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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. 516 (page 987)

In the following exercises, (a) rewrite each function in $f (x) = a {(x - h)}^{2} + k$ form and (b) graph it using properties.
$f (x) = 2 x^{2} - 12 x + 7$

Short Answer

Expert verified

(a) The function can be written as $f (x) = 2 {(x - 3)}^{2} - 11$ .

(b) The graph of function, axis of symmetry $x = 3$ and vertex $(3, - 11)$ shown.

Step by step solution

01

Step 1. Given information.

We have:

$f (x) = 2 x^{2} - 12 x + 7$

02

Part (a). Step 1. Rewrite the function in standard form.

The function is $f (x) = 2 x^{2} - 12 x + 7$ .

Add the same value to both sides:

$f (x) = 2 (x^{2} - 6 x) + 7$

Add $2 脳 9$ to the both sides:

$f (x) + 18 = 2 (x^{2} - 6 x + 9) + 7 f (x) + 18 = 2 {(x - 3)}^{2} + 7 f (x) = 2 {(x - 3)}^{2} + 7 - 18 f (x) = 2 {(x - 3)}^{2} - 11$

03

Part (b). Step 1. Find axis of symmetry.

The function is $f (x) = 2 x^{2} - 12 x + 7$ .

Identify the coefficients by comparing the function with $y = a x^{2} + b x + c$ , it follows:

$a = 2, b = - 12 x = - \frac{b}{2 a} x = - \frac{- 12}{2 路 2} x = 3$

04

Part (b). Step 2. Find vertex.

The vertex of an up-down facing parabola of the form $y = a x^{2} + b x + c 鈥�$ is $x_{v} = - \frac{b}{2 a}$ .

The parabola params are:

$a = 2, 鈥� b = - 12, 鈥� c = 7 x_{v} = - \frac{b}{2 a} x_{v} = - \frac{(- 12)}{2 路鈥� 2} x_{v} = 3$

Substitute $x_{v} = 3$ into the original function to find $y_{v} :$

$y_{v} = - 11$

Therefore, the parabola vertex is:

$(3, 鈥� - 11)$

05

Part (b). Step 3. Find the graph using properties.

Plot the vertex $(3, - 11)$ . Plot the axis of symmetry $x = 3$ .

Draw the smooth curve through the points. The points represents the function $f (x) = 2 x^{2} - 12 x + 7$ :

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