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Chapter 9: Q. 515 (page 987)

In the following exercises, (a) rewrite each function in f(x)=ax-h2+kform and (b) graph it using properties.

f(x)=-3x2-12x-5

Short Answer

Expert verified

(a) The function can be written as f(x)=-3x-22+7.

(b) The graph of the function, axis of symmetry x=-2and the vertex -2,7shown:

Step by step solution

01

Step 1. Given information.

We have:

fx=-3x2-12x-5

02

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

The function is fx=-3x2-12x-5.

Add the same value to both sides:

f(x)=-3x2+4x+5

Subtract3脳4to the both sides:

role="math" localid="1645554479048" f(x)-3路4=-3x2+4x+4-5f(x)-12=-3x+22-5f(x)=-3x+22-5+12f(x)=-3x+22+7

03

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

The function is fx=-3x2-12x-5.

Identify the coefficients by comparing the function with y=ax2+bx+c, it follows:

x=-b2ax=--122-3x=-2

04

Part (b). Step 2. Find vertex. 

The vertex of an up-down facing parabola of the form y=ax2+bx+cis xv=-b2a.

xv=-b2axv=--122-3xv=-2

Substitute xv=-2into the original equation to get yv:

yv=7

Therefore, the parabola vertex isyv=7.

05

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

Draw the smooth curve through the points. The points represents the function:

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