Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 8 - Section 8.3 - Quadratic Functions and Their Graphs - Exercise Set - Page 626: 31

Answer

The graph is shown below. Range $(-\infty,4]$.

Work Step by Step

The given function is a quadratic function: $f(x)=2x-x^2+3$ Rewrite as $f(x)=-x^2+2x+3$ The standard form of the quadratic function is $f(x)=ax^2+bx+c$ Compare both equations $a=-1,b=2$ and $c=3$. Step 1:- Parabola opens. $a<0$, the parabola open downward. Step 2:- Vertex. $x-$coordinate of the vertex is $x=-\frac{b}{2a}=-\frac{2}{2(-1)}=1$. Substitute the value of $x$ into the function. $\Rightarrow f(1)=-(1)^2+2(1)+3$ Simplify. $\Rightarrow f(1)=-1+2+3$ $\Rightarrow f(1)=4$ The vertex is $(1,4)$. Step 3:- $x-$intercepts. Replace $f(x)$ with $0$ into the given function. $\Rightarrow 0=-x^2+2x+3$ Use quadratic formula, we have $a=-1,b=2$ and $c=3$. $\Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ Substitute all values. $\Rightarrow x=\frac{-(2)\pm\sqrt{(2)^2-4(-1)(3)}}{2(-1)}$ Simplify. $\Rightarrow x=\frac{-2\pm\sqrt{4+12}}{-2}$ $\Rightarrow x=\frac{-2\pm\sqrt{16}}{-2}$ $\Rightarrow x=\frac{-2\pm4}{-2}$ Separate the fractions. $\Rightarrow x=\frac{-2+4}{-2}$ or $x= \frac{-2-4}{-2}$ Simplify. $\Rightarrow x=\frac{2}{-2}$ or $ x=\frac{-6}{-2}$ $\Rightarrow x=-1$ or $ x=3$ The $x-$intercepts are $-1$ and $3$. The parabola passes through $(-1,0)$ and $(3,0)$. Step 4:- $y-$intercept. Replace $x$ with $0$ in the given function. $\Rightarrow f(0)=-(0)^2+2(0)+3$ Simplify. $\Rightarrow f(0)=3$ The $y-$intercept is $3$. The parabola passes through $(0,3)$. Step 5:- Graph. Use the points:- vertex, $x-$intercepts and $y-$intercept to draw a parabola. The axis of symmetry is $x=1$. $A=(1,4)$ $B=(-1,0)$ $C=(3,0)$ $D=(0,3)$. From the graph the range of the function is $(-\infty,4]$.
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