Q. 41 - Precalculus Enhanced with Graphing Utilities

Question

(a).  Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any. 

(b).  Determine the domain and the range of the function. 

(c).  Determine where the function is increasing and where it is decreasing. 

Verify your results using a graphing utility.

Function -  $f\left(x\right)=-2x^2+2x-3$

Step-by-Step Solution

Answer

(a). 

• The graph is open downwards.
• The vertex of graph is - (0.5, -2.5)
• The axis of symmetry passes through the vertex of the function and is parallel to the y-axis.

(b). 

• The domain is - $\mathrm{ℝ}$
• The range is  -   $y\in (-\infty ,-2.5]$
  

(c).

• Function increasing for -   $x\in (-\infty ,0.5]$
• Function decreasing for -  $x\in [0.5,+\infty )$
  

Part (a). Step 1. Given data

Function -  $f\left(x\right)=-2x^2+2x-3$

Part (a). Step 2. To Find

Graph each quadratic function by determining whether its graph opens up or down and by finding its vertex, axis of symmetry, y-intercept, and x-intercepts, if any.

Part (a). Step 3. Explanation

Remind yourself that the quadratic function is given by -    $f\left(x\right)=a{x}^2+bx+c, a\ne 0$

Since $a<0$,We can conclude that the function open downwards. we are looking the vertex as $[-\frac{b}{2a},f(-\frac{b}{2a}\left)\right]$, Lets us find the vertex.

$-\frac{b}{2a}=-\frac{2}{2(-2)}$

$-\frac{b}{2a}=0.5$

$$\begin{gathered} f(-\frac{b}{2a})=f(0.5) \\ f(-\frac{b}{2a})=-2\times 0.5^2+2\times 0.5-3 \\ f(-\frac{b}{2a})=-0.5+1-2 \\ f(-\frac{b}{2a})=-2.5 \end{gathered}$$

Vertex is -   (0.5, -2.5)

Determine the Axis of symmetric -

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{2}{2(-2)} \\ x=0.5 \end{gathered}$$

The axis of symmetric passes through the vertex of the function and is parallel to the y-axis.

Determine the point at which the function intersects the x-axis -

$$\begin{gathered} f\left(x\right)=0 \\ -2x^2+2x-3=0 \end{gathered}$$

This function does not intersect the x-axis.

Determine the point at which the function intersects the y-axis -

$$\begin{gathered} f\left(0\right)=-2\times 0^2+2\times 0-3 \\ f\left(0\right)=-3 \end{gathered}$$

The point at which the function intersect the y-axis is -  

$y=-3$

Part (b). Step 1. Given data

Function -   $f\left(x\right)=-2x^2+2x-3$

Part (b). Step 2. To Find

Determine the domain and the range of the function.

Part (b). Step 3. Explanation

• The Domain is -  $\mathrm{ℝ}$
• And the range is  -   $y\in (-\infty ,-2.5]$
  

Part (c). Step 1. Given data

Function -   $f\left(x\right)=-2x^2+2x-3$

Part (c). Step 2. To Find

Determine where the function is increasing and where it is decreasing.

Part (c). Step 3. Explanation

Function increasing for  -  $x\in (-\infty ,0.5]$

Function decreasing for  -  $x\in [0.5,+\infty )$

Graph

The graph of function will be -