QUADRATIC FUNCTIONS AND THEIR ZEROS

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SECTION 2.3

• QUADRATIC FUNCTIONS AND THEIR ZEROS

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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.
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QUADRATIC FUNCTIONS
We will discuss finding the zeros of a quadratic
function using four different methods 1. Factorin
g 2. Square Root Method 3. Completing the
Square 4. Quadratic Formula
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FINDING THE ZEROS OF A QUADRATIC FUNCTION BY
FACTORING
Example f(x) x 2 x - 12 f(x) x2 6x
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Check on calculator!
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FINDING THE ZEROS OF A QUADRATIC FUNCTION BY
SQUARE ROOT METHOD
Example f(x) x 2 - 5 f(x) (x 2)2 - 16
Check on calculator!
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FINDING THE ZEROS OF A QUADRATIC FUNCTION BY
COMPLETING THE SQUARE
Example f(x) x 2 5x 4
Check on calculator!
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QUADRATIC FORMULA
This formula is derived by completing the square
on the standard form of a quadratic equation.
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Find the zeros of the quadratic function using
the Quadratic Formula f(x) 3x 2 - 5x 1
Exact Solutions
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Calculator Solutions
x ? 1.43, .23
Check Intercepts!
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Find the zeros using the Quadratic Formula
25x 2 - 60x 36 0
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Exact Solution
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Calculator Solution
Check Intercepts!
x 1.2
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Find the zeros using the Quadratic Formula
Use the Quadratic Formula to solve f(x) 3x 2
2 - 4x 3x 2 - 4x 2 0
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No real solution. Check Intercepts.
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DISCRIMINANT
If b 2 - 4ac gt 0 , 2 unequal real solutions If b
2 - 4ac 0, 1 real solution If b 2 - 4ac lt 0, no
real solutions
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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.
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FINDING THE POINT OF INTERSECTION OF TWO FUNCTIONS
Sometimes were 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) x2 5x 3 and g(x) 2x 1
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FINDING THE ZEROS OF A FUNCTION WHICH IS
QUADRATIC IN FORM
Ex f(x) (x 2)2 11(x 2) 12
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• CONCLUSION OF SECTION 2.3