Quadratic Functions and Applications
Graph a Quadratic Function Written in Vertex Form
The graph of the quadratic function f  x   ax 2  bx  c (a ≠ 0) is a parabola.
The parabola opens upward for a > 0 and opens downward for a < 0
The function may be written in vertex form as f  x   a  x  h   k . The vertex is (h, k).
2
a<0
vertex (h, k)
a>0
axis of symmetry: x = h
1.
Graph f  x    x  3  9 . Identify the vertex, x-and y-intercepts, and axis of symmetry.
2.
Graph f  x     x  2  1 . Identify the vertex, x-and y-intercepts, and axis of symmetry.
2
2
Section 3.1
2
Write f  x   ax  bx  c (a ≠ 0) in Vertex Form
3.
Use the technique of completing the square to write f  x   2 x 2  20 x  42 into vertex form. Identify the
vertex.
Factor out the leading coefficient of the x2
term from the two terms containing x.
Complete the square within parentheses.
Remove –25 from within the parentheses
along with a factor of 2.
4.
Write f  x    x 2  2 x  3 in vertex form and graph the function. Identify the vertex, x-and y-intercepts, and
axis of symmetry.
Identify the domain and range.
Find the Vertex of a Parabola by Using the Vertex Formula
For f  x   ax 2  bx  c (a ≠ 0) the x-coordinate of the vertex is given by x  
 b 
.
 2a 
To find the y-coordinate, evaluate f  

b
,
 2a
The vertex is given by  
 b 
f  
 2a  
b
.
2a
5.
Given f  x   3x 2  48x  165 determine the vertex by using the vertex formula and by writing in vertex
form.
Solve Applications Involving Quadratic Functions
6.
At the end of his career, Sherlock Holmes retired to small farm in the country and took up a hobby of
beekeeping. Being of an analytical mind, he could not just watch a bee fly from flower to flower, he
formulated a mathematic model for the parabolic path the bee follows from flower A to flower B.
The path is given by
f  x   0.03x 2  1.8 x  24
where x is the distance in inches along the ground from Sherlock.
a.
What is the maximum height the bee reaches as it flies from A to B?
b.
How far away from Sherlock are flowers A and B?