• Parabolas
• Quadratic equations and functions
• Graphs of quadratic functions
• Applications
The standard form for a quadratic function is
f(x) = ax2+bx+c.
The graph of a quadratic function is a parabola.
Vertex of the parabola , maximum and minimum
f(x) = ax2+bx+c.
The vertex of the parabola is found at the point where x = -b/2a.
Often the vertex is denoted (h,k). In this case, h = -b/2a and
k can be de termined by the equation .
Example:
What are the coordinates of the vertex of the graph of
f(x) = 3x2 +6x-5? (h,k) = ( , )
Vertex-axis form for a quadratic function
• f(x) = ax2 +bx+c (standard form)
• f(x) = a(x-h)2 +k (vertex-axis form)
Since
a(x-h)2 +k = ax2 -2ahx+ah2 +k,
b = -2ah, or h = -b/2a.
Example and Exercise: Remember: h = -b/2a.
Convert the standard form into the vertex-axis form:
f(x) = 2x2 +12x+13.
a = 2,b = 12,c = 13, so h = -3 and
a(x-h)2 +k = 2(x+3)2 +k = 2(x2 +6x+9)+k
= 2x2 +12x+18+k. So k = -5.
f(x) = 2(x+3)2-5.
Exercise: Convert the standard form into the vertex-axis form:
f(x) = -x2 +3x-7.
Graphing a quadratic function
y = 2x2 +12x-5
The vertex is the point at
The axis of symmetry is the vertical line
Exercise: Graph the parabola: f(x) = x2-6x+10
| Opens:
Vertex:
y-intercept:
|
 |
Exercise: Graph the parabola: f(x) = 3x2 +6x+1
| Opens:
Vertex:
y-intercept:
|
|
The quadratic formula
The general quadratic function:
f(x) = ax2+bx+c.
The quadratic formula tells you the solutions to f (x) = 0,
which is the same as locating the x- intercepts on the graph :
The quadratic formula
Example: Solve
2x2-5x-3 = 0,
for x.
a = 2, b = -5,c = -3
So x = 3 and x = -1/2 are the solutions.
The quadratic formula
Exercise: Graph y = x2-5x-6 and solve x 2-5x-6 = 0, for
x.
a = , b = ,c =
Summary: Quadratic Functions
f(x) = ax2+bx+c.
where a is not equal to zero
• if a > 0, the graph opens
• if a < 0, the graph opens
• x-coordinate of vertex h =
• f(h) = k is minimum if a > 0
• f(h) = k is maximum if a < 0
• Domain: set of all real numbers
| •Range: : |
(-∞,k], if a<0 |
| [k,∞), if a>0 |
Application
The CSUN bookstore pays $1.00 per hundred copies of Northridge
Today and $14.00 shipping costs, no matter how many copies
it orders . The price-demand equation is estimated to be p(x) =
10-x, where x is the number of copies sold, in hundreds. How
many copies must be the bookstore sell to break even?
