Use a graphing calculator to graph f(x) = (4x^2 - 1) / (x^2 - 9) and then select the response which is true.

Question

f has a horizontal tangent line

f increases without bound

f has no tangent line

none of these responses

	f has a horizontal tangent line
		f increases without bound
		f has no tangent line
		none of these responses

Answer 1

Mark gave you the analytical solution.  I will give you the solution using the graph on your calculator.

 

 

Since you are dealing with a rational function, the graph will be a curved shape.  Curved shapes always has a tangent line at the curve.  So by process of elimination, "f has no tangent line" is not an answer choice here.  Only straight lines have "no tangent lines".

 

If you see a maximum (peak) or minimum (depression) in the shape of the graph, then "f has a horizontal tangent line".  To verify this, go to 2ND, CALC, and scroll down to select HORIZONTAL on your calculator.  Then, move that HORIZONTAL to a point on that graph where the graph does not intersect the horizontal line.  If there is such a point, then "f has a horizontal tangent" is a choice.

 

You can use the table of values on your calculator.  If f increases infinitely as x increases infinitely, then "f increases without bound" is a choice.  Also, you know that rational functions have vertical asymptotes as well as horizontal asymptotes, so there is a boundary.

 

 

 

Answer 2

Hi Veronica,

 

I don't know about the graphing calculator statement of this problem. But, we should be able to answer it without a calculator.

 

Let's look at the first option. If f has a horizontal tangent line anywhere, this means that its first derivative somewhere must be 0. Let's calculate the first derivative, using the quotient rule:

 

f(x) = (4x² - 1) / (x² - 9)

 

First, let's note that the function has divide-by-zero problems, or vertical asymptotes at x = +/-3.

 

df/dx = [(x² - 9)(8x) - (4x² - 1)(2x)]/(x2 - 9)² = 0

 

We can cross multiply the above to get

 

(x² - 9)(8x) - (4x² - 1)(2x) = 0

 

Then

 

8x³ - 72x - 8x³ +2x = 0

 

The cubic terms cancel and we can write

 

-70x = 0

 

Or

 

x = 0

 

So, yes, there is a horizontal tangent line at x = 0. I choose this answer.

 

Since we found a horizontal tangent line, then certainly the response "f has no tangent line" cannot be right.

 

What about the response "f increases without bound"?

 

We have

 

f(x) = (4x² - 1) / (x² - 9)

 

When x gets very large, the square terms in the numerator and the denominator dominate, and the function goes like:

 

f(x) = (4x²)/(x²) → 4

 

So, as x gets very large, f(x) shall approach a horizontal asymptote of f(x) = 4, hence it does not grow without bound.

 

Therefore, your best answer is "f has a horizontal tangent line"