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## The Graph of f(x) = Ax2 + 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) = x2. 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) = x2 (x ,y) -2-1 0 1 2 f(-2) = (-2)2 = 4f(-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) = Ax2 + Bx + C, has a distinctive shape called a parabola. The sign of the coefficient of x2, 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) = x2 + 3

b. f(x) = x2 - 5

Solution

a. Graph The graph of f(x) = x2 + 3 is related to the graph of y = x2:

For each input value x, the output of f(x) = x2 + 3 is 3 more than the output of f(x) = x2.

Thus, the graphs have the same shape, but the graph of f(x) = x2 + 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) = x2 - 5, we note that for each input value x, the output of f(x) = x2 - 5 is 5 less than the output of f(x) = x2. 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, +).