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Unformatted text preview: ∞, 3]
and decreasing on [3, ∞). The relative maximum occurs at the point (3, 1) with 1 being both the
relative and absolute maximum value of g .
Hopefully the previous examples have reminded you of some of the basic characteristics of the
graphs of quadratic equations. First and foremost, the graph of y = ax2 + bx + c where a, b, and
c are real numbers with a = 0 is called a parabola. If the coeﬃcient of x2 , a, is positive, the
parabola opens upwards; if a is negative, it opens downwards, as illustrated below.1
Graphs of y = ax2 + bx + c. The point at which the relative minimum (if a > 0) or relative maximum (if a < 0) occurs is called
the vertex of the parabola. Note that each of the parabolas above is symmetric about the dashed
vertical line which contains its vertex. This line is called the axis of symmetry of the parabola.
As you may recall, there are two ways to quickly ﬁnd the vertex of a parabola, depending on which
form we are given. The results are summarized below.
Equation 2.4. Vertex Formulas for Quadratic Functions: Suppose a, b, c, h, and k are
real numbers with a = 0.
• If f (x) = a(x − h)2 + k , the vertex of the graph of y = f (x) is the point (h, k ).
• If f (x) = ax2 + bx + c, the vertex of the graph of y = f (x) is the point − b
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