that models the height of the ball t seconds after it was thrown. Sep 15 3:28 PM b.) What is the maximum height the ball will reach and how many seconds will it take to reach that height? c.) How long will the ball be in the air? Sep 15 3:40 PM 4

5 Average Rate of Change Examples Compute the average rate of change on the given intervals for the given functions. 1.) f(x) = 3x + 2 on [0,2] May 27 12:38 PM 2.) g(x) = x 3 on [a,b] 3.) h(x) = ax 2 + bx + c from x = -1 to x = 1 May 27 12:40 PM 5

6 2.1 Homework Name: Determine which of the following are polynomials. For those that are, state the degree and leading coefficient. For those that aren't, explain why. 1.) 2.) 3.) 4.) Write a linear equation f that satisfies f( 5) = 1 and f(2) = 4. Graph the line. Sep 17 1:40 PM 5.) State the vertex and the axis of symmetry. 6.) Given, find the following: a.) Vertex b.) Axis of symmetry c.) Rewrite f(x) in vertex form by completing the square. Sep 17 1:45 PM 6

7 7.) Given, find the following: a.) Vertex b.) Axis of symmetry c.) Rewrite f(x) in vertex form by completing the square. 8.) Write an equation for the parabola whose vertex is (1,3) and passes through (0,5). Sep 17 1:49 PM 9.) Write an equation for the parabola shown. 10.) (Use your calculator.) Sep 17 1:50 PM 7

8 11.). a.) b.) c.) Compute the average rate of change of f from x = c to x = d. Sep 17 1:54 PM 12.) a.) b.) Sep 17 1:57 PM 8