I have the question what is the equation of the function in the form f(x) = 1/x-c for the function with a vertical asymptote at x=1? sketch the graph.
I am uncertain how I plug this into a graphing calculator. Could I substitute 1 for x and then put an x in for c to input this into my graphing calculator?
I know this is a reciprocal graph and it will also be a linear line.
To find the equation of the function in the given form, f(x) = 1/x - c, with a vertical asymptote at x = 1, you need to determine the value of c.
Since the given function has a vertical asymptote at x = 1, this means that the function approaches positive or negative infinity as x approaches 1 from both sides.
When x approaches 1 from the left side (x < 1), the function will approach negative infinity. Therefore, we can write:
lim x→1- (1/x - c) = -∞
To find the value of c, we need to plug in x = 1 in our function and equate it to -∞.
1/1 - c = -∞
Since dividing by zero gives us an undefined value, we can say that c must be such that 1 - c = 0.
Solving for c, we get c = 1.
Therefore, the equation of the function with a vertical asymptote at x = 1 is:
f(x) = 1/x - 1
To graph the function on a graphing calculator, you can indeed substitute 1 for x and then vary the value of c to observe the changes in the graph.
Start by entering the equation f(x) = 1/x - 1 into your graphing calculator.
Then, using the graphing capabilities of your calculator, plot points by inputting various x-values and observing the corresponding y-values. You can start with x = -10, -5, -3, -2, -1.5, -1.2, -1.1, -1.05, -1.01, -1.001, -1.0001, -0.9999, -0.999, -0.99, -0.95, -0.9, -0.8, -0.7, -0.5, -0.2, -0.1, 0.1, 0.2, 0.5, 0.7, 0.8, 0.9, 0.95, 0.99, 1.0001, 1.001, 1.01, 1.05, 1.1, 1.2, 1.5, 2, 3, 5, and 10. These points will help you sketch the graph of the function.
Remember that the function will have a vertical asymptote at x = 1, so you need to make sure that the graph shows a vertical line at x = 1, where the function approaches infinity.
By observing the pattern of points and the vertical asymptote, you can sketch the graph of the function.
is it
y = 1/x - c , like you typed it, or
y = 1/(x-c)
comparing it to y = 1/x
for the first interpretation the graph of y = 1/x would simply be translated vertically by c
for the 2nd interpretation the graph of y = 1/x would be translated horizontally by -c , e.g
if y = 1/(x+2) is a shift of 2 to the left
if y = 1/(x-5) is a shift of 5 to the right