SECTION 2.3 QUADRATIC FUNCTIONS AND THEIR ZEROS QUADRATIC FUNCTIONS AND THEIR ZEROS

QUADRATIC FUNCTIONS A quadratic function is a function of the form f(x) = ax 2 + bx + c where a, b & c are real numbers and a  0 The domain of a quadratic function consists of all real numbers.

QUADRATIC FUNCTIONS We will discuss finding the zeros of a quadratic function using four different methods: 1.Factoring 2.Square Root Method 3.Completing the Square 4.Quadratic Formula

FINDING THE ZEROS OF A QUADRATIC FUNCTION BY FACTORING Example:f(x) = x 2 + x - 12 f(x) = x 2 – 6x + 9 Check on calculator!

FINDING THE ZEROS OF A QUADRATIC FUNCTION BY SQUARE ROOT METHOD Example:f(x) = x f(x) = (x – 2) Check on calculator!

FINDING THE ZEROS OF A QUADRATIC FUNCTION BY COMPLETING THE SQUARE Example:f(x) = x 2 + 5x + 4 Check on calculator!

QUADRATIC FORMULA This formula is derived by completing the square on the standard form of a quadratic equation.

Find the zeros of the quadratic function using the Quadratic Formula: f(x) = 3x 2 - 5x + 1 Exact Solutions:

Calculator Solutions: Check Intercepts! x  1.43,.23

25x x + 36 = 0 Find the zeros using the Quadratic Formula

Exact Solution:

Calculator Solution: x = 1.2 Check Intercepts!

Use the Quadratic Formula to solve: f(x) = 3x x 3x 2 - 4x + 2 = 0 Find the zeros using the Quadratic Formula

No real solution. Check Intercepts.

DISCRIMINANT If b 2 - 4ac > 0, 2 unequal real solutions If b 2 - 4ac = 0, 1 real solution If b 2 - 4ac < 0, no real solutions

STEPS FOR FINDING THE ZEROS OF A QUADRATIC FUNCTION STEP 1:Put the function in standard form. STEP 2:Identify a, b, and c. STEP 3:If the discriminant is negative, the function has no real zeros. STEP 4:If the discriminant is nonnegative and a perfect square, solve by factoring. If it is nonnegative and not a perfect square, use the Quadratic Formula, Square Root Method, or Completing the Square.

FINDING THE POINT OF INTERSECTION OF TWO FUNCTIONS Sometimes we’re interested in when two functions are equal to each other. For example, if R(x) is Revenue and C(x) is Cost, the point at which they are equal would be the “break-even” point. Ex: f(x) = x 2 + 5x – 3 and g(x) = 2x + 1

FINDING THE ZEROS OF A FUNCTION WHICH IS QUADRATIC IN FORM Ex: f(x) = (x + 2) (x + 2) – 12

CONCLUSION OF SECTION 2.3 CONCLUSION OF SECTION 2.3