Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 2 - Section 2.2 - Quadratic Functions - Exercise Set - Page 330: 37

Answer

The required parabola is as shown below.

Work Step by Step

So, for the provided quadratic equation $f\left( x \right)=2x-{{x}^{2}}-2$ , when compared with the standard form of the quadratic equation $f\left( x \right)=a{{x}^{2}}+bx+c$ , one gets: $\begin{align} & a=-1, \\ & b=2, \\ & c=-2 \end{align}$ And use the steps shown below to determine the graph of the quadratic equation. Step 1: Determine how the parabola opens: Note that a, the coefficient of ${{x}^{2}}$ , is -1. If $a>0$ , the parabola opens in the downward direction. Also, if $\left| a \right|$ is small, the parabola opens more flatly than if $\left| a \right|$ is large. Now, from the provided equation of the function, it is observed that the graph opens downwards as $a<0$. Step 2: Calculate the vertex: The x-coordinate can be calculated as: $\begin{align} & x=-\frac{b}{2a} \\ & =-\frac{-2}{2\times \left( -1 \right)} \\ & =1 \end{align}$ The y-coordinate can be calculated as: $\begin{align} & y=2\left( 1 \right)-{{\left( 1 \right)}^{2}}-2 \\ & =2-1-2 \\ & =-1 \end{align}$ So, the vertex is $\left( 1,-1 \right)$. Step 3: And above steps lead to the parabola that opens downwards and has a vertex at $\left( 1,-1 \right)$ and y-axis at $-2$. So, the required parabola is as shown above.
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