\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the domain and range of \(\displaystyle f(x)=2 x^{2} + 32 x + 142 \)
The domain of every polynomial is all real numbers. The sign of \(\displaystyle a\) is positive so the function has a minimum value at the vertex. The vertex is located at \(\displaystyle h=\frac{-b}{2a}=-8\) and \(\displaystyle k=f(h)=14\). This gives the range as \(\displaystyle [14,\infty)\)
\begin{question}Find the domain and range of $f(x)=2 x^{2} + 32 x + 142 $
\soln{9cm}{The domain of every polynomial is all real numbers. The sign of $a$ is positive so the function has a minimum value at the vertex. The vertex is located at $h=\frac{-b}{2a}=-8$ and $k=f(h)=14$. This gives the range as $[14,\infty)$}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the domain and range of <img class="equation_image" title=" \displaystyle f(x)=2 x^{2} + 32 x + 142 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D2%20x%5E%7B2%7D%20%2B%2032%20x%20%2B%20142%20%20" alt="LaTeX: \displaystyle f(x)=2 x^{2} + 32 x + 142 " data-equation-content=" \displaystyle f(x)=2 x^{2} + 32 x + 142 " /> </p> </p><p> <p>The domain of every polynomial is all real numbers. The sign of <img class="equation_image" title=" \displaystyle a " src="/equation_images/%20%5Cdisplaystyle%20a%20" alt="LaTeX: \displaystyle a " data-equation-content=" \displaystyle a " /> is positive so the function has a minimum value at the vertex. The vertex is located at <img class="equation_image" title=" \displaystyle h=\frac{-b}{2a}=-8 " src="/equation_images/%20%5Cdisplaystyle%20h%3D%5Cfrac%7B-b%7D%7B2a%7D%3D-8%20" alt="LaTeX: \displaystyle h=\frac{-b}{2a}=-8 " data-equation-content=" \displaystyle h=\frac{-b}{2a}=-8 " /> and <img class="equation_image" title=" \displaystyle k=f(h)=14 " src="/equation_images/%20%5Cdisplaystyle%20k%3Df%28h%29%3D14%20" alt="LaTeX: \displaystyle k=f(h)=14 " data-equation-content=" \displaystyle k=f(h)=14 " /> . This gives the range as <img class="equation_image" title=" \displaystyle [14,\infty) " src="/equation_images/%20%5Cdisplaystyle%20%5B14%2C%5Cinfty%29%20" alt="LaTeX: \displaystyle [14,\infty) " data-equation-content=" \displaystyle [14,\infty) " /> </p> </p>