Chapter 2 Polynomial and Rational Functions 2.1 Quadratic Functions

1. Chapter 2 Polynomial and Rational Functions2.1 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 4th degree polynomial

2. y 2 x –2 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.

3. y 2 x –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.

4. 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 … ?

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

6. y 2 x –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 , ,

7. 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!

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

9. y 2 x –2 Identifying the vertex of a quadratic function One way to find the vertex is to put the quadratic function in standard form by completing the square. Where is the vertex? ( , )

10. 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

11. 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

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

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

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

15. 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:

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

17. THE END OF CHAPTER 2.1