Question

I need some help please. Given the functions, complete the sections. a) Find the intercepts with the axes. b) Indicate the basic function that you will use to graph it. c) Identify the transformations that your graph will undergo, starting from its basic function. d) Draw the sketch showing the transformations, taking into account all the previous sections. Highlight f(x) with a pen or marker. e) Determine its domain and range.$f(x) = - \frac{1}{4} \sqrt{8 - 4x} + 1$

Answer 1

$f(x)=-\frac{1}{4}\sqrt[]{8-4x}+1$

a)

The x-intercept can be found as:

$\begin{gathered} f(x)=0 \\ so\colon \\ -\frac{1}{4}\sqrt[]{8-4x}+1=0 \\ -\frac{1}{4}\sqrt[]{8-4x}=-1 \\ \sqrt[]{8-4x}=4 \\ 8-4x=16 \\ 4x=8-16 \\ 4x=-8 \\ x=-\frac{8}{4} \\ x=-2 \end{gathered}$

Therefore, the x-intercept is: (-2,0)

The y-intercept can be found evaluating the function for x = 0, so:

$\begin{gathered} f(0)=-\frac{1}{4}\sqrt[]{8-4(0)}+1 \\ f(0)=1-\frac{\sqrt[]{2}}{2} \\ f(0)\approx0.29 \end{gathered}$

b) The parent function for this is given by:

$g(x)=\sqrt[]{x}$

c)

1st: A reflection over y-axis:

$\begin{gathered} y=g(-x) \\ y=\sqrt[]{-x} \end{gathered}$

2nd: A horizontal compression:

$\begin{gathered} y=g(-4x) \\ y=\sqrt[]{-4x} \end{gathered}$

3rd: A horizontal translation 8 units to the left:

$\begin{gathered} y=g(x+8) \\ y=\sqrt[]{-4x+8}=\sqrt[]{8-4x} \end{gathered}$

4th: A reflection over y-axis:

$\begin{gathered} y=-g(x) \\ y=-\sqrt[]{8-4x} \end{gathered}$

5th: A vertical compression:

$\begin{gathered} y=\frac{1}{4}g(x) \\ y=-\frac{1}{4}\sqrt[]{8-4x} \end{gathered}$

6th: A vertical translation 1 unit up:

$\begin{gathered} y=g(x)+1 \\ y=-\frac{1}{4}\sqrt[]{8-4x}+1 \end{gathered}$

d)

Where the blue graph is the parent function:

$g(x)=\sqrt[]{x}$

And the red graph is the function after the transformations:

$f(x)=-\frac{1}{4}\sqrt[]{8-4x}+1$

e)

The domain and the range are:

$\begin{gathered} D\colon\mleft\lbrace x\in\R\colon x\le2\mright\rbrace \\ R\colon\mleft\lbrace y\in\R\colon y\le1\mright\rbrace \end{gathered}$