Q. 45 - 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 -  $f\left(x\right)=-4x^2-6x+2$

Step-by-Step Solution

Answer

(a). 

• The graph is open downwards.
• The vertex of graph is - (-0.75, 4.25)
• 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  -   $y\in (-\infty , 4.25]$

(c).

• Function increasing for -  $x\in (-\infty , -0.75]$
• Function decreasing for -  $x\in [-0.75, +\infty )$

Part (a). Step 1. Given data

Function -  $f\left(x\right)=-4x^2-6x+2$

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.

$$\begin{gathered} -\frac{b}{2a}=-\frac{-6}{2\times (-4)} \\ -\frac{b}{2a}=-0.75 \end{gathered}$$

$$\begin{gathered} f(-\frac{b}{2a})=f(-0.75) \\ f(-\frac{b}{2a})=-4\times {(-0.75)}^2-6\times (-0.75)+2 \\ f(-\frac{b}{2a})=-2.25+4.5+2 \\ f(-\frac{b}{2a})=4.25 \end{gathered}$$

Vertex is -   (-0.75, 4.25)

Determine the Axis of symmetric -

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{-6}{2\times (-4)} \\ x=-0.75 \end{gathered}$$

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 -

$$\begin{gathered} f\left(x\right)=0 \\ -4x^2-6x+2=0 \\ {x}_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ {x}_{1,2}=\frac{-(-6)\pm \sqrt{{(-6)}^2-4\times (-4)\times 2}}{2\times (-4)} \\ {x}_{1,2}=\frac{6\pm 8.24}{-8} \end{gathered}$$

The point where the function intersect the x-axis are -  

$\mathbf{(}{\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{ }\mathbf{0}\mathbf{.}\mathbf{281}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{781}\mathbf{)}\mathbf{ }$

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

$$\begin{gathered} f\left(0\right)=-4\times 0^2-6\times 0+2 \\ f\left(0\right)=2 \end{gathered}$$

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

$y=2$

Part (b). Step 1. Given data

Function -  $f\left(x\right)=-4x^2-6x+2$

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 -  $y\in (-\infty , 4.25]$

Part (c). Step 1. Given data

Function -  $f\left(x\right)=-4x^2-6x+2$

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  -   $x\in (-\infty , -0.75]$
• Function decreasing for  -   $x\in [-0.75, +\infty )$

Graph

The graph of function will be -