2.2 Quadratic Functions
Anatomy of a Quadratic Function (two possible forms)
A quadratic function can be represented in one of two forms:
1. Standard Form: f(x) = a(x − h)2 + k
Handy because ...
‐ it allows you to use the transformations you learned in section 1.6 to
make its graph
‐ the values of ‘h’ and ‘k’ represent the vertex of the parabola
2. General Form: f(x) = ax2 + bx + c
Not as handy as the standard form. Only real perk is you can get your
y‐intercept easily by plugging in x = 0 ... f(0) = c
NOTICE! In both forms, the value of ‘a’ is the same!
The graph of a quadratic function is a parabola.
If a > 0, the parabola opens upward from the vertex.
If a < 0, the parabola opens downward from the vertex.
The graph of a parabola is always symmetric on each side of its vertex.
The vertical line through the vertex is the axis of symmetry.
1. Standard form (best for graphing using 1.6 techniques)
f ( x) = a(x − h)2 + k
ex) Graph f (x) = 2(x − 1)2 − 8
a) Vertex ________
b) Domain ________
c) Range ________
d) x‐intercepts ___________
e) y‐intercept ____________
ex) Graph f (x) = − 12 (x + 3)2 + 8
a) Vertex ________
b) Domain ________
c) Range ________
d) x‐intercepts ___________
e) y‐intercept ____________
2. General Form f ( x) = ax 2 + bx + c
VERY VERY VERY VERY IMPORTANT CALCULATION!
To get vertex from the general form of a quadratic function:
x = −b /2a and y = f (−b /2a) (can rewrite in standard )
KNOW THESE FORMULAS!!!
ex) For f ( x) = 2 x 2 − 4 x + 4
a) Determine the vertex
b) Rewrite in standard form
c) Sketch its graph
d) Determine the EXACT x‐intercepts
algebraically
e) y‐intercept
ON QUADRATIC FUNCTIONS,
THE ABSOLUTE MAXIMUM/MINIMUM
VALUES ALWAYS OCCUR AT THE VERTEX!
ex) For the function f ( x) = −0.32 x 2 + 122.6 x − 13.6 , determine its
absolute maximum or minimum.
ex) A quarterback snaps to the receiver and the function
h( x) = −0.04 x 2 + 2.1 x + 6.1
models the pass where x (input) represents the number of
horizontal yards from the quarterback the ball has traveled
and y (output) represents the height of the ball in feet.
a) Find the vertex for this quadratic function.
b) What is the maximum height the ball reaches?
c) How far downfield does this occur?
d) If the receiver misses the pass, how far downfield does the
ball land?
ex) You have 1200 feet of fencing to enclose a rectangular pen
next to a straight river. There will be two dividing walls in the
enclosure to make 3 equal sized spaces.
a) Create a function modeling the
enclosed area in terms of ‘x’
b) Find the vertex of the quadratic
function from part (a).
c) What is the maximum enclosed area?
d) What should the dimensions of the rectangle be to get the
most area?