:188 Linear and Quadratic Functions

2.3 Quadratic Functions

You may recall studying quadratic equations in Intermediate Algebra. In this section, we review

those equations in the context of our next family of functions: the quadratic functions.

Definition 2.5. A quadratic function is a function of the form

f(x) = ax2 + bx + c,

where a, b and c are real numbers with a 6= 0. The domain of a quadratic function is (−∞, ∞).

The most basic quadratic function is f(x) = x

2

, whose graph appears below. Its shape should look

familiar from Intermediate Algebra – it is called a parabola. The point (0, 0) is called the vertex

of the parabola. In this case, the vertex is a relative minimum and is also the where the absolute

minimum value of f can be found.

(−2, 4)

(−1, 1)

(0, 0)

(1, 1)

(2, 4)

x

y

−2 −1 1 2

1

2

3

4

f(x) = x

2

Much like many of the absolute value functions in Section 2.2, knowing the graph of f(x) = x

2

enables us to graph an entire family of quadratic functions using transformations.

Example 2.3.1. Graph the following functions starting with the graph of f(x) = x

2 and using

transformations. Find the vertex, state the range and find the x- and y-intercepts, if any exist.

1. g(x) = (x + 2)2 − 3 2. h(x) = −2(x − 3)2 + 1

Solution.

1. Since g(x) = (x + 2)2 − 3 = f(x + 2) − 3, Theorem 1.7 instructs us to first subtract 2 from

each of the x-values of the points on y = f(x). This shifts the graph of y = f(x) to the left

2 units and moves (−2, 4) to (−4, 4), (−1, 1) to (−3, 1), (0, 0) to (−2, 0), (1, 1) to (−1, 1) and

(2, 4) to (0, 4). Next, we subtract 3 from each of the y-values of these new points. This moves

the graph down 3 units and moves (−4, 4) to (−4, 1), (−3, 1) to (−3, −2), (−2, 0) to (−2, 3),

(−1, 1) to (−1, −2) and (0, 4) to (0, 1). We connect the dots in parabolic fashion to get