Graph the function. Plot two points and the asymptotes of the graph of the function
g(x)=3/2log(3 is lower next to log)
(x-2)-1
We are not able to draw graphs on these boards.
Here is a simple graphing program.
http://www.coolmath.com/graphit/
enter the function value without the y = part
e.g. for y = x^2 - 4
just enter x^2-4, then click on the "eval"
at the bottom you will see the coordinates of the cursor.
To graph the function g(x) = (3/2)log(x-2)-1, we can follow these steps:
Step 1: Determine the asymptotes:
Since the function involves a logarithm, we need to find the vertical asymptote and the horizontal asymptote.
Vertical asymptote: The vertical asymptote occurs when the argument of the logarithm becomes zero. Therefore, set (x-2) = 0, and solve for x:
x - 2 = 0
x = 2
So, the vertical asymptote is x = 2.
Horizontal asymptote: To find the horizontal asymptote, we need to analyze the behavior of the function as x approaches positive and negative infinity. The coefficient in front of the logarithm determines the horizontal asymptote. In this case, the coefficient is (3/2), so the horizontal asymptote is y = (3/2).
Step 2: Plot the points:
Choose some x-values and calculate the corresponding y-values to plot the points on the graph. Let's select x = 3 and x = 4.
For x = 3:
g(3) = (3/2)log(3-2)-1 = (3/2)log(1)-1 = (3/2)(0)-1 = -1
Therefore, the first point is (3, -1).
For x = 4:
g(4) = (3/2)log(4-2)-1 = (3/2)log(2)-1
To calculate the value of log(2), you can use a calculator to find that log(2) ≈ 0.301.
g(4) = (3/2)(0.301)-1 = 0.451-1 = -0.549
Therefore, the second point is (4, -0.549).
Step 3: Plot the points on the graph:
Now, draw a graph with the vertical asymptote at x = 2 and the horizontal asymptote at y = (3/2). Plot the two points (3, -1) and (4, -0.549).
The graph should resemble an upward curve approaching the vertical asymptote and moving towards the horizontal asymptote.
Note: Since the logarithm is undefined for x ≤ 2, the graph is only valid for x > 2.
To graph the function g(x) = (3/2)log(x-2)-1, you can follow these steps:
Step 1: Determine the vertical asymptote:
The vertical asymptote is the x-value at which the function approaches infinity. In this case, the vertical asymptote occurs when the argument of the logarithm, (x-2), becomes equal to zero. So, set x-2 = 0 and solve for x:
x - 2 = 0
x = 2
Therefore, the vertical asymptote is x = 2.
Step 2: Find two points on the graph:
To find two points on the graph, you can choose any x-values (except for x = 2, which is the vertical asymptote) and calculate the corresponding y-values.
Let's choose x = 0:
g(0) = (3/2)log(0-2)-1
g(0) = (3/2)log(-2)-1
Since log(-2) is undefined, we cannot use x = 0.
Let's choose x = 3:
g(3) = (3/2)log(3-2)-1
g(3) = (3/2)log(1)-1
g(3) = (3/2)(0)-1
g(3) = -1
So, one point on the graph is (3, -1).
Step 3: Plot the points and asymptote:
Plot the point (3, -1) on the graph. Since the vertical asymptote is x = 2, draw a vertical dashed line at x = 2.
Now, you have a point (3, -1) on the graph and the vertical asymptote x = 2. You can sketch the graph by connecting the points and following the shape of the logarithmic function.
Note: The graph of a logarithmic function typically approaches the asymptote as x approaches negative infinity and positive infinity.