Graphs of Quadratic Functions
The graph of any quadratic function is called a parabola.
Parabolas are shaped like cups , as shown in the graph below. If
the coefficient of x ^{2} is positive , the parabola opens upward;
otherwise, the parabola opens downward. The vertex (or turning
point) is the minimum or maximum point.
The Standard Form of a
Quadratic Function
• The quadratic function
• f (x) = a( x  h)^{2} + k, a ≠ 0
• is in standard form. The graph of f is a parabola
whose vertex is the point (h, k). The parabola is
symmetric to the line x = h. If a > 0, the parabola
opens upward; if a < 0, the parabola opens
downward.
Graphing Parabolas With Equations
in Standard Form
• To graph f (x) = a(x  h)^{2} + k:
2. De termine whether the parabola opens upward or
downward. If a > 0, it opens upward. If a < 0, it opens
downward.
3. Determine the vertex of the parabola. The vertex is (h,
k).
4. Find any xintercepts
by replacing f (x) with 0. Solve
the resulting quadratic equation for x .
5. Find the yintercept
by replacing x with zero .
6. Plot the intercepts and vertex . Connect these points with
a smooth curve that is shaped like a cup.
Text Example
• Graph the quadratic function f (x) = 2(x  3)^{2} + 8.
Solution We can graph this function by fol lowing the steps
in the preceding
box. We begin by identifying values for a , h, and k.
Standard form 

Given equation 
Step 1 Determine how the parabola opens. Note that a, the
coefficient of
x ^{2}, is 2.Thus, a < 0; this negative value tells us that the parabola opens
downward.
Step 2 Find the vertex. The vertex of the parabola is at
(h, k). Because h =
3 and k = 8, the parabola has its vertex at (3, 8).
Step 3 Find the xintercepts.Replace f (x) with 0 in f (x) = 2(x  3)^{2} + 8.
0 = 2(x  3)^{2} + 8 
Find xintercepts,
setting f (x) equal to zero. 
2(x  3)^{2} = 8 
Solve for x . Add 2(x  3)^{2} to both sides of
the equation. 
(x  3)^{2} = 4 
Divide both sides by 2. 
(x  3) = ±2 
Apply the square root method . 
x  3 = 2 or x  3 = 2 
Express as two separate equations. 
x = 1 or x = 5 
Add 3 to both sides in each equation. 
The xintercepts
are 1 and 5. The parabola passes through (1, 0) and (5, 0).
Step 4 Find the yintercept.
Replace x with 0 in f (x) = 2(x  3)^{2} + 8.
The yintercept
is –10. The parabola passes through (0, 10).
Step 5 Graph the parabola. With a vertex at (3, 8), xintercepts
at 1 and 5,
and a yintercept
at –10, the axis of symmetry is the vertical line whose
equation is x = 3.
The Vertex of a Parabola Whose
Equation Is f (x) = ax^{2} + bx + c
• Consider the parabola defined by the
quadratic function
• f (x) = ax^{2} + bx + c. The parabola's vertex is
at
Example
Graph the quadratic function f (x) = x^{2 } + 6x 2.
Solution:
Step 1 Determine how the parabola opens. Note that a, the
coefficient of x^{2}, is 1.
Thus, a < 0; this negative value tells us that the
parabola opens downward.
Step 2 Find the vertex. We know the x coordinate
of the vertex is x =
b/(2a). We identify a, b, and c to substitute the values into the
equation for the xcoordinate:
x = b/(2a) = 6/2(1)=3.
The xcoordinate
of the vertex is 3. We substitute 3 for x in the equation
of the function to find the ycoordinate:
y=f(3) = (3)^2+6(3)2=9+182=7, the parabola has its vertex at (3,7).
• Step 3 Find the xintercepts.
Replace f (x) with 0 in f (x) = x^{2} + 6x
 2.
a = 1,b = 6,c = 2
• 0 = x^{2} + 6x  2
• Step 4 Find the yintercept. Replace x with 0 in f (x) = x^{2} + 6x 
2.
The yintercept
is –2. The parabola passes through (0, 2).
Step 5 Graph the parabola.
Minimum and Maximum:
Quadratic Functions
• Consider f(x) = ax^{2} + bx +c.
2. If a > 0, then f has a minimum that occurs at
x = b/(2a). This minimum value is f(b/(2a)).
3. If a < 0, the f has a maximum that occurs at
x = b/(2a). This maximum value is f (b/(2a)).