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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q.480. (page 985)

Find (a) the equation of the axis of symmetry and (b) the vertex.
$y = - x^{2} + 6 x + 8$

Short Answer

Expert verified

Part (a). The equation of the axis of symmetry is $x = 3$ .

Part (b). The vertex is $(3, 17)$ .

Step by step solution

01

Part (a). Step 1. Given information.

The given equation is $y = - x^{2} + 6 x + 8$ .

02

Part (a). Step 2. Determine the equation of the axis of symmetry.

The axis of symmetry is the vertical line given by $x = - \frac{b}{2 a}$ .

Compare the given equation $y = - x^{2} + 6 x + 8$ with $y = a x^{2} + b x + c$ .

So, here $a = - 1, b = 6, c = 8$ .

The required equation of symmetry is:

$\begin{array}{rcl} x & = & - \frac{b}{2 a} \\ = & - \frac{6}{2 (- 1)} \\ = & - \frac{6}{- 2} \\ = & 3 \end{array}$

03

Part (b). Step 1. Given information.

The given equation is $y = - x^{2} + 6 x + 8$ .

04

Part (b). Step 2. Determine the vertex.

Substitute $x = 3$ in the given equation $y = - x^{2} + 6 x + 8$ .

$\begin{array}{rcl} y (3) & = & - {(3)}^{2} + 6 (3) + 8 \\ = & - 9 + 18 + 8 \\ = & - 1 + 18 \\ = & 17 \end{array}$

Hence, the required vertex is $(3, 17)$ .

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