Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions Definition of a polynomial function Let n be a nonnegative integer so n={0,1,2,3…} Let be real numbers with The function given by Is called a polynomial function of x with degree n Example: This is a 4 th degree polynomial

Polynomial Functions are classified by degree For example: In Chapter 1 Polynomial function, with Example: This function has degree 0, is a horizontal line and is called a constant function. y x –2 2

Polynomial Functions are classified by degree In Chapter 1 A Polynomial function, is a line whose slope is m and y-intercept is (0,b) Example: This function has a degree of 1,and is called a linear function. y x –2 2

Section 2.1 Quadratic Functions Definition of a quadratic function Let a, b, and c be real numbers with. The function given by f(x)= Is called a quadratic function This is a special U shaped curve called a … ?

Parabola ! Parabolas are symmetric to a line called the axis of symmetry. The point where the axis intersects with the parabola is the vertex. y x –2 2

The simplest type of quadratic is When sketching Use as a reference. (This is the simplest type of graph) a>1 the graph of y=af(x) is a vertical stretch of the graph y=f(x) 0<a<1 the graph of y=af(x) is a vertical shrink of the graph y=f(x) Graph on your calculator,, y x –2 2

Standard Form of a quadratic Function The graph of f(x) is a parabola whose axis is the vertical line x=h and whose vertex is the point (, ). - shifts the graph right or left - shifts the graph up or down For a>0 the parabola opens up a<0 the parabola opens down NOTE!

Example of a Quadratic in Standard Form Graph : Where is the Vertex? (, ) Graph: Where is the Vertex? (, ) y x –2 2

Identifying the vertex of a quadratic function Another way to find the vertex is to use the Vertex Formula If a>0, f has a minimum x If a<0, f has a maximum x a b c NOTE: the vertex is: (, ) To use Vertex Formula- To use completing the square start with to get

Identifying the vertex of a quadratic function (Example) Find the vertex of the parabola (, ) The direction the parabola opens?________ By completing the square? By the Vertex Formula

Identifying the x-Intercepts of a quadratic function The x-intercepts are found as follows

Identifying the x-Intercepts of a quadratic function (continued) Standard form is: Shape:_______________ Opens up or down?_____ X-intercepts are y x –2 2

Identifying the x-Intercepts of a Quadratic Function (Practice) Find the x-intercepts of y x –2 2

Writing the equation of a Parabola in Standard Form Vertex is: The parabola passes through point *Remember the vertex is Because the parabola passed through we have:

Writing the equation of a Parabola in Standard Form (Practice) Vertex is: The parabola passes through point Find the Standard Form of the equation.

Minimum and Maximum Values of Quadratic Functions 1. If a>0, f has a minimum value at 2. If a<0, f has a maximum value at

Baseball A baseball is hit at a point 3 feet above the ground at a velocity of 100 feet per second and at an angle of 45 degrees with respect to the ground. The path of the baseball is given by the function f(x)= x 2 + x + 3, where f(x) 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 baseball?

Cost A soft drink manufacturer has daily production costs of where C is the total cost (in dollars) and x is the number of units produced. Estimate numerically the number of units that should be produced each day to yield a minimum cost.

Grants The numbers g of grants awarded from the National Endowment for the Humanities fund from 1999 to 2003 can be approximated by the model 9≤t≤13 where t represents the year, with t=9 corresponding to Using this model, determine the year in which the number of grants awarded was greatest.

Homework Page all, 6, 8-20 even, 27,28,29-33 odd, even, 55,57,61