Q. 5 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 4 and 5,

(a) Determine whether the graph opens up or down.

(b) Determine the vertex of the graph of the quadratic function.

(c) Determine the axis of symmetry of the graph of the quadratic

function.

(d) Determine the intercepts of the graph of the quadratic function.

(e) Use the information in parts (a)–(d) to graph the quadratic

function.

(f) Based on the graph, determine the domain and the range of

the quadratic function.

(g) Based on the graph, determine where the function is

increasing and where it is decreasing.

$g\left(x\right)=-2x^2+4x-5$

Step-by-Step Solution

Answer

(a) Graph is open down.

(b) Vertex$=(1,-3)$

(c) Axis of symmetry $x=1$

(d) y-intercept $(0,-5)$. No x-intercept.

(f) Domain$=(-\infty ,\infty )$ and Range$=(-\infty ,-3]$

(g) The function is decreasing in $[1,\infty )$ and increasing in .

(e) 

Part (a) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (a) Step 2. Calculation

Comparing the given function with $f\left(x\right)=a{x}^2+bx+c$ we get,

$a=-2,b=4 \& c=-5$

Hence $a<0$ then the graph is open down.

Part (b) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (b) Step 2. Calculation

Since $g\left(x\right)=-2x^2+4x-5$

$=-2{(x-1)}^2-3$

Therefore vertex is at $(1,-3)$

Part (c) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (c) Step 2. Calculation

Since vertex is at $(1,-3)$

The graph of the function is symmetric about $x=1$

Part (d) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (d) Step 2. Calculation

For y-intercept $x=0$,

Then $f\left(0\right)= -2.0+4.0-5$

$=-5$

Therefore y-intercept is $(0,-5)$

For x-intercept, $16-4.(-2).(-5)<0$

Hence there is no x-intercept.

Part (e) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (e) Step 2. Graph

Part (f) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (f) Step 2. Calculation

The domain of the given function is $(-\infty ,\infty )$.

The range of the function is $(-\infty ,-3]$

Part (g) Step 1. Given Information

$g\left(x\right)=-2x^2+4x-5$

Part (g) Step 2. Calculation

By the graph, the function is increasing in the interval $(-\infty ,1]$ and the function is decreasing in the interval $[1,\infty )$.