Use a graphing calculator to graph f(x)=(2x^2-8)/(x^2-6) and then select the response which is true
a) f increases without bound
b) f has no tangent line
c) f has a horizontal tangent line
d) none of these responses
as x becomes very large , the -8 and -6 become inconsequential
the function approaches 2
of course, as x -> ±√6 f increases and decreases without bound, but that's probably not what you are looking for.
To graph the function f(x) = (2x^2 - 8)/(x^2 - 6) using a graphing calculator, follow these steps:
1. Turn on your graphing calculator and navigate to the graphing function screen.
2. Enter the equation f(x) = (2x^2 - 8)/(x^2 - 6) into the calculator.
3. Make sure your calculator is set to the appropriate window, which should cover a suitable range of x-values.
4. Press the "Graph" button to generate the graph.
After plotting the graph, you will be able to determine which of the given responses (a, b, c, or d) is true.
Analyzing the graph, we can make the following observations:
a) f increases without bound: Look at the behavior of the graph as x approaches positive or negative infinity. If the graph trend continuously moves upward or downward without reaching a finite value, then the response may be true. Examine the behavior of the graph in these scenarios.
b) f has no tangent line: A function may have no tangent line at certain points if the function has a sharp corner, vertical asymptote, or a hole in the graph. Check for such features in the graph to determine if the response may be true.
c) f has a horizontal tangent line: A function may have a horizontal tangent line at a certain point if the slope of the function at that point is zero. This is typically indicated by a flat portion of the graph. Look for areas where the graph appears to flatten out horizontally to evaluate if this response is true.
d) none of these responses: If the graph does not exhibit any of the characteristics described in options a, b, or c, then the correct response may be "none of these."
By carefully observing the graph, compare it to the criteria outlined above to determine which response is true.