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David Terr
Ph.D. Math, UC Berkeley

2.3. Quadratic Functions and Their Graphs

In section 1.10 we considered quadratic equations, i.e. equations of the form ax2 + bx + c = 0, where a, b, and c are real-valued constants with a nonzero. The left side of this equation is a quadratic function, namely a function of the form f(x) = ax2 + bx + c. The graph of a quadratic function is a special kind of curve known as a parabola. The simplest parabola is given by the function f(x) = x2. Its graph is shown in Figure 1.6.1. All other parabolas have the same shape, i.e. they can be made to look the same by the appropriate translations, rotations, reflections, and scale transformations.

Consider the quadratic function f(x) = ax2 + bx + c. The roots of this function are the solutions to the equation f(x) = 0. As we have seen, they are given by the quadratic formula, Equation (P.10.1), which we restate below:

• (P.10.1) x = (-b ± √b2 - 4ac) / 2a.

The quantity D = b2 - 4ac inside the radical is known as the discriminant of f(x). The sign of the discriminant determines the number of real roots of f. If D is positive, then f has two distinct real roots, if D is zero, then f has just one real root, and if D is negative, then f has no real roots.

When graphing a quadratic function, it is a good idea to first compute its roots.

• Example 1: Graph the quadratic function f(x) = x2 + x - 2.
• Solution: By trial and error, we see that f(x) factors as (x - 1)(x + 2). Thus the roots of f are 1 and -2. We compute values of f(x) for values of x near these roots. The following table summarizes the results:
x f(x)
-3
4
-2
0
-1
-2
0
-2
1
0
2
4
3
10

Now we plot a smooth curve connecting these points as usual, obtaining the following graph:

Below we give a slightly more complicated example.

• Example 2: Graph the quadratic function f(x) = 2 + 2x - x2.
• Solution: To find the roots of f, we apply the quadratic formula with a=-1, b=2, and c=2, obtaining

x = (-2 ± √42 -(4)(-1)(2)) / (2)(-1)

= (-2 ± √12)/-2

= 1 ± √3.

Thus the roots of f are 1 - √3 ≈ -0.73 and 1 + √3 ≈ 2.73. Once again, we compute values of f(x) for values of x near these roots.

x f(x)
-2
-6
-1
-1
-0.73
0
0
2
1
3
2
2
2.73
0
3
-1
4
-6

Finally, we plot these points and draw a smooth curve connecting them, obtaining the following graph: