PreCalculus Class Notes QF7 Quadratic Functions
Quadratic equation ax 2 + bx + c = 0 , x =
−b ± b 2 − 4ac
2a
Nature of the discriminant
two real solutions
y
Graph
r1
one real solution
y
y
r2 x
r
Discriminant
b 2 − 4ac
no real solutions
Positive
Not a
Perfect
perfect
square,
square,
zeros are
zeros are
rational
irrational
x
Equals 0
x
Negative
(solutions are complex)
1 2
x + 2 x + 4 = 0 . Then solve
4
the equation by using the quadratic formula. Support your answer graphically.
Example: Use the discriminant to find the number of solutions to
Functions with Quadratics
Rational Functions Q ( x ) =
f ( x)
; g ( x) ≠ 0
g ( x)
Domain Rule: No division by zero. Set denominator equal to 0, then exclude the solutions from
the domain of the rational function.
Example: Find the domain of the function f ( x ) =
2x −1
. Write the domain in set notation
x − 5x + 4
2
and in interval notation.
Functions: solve for y and determine if y is a function of x
Recall: How do you determine if an equation represents a function in y?
Example: Solve for y and determine if y is a function of x,
x2 − 4 y
=
2y
x
2
2
Example: Solve for y and determine if y is a function of x, ( x + 4 ) + ( y − 3) = 25
Bonus question: Describe the graph of this equation.
Example: Solve for y and determine if y is a function of x, x = 2 y 2 − 6 y + 5