Q. 34 - 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 -  $f\left(x\right)=-x^2+4x$

Step-by-Step Solution

Answer

(a). 

• The graph is open downwards.
• The vertex of graph is - (2, 4)
• 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  -  $y\in (-\infty ,4]$
  

(c).

• Function increasing for   $x\in (-\infty ,2]$
• Function decreasing for  $x\in [2,+\infty )$

Part (a). Step 1. Given data

Function -  $f\left(x\right)=-x^2+4x$

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.

$-\frac{b}{2a}=-\frac{4}{2(-1)}$

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

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

$f(-\frac{b}{2a})=-{\left(2\right)}^2+4\times 2$

$f(-\frac{b}{2a})=4$

Vertex is -   (2, 4)

Determine the Axis of symmetric -

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{4}{2(-1)} \\ x=2 \end{gathered}$$

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 -

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

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

(x= 0 and x= 4)

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

$f\left(0\right)=-0^2+4\times 0$

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

$y=0$

Part (b). Step 1. Given data

Function -  $f\left(x\right)=-x^2+4x$

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 -  $y\in (-\infty ,4]$

Part (c). Step 1. Given data

Function -  $f\left(x\right)=-x^2+4x$

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 -  $x\in (-\infty ,2]$
• Function decreasing for  -  $x\in [3,+\infty )$
  

Graph

The graph of function $f\left(x\right)=-x^2+4x$ will be -