Unformatted text preview: f
2a − b
2a . Example 2.3.2. Use Equation 2.4 to ﬁnd the vertex of the graphs in Example 2.3.1.
1. The formula f (x) = x2 − 4x + 3 is in the form f (x) = ax2 + bx + c. We identify a = 1, b = −4,
and c = 3, so that
f − b
2a = f (2) = −1, so the vertex is (2, −1) as previously stated.
1 We will justify the role of a in the behavior of the parabola later in the section. 142 Linear and Quadratic Functions 2. We see that the formula g (x) = −2(x − 3)2 + 1 is in the form g (x) = a(x − h)2 + k . We identify
a = −2, x − h as x − 3 (so h = 3), and k = 1 and get the vertex (3, 1), as required.
The formula f (x) = a(x − h)2 + k , a = 0 in Equation 2.4 is sometimes called the standard form
of a quadratic function; the formula f (x) = ax2 + bx + c, a = 0 is sometimes called the general
form of a quadratic function.
To see why the formulas in Equation 2.4 produce the vertex, let us ﬁrst consider a quadratic function
in standard form. If we consider the graph of the equation y = a(x − h)2 + k we see that when
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