1. The Quadratic Function A Quadratic function is a function of the form where a, b and c are real numbers and

2. The Quadratic Equation A Quadratic equation is an equation equivalent of the form where a, b and c are real numbers and Examples of Quadratic Equations

3. Second Degree Equation A Quadratic equation is also a Second Degree equation because the highest exponent is two. The Standard Form of a quadratic equation is where a, b and c are real numbers and

4. Zeros of a Quadratic Function The zeros of a quadratic function is the solution of the equation f(x) = 0 Since x = 4 and x= -3 are solutions of the zeros of are x = 4 and x= -3. We note that

5. Methods for Solving Quadratic Equations Factoring Taking the square root Using the quadratic formula Completing the square

6. The Quadratic Formula For any Quadratic equation in standard form The Quadratic formula is given by We have derived this equation in class

7. The Discriminant The expression is called the discriminant of the quadratic equation

8. The Discriminant The Discriminant of a Quadratic Equation tells whether the quadractic equation has any real solutions It also tells how many real zeros or x- intercepts the equation has.

9. The Discriminant Two distinct real solutions The quadratic equation has 2 distinct unequal real solutions if

10. The Discriminant One distinct Solutions The quadratic equation has 1distinct or 2 multiply real solutions if

11. No real solutions The quadratic equation has two non - real solutions if

12. Application (Second degreee equation) Page 143 #87 Side of square metal sheet =x Subtract 1 foot from each end of each side Result - new side is x – 2 Depth of box is the inch taken from the sides Area of bottom of box is (x – 2)(x – 2) Volume of box is (side) (side) (depth) Volume is (x – 2)(x – 2)(1) Since box is to hold 4 cubic feet Volume of box is 4 feet V (x) = (x – 2)(x – 2)(1)= (x-2) = 4

13. Application (Second degree equation) Page 142 #90 Height of Object Initial height is 0, since the object is thrown from the ground. When object is 15 meters above the ground, the height is Using the quadratic formula, we get t =.99 or t = 3.09

14. Page 142 #90(cont.) b. When the object strikes the ground the height is Thus t = 0 or t = 4.08. Thus the object will strike the ground after 4.08 sec. c. When the object is 100 meter above the ground, the height function is There is no real solution so the object never reaches 100 meters.

15. The Quadratic function Opens up when a>o axis of symmetry opens down a < 0 Vertex