Graphing a function and finding the domain and range

[land_of_ice]

Homework Statement

y= -4(√4x-1) -1
in case it's not clear, the 4x-1 part is all under the radical
the question is to graph and find the domain and range

Homework Equations

The Attempt at a Solution

This is probably what your supposed to do , but not sure, OK
To get the domain set everything under the radical to zero and solve for x, x is 1/4 so that's the domain, but how to get the range??
To graph this, first graph the basic graph of y = √
Then, does it matter what order you do the other things in ?
Some of the other things to do apparently, are:
With regards to the -1 at very right side of the problem means to move the graph to the right along the x axis?
The -4 means to reflect about the y-axis and shrink by 4 ?

Confused - There are more steps to this or different steps or in different order right?

 

[vela]

Homework Statement

y= -4(√4x-1) -1
in case it's not clear, the 4x-1 part is all under the radical
the question is to graph and find the domain and range

Homework Equations

The Attempt at a Solution

This is probably what your supposed to do , but not sure, OK
To get the domain set everything under the radical to zero and solve for x, x is 1/4 so that's the domain, but how to get the range??

Not quite. The domain consists of all real values of x for which the function is defined. You've only specified one point. When you solve the equation 4x-1=0, you're finding the boundary between where y(x) is defined and not defined. You still need to figure out which side is which. The function is defined only when 4x-1≥0. If you solve the inequality for x, you get x≥1/4. That's the domain.

To graph this, first graph the basic graph of y = √
Then, does it matter what order you do the other things in ?

Yes, the order matters, but there's not just one ordering that works. You have some leeway here.

Some of the other things to do apparently, are:
With regards to the -1 at very right side of the problem means to move the graph to the right along the x axis?

No, it shifts the function down by 1 in the vertical direction.

The -4 means to reflect about the y-axis and shrink by 4?

The negative sign does indicate a reflection, but the factor of 4 expands, not shrinks, the function in the vertical direction.

Confused - There are more steps to this or different steps or in different order right?

Yes. For one thing, you haven't accounted for what's under the radical yet.

Start from the graph of $$y=\sqrt{x}$$ and do one thing at a time until you get to the original function. At each step, figure out what happens to the graph. For instance, when you go from

$$y=\sqrt{x}$$

to

$$y=\sqrt{4x}$$,

you compress the graph in the horizontal direction. Now figure out what happens to the new graph when you do another transformation. Keep adding transformations until you get to the desired function. For instance, the following progression will get you to the desired graph:

$$\begin{align*} y & = \sqrt{4(x-1/4)} = \sqrt{4x-1} \\ y & = 4\sqrt{4x-1} \\ y & = -4 \sqrt{4x-1} \\ y & = -4 \sqrt{4x-1}-1 \end{align*}$$

Explain what happens at each step.

 

[land_of_ice]

Not quite. The domain consists of all real values of x for which the function is defined. You've only specified one point. When you solve the equation 4x-1=0, you're finding the boundary between where y(x) is defined and not defined. You still need to figure out which side is which. The function is defined only when 4x-1≥0. If you solve the inequality for x, you get x≥1/4. That's the domain.

Yes, the order matters, but there's not just one ordering that works. You have some leeway here.

No, it shifts the function down by 1 in the vertical direction.

The negative sign does indicate a reflection, but the factor of 4 expands, not shrinks, the function in the vertical direction.

Yes. For one thing, you haven't accounted for what's under the radical yet.

Start from the graph of $$y=\sqrt{x}$$ and do one thing at a time until you get to the original function. At each step, figure out what happens to the graph. For instance, when you go from

$$y=\sqrt{x}$$

to

$$y=\sqrt{4x}$$,

you compress the graph in the horizontal direction. Now figure out what happens to the new graph when you do another transformation. Keep adding transformations until you get to the desired function. For instance, the following progression will get you to the desired graph:

$$\begin{align*} y & = \sqrt{4(x-1/4)} = \sqrt{4x-1} \\ y & = 4\sqrt{4x-1} \\ y & = -4 \sqrt{4x-1} \\ y & = -4 \sqrt{4x-1}-1 \end{align*}$$

Explain what happens at each step.

Thanks, you answered every little thing that was important about the question - Excellent :) Once again, thanks :) :)