Graphing Quadratic Functions
The Graph of f(x) = Ax^{2} + Bx + C
To graph a quadratic function, first calculate several ordered pairs. Then,
plot the corresponding points on a Cartesian coordinate system. Finally,
connect the points with a smooth line.
Example 1
Make a table of five ordered pairs that satisfy the function f(x) = x^{2}. Then,
use the table to graph the function.
Solution
To make a table, select 5 values for x. We will let x = 2, 1, 0, 1, and 2.
Substitute those values of x into the function and simplify.
x 
f(x) = x^{2} 
(x ,y) 
2 1
0
1
2 
f(2) = (2)^{2} = 4 f(1) = (1)^{2} =
1
f(0) = (0)^{2} = 0
f(1) = (1)^{2} = 1
f(2) = (2)^{2} = 4 
(2, 4) (1, 1)
(0, 0)
(1, 1)
(2, 4) 
Now, plot the points and connect them with a smooth curve.
The graph of a quadratic function, f(x) = Ax^{2} + Bx + C, has a distinctive
shape called a parabola. The sign of the coefficient of x^{2}, A, determines whether the graph opens up or down.
â€¢ When A is positive, the parabola opens up.
â€¢ When A is negative, the parabola opens down.
Graph the functions and state the domain and range of each.
a. f(x) = x^{2} + 3
b. f(x) = x^{2}  5
Solution
a. Graph The graph of f(x) = x^{2} + 3 is related to the graph of y
= x^{2}:
For each input value x, the output of f(x) = x^{2} + 3 is 3 more than the
output of f(x) = x^{2}.
Thus, the graphs have the same shape, but the graph of f(x) = x^{2} + 3 is shifted up 3 units.
Domain Since we can square any real number, the domain is all real
numbers or (∞, +∞).
Range We can see from the graph that the smallest value of y is 3.
Therefore, the range is y ≥ 3 or [3, +∞).
b. Graph To graph f(x) = x^{2}  5, we note that for each input value x, the
output of f(x) = x^{2}  5 is 5 less than the output of f(x) = x^{2}. Thus, the graph is shifted down 5 units.
Domain Since we can square any real number, the domain is all real
numbers or (∞, +∞).
Range We can see from the graph that the smallest value of y is 5.
Therefore, the range is y ≥ 5 or [5, +∞).
