Background
The graph of a quadratic expression has a Ushaped graph called a
parabola. We can write a quadratic expression in the form ax^{2}
+ c, where a is called the leading
coefficient. The way the parabola opens depends on the sign of the leading
coefficient. In fact, if a is positive, the parabola opens up; if a is negative, the parabola opens down. If the
absolute value of a is greater than 1, then the
parabola is narrower than y = x^{2}. If the absolute
value of a is less than 1, then the parabola is wider
than y = x^{2}.
A quadratic function can be written in different
forms. Each form gives information about the graph.
Standard Form (no linear term) 
WarmUp
y = 3x^{2} + 4 matches
.
â€œThis quadratic function is written in standard form.
Because the leading coefficient is negative, the
graph opens down. The 3 causes the graph to be
narrower than y = x^{2}. The vertex is at (0, 4),
because the graph has been shifted upward 4 units.
You can test the ordered pair (1, 1). When x = 1, y = 3(1)^{2} + 4,
which is 1.â€

y = ax^{2} + c 
Graphical Information
c > 0:
vertical shift
upward of c units
c < 0:
vertical shift
downward of
c units 
WarmUp
y = (x + 3)^{2} + 1 matches
.
â€œThis quadratic function can be written in vertex
form as The vertex is at
The graph opens up because a is positive.â€

y = a(x  h)^{2} + k 
Graphical Information vertex at
(h, k) 
WarmUp
matches
.
â€œThis quadratic function is written in factored form.
The graph opens down because a is less than 0.
Because
the graph is wider than
y = x^{2}.
The intercepts are at (5, 0) and (2, 0)â€
