Graph the rational function
F(x)=6/-x-3
Draw the vertical and horizontal asymptotes then plot two points on each side of the graph
To graph the rational function f(x) = 6/(-x - 3), we need to find the vertical and horizontal asymptotes first.
Vertical Asymptote:
The vertical asymptote occurs when the denominator is equal to zero.
Set -x - 3 = 0 and solve for x:
-x = 3
x = -3
So, the vertical asymptote is x = -3.
Horizontal Asymptote:
To find the horizontal asymptote, we examine the degree of the numerator and denominator.
The degree of the numerator is 0, and the degree of the denominator is 1.
Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y = 0.
Now, let's plot two points on each side of the graph:
When x = -4:
f(-4) = 6/(-(-4) - 3) = 6/(-1) = -6
So, we have the point (-4, -6).
When x = -2:
f(-2) = 6/(-(-2) - 3) = 6/(-5) = -6/5
So, we have the point (-2, -6/5).
When x = -8:
f(-8) = 6/(-(-8) - 3) = 6/5
So, we have the point (-8, 6/5).
When x = -6:
f(-6) = 6/(-(-6) - 3) = -6/5
So, we have the point (-6, -6/5).
Here is the graph of the rational function with the vertical and horizontal asymptotes, as well as the plotted points:
{{{ graph( 300, 300, -10, 10, -10, 10, 6/(-x - 3)) }}}
Note: The graph has been shifted to the left by 3 units due to the negative sign in front of the x in the denominator.
Use the properties of logarithms to expand logzx^4 assume all variables are positive
Using the properties of logarithms, we can expand log_z(x^4) as follows:
log_z(x^4) = 4log_z(x)
So, log_z(x^4) expands to 4log_z(x).
Solve for x
Log base 10 x=-1
To solve for x, we need to rewrite the equation using exponential form.
In logarithmic form, the equation log base 10 x = -1 means that 10 raised to the power of -1 is equal to x.
Thus, x = 10^(-1).
Evaluating 10^(-1), we find that x = 1/10 or 0.1.
To graph the rational function F(x) = 6/(-x - 3), we can follow these steps:
Step 1: Determine the vertical asymptote(s):
The vertical asymptote(s) occur when the denominator of the rational function equals zero. In this case, the denominator is -x - 3.
Setting -x - 3 = 0, we find x = -3. So, the vertical asymptote is x = -3.
Step 2: Determine the horizontal (or slant) asymptote:
To find the horizontal asymptote, we need to compare the degrees of the numerator and denominator polynomial.
The degree of the numerator is 0 (constant 6), and the degree of the denominator is 1 (linear term -x).
Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
Step 3: Plot points on each side of the graph:
To plot points on each side of the graph, we can choose x-values that are slightly greater than and less than -3.
When x approaches -∞ (negative infinity):
Let's choose x = -4. Plugging this value into the function:
F(-4) = 6/(-(-4) - 3) = 6/1 = 6.
When x approaches +∞ (positive infinity):
Let's choose x = -2. Plugging this value into the function:
F(-2) = 6/(-(-2) - 3) = 6/-1 = -6.
Now, let's plot these points on a graph:
- Asymptotes:
The vertical asymptote is x = -3 (a vertical line passing through x = -3)
The horizontal asymptote is y = 0 (a horizontal line passing through y = 0)
- Points:
(-4, 6) and (-2, -6)
Final graph:
|
6 |
\ |
\ |
\ |
-------|-------
| -3
To graph the rational function f(x) = 6/(-x-3), we can follow a step-by-step process:
Step 1: Determine the vertical asymptote(s)
For a rational function, vertical asymptotes occur where the denominator equals zero. In this case, the denominator -x-3 equals zero when x = -3. Hence, we have a vertical asymptote at x = -3.
Step 2: Determine the horizontal asymptote(s)
To find the horizontal asymptote(s) of a rational function, we examine the degrees of the numerator and denominator. In this case, the degree of the numerator is 0 and the degree of the denominator is 1. Since the degree of the numerator is lower than the degree of the denominator, the horizontal asymptote is y = 0.
Step 3: Plot two points on each side of the graph
To plot points on the graph, choose x-values that are slightly greater and slightly smaller than the vertical asymptote (-3) and plug them into the function to find the corresponding y-values.
Let's choose x = -4 and x = -2 for the left side of the graph:
For x = -4: f(-4) = 6/(-(-4)-3) = 6/1 = 6. So, we have the point (-4, 6).
For x = -2: f(-2) = 6/(-(-2)-3) = 6/1 = 6. So, we have the point (-2, 6).
Similarly, let's choose x = -1 and x = -5 for the right side of the graph:
For x = -1: f(-1) = 6/(-(-1)-3) = 6/4 = 3/2. So, we have the point (-1, 3/2).
For x = -5: f(-5) = 6/(-(-5)-3) = 6/2 = 3. So, we have the point (-5, 3).
Step 4: Draw the graph
Using the information obtained from the above steps, we can now plot the graph of f(x).
1. Mark the vertical asymptote at x = -3 (denoted by a vertical line).
2. Draw the horizontal asymptote y = 0 (denoted by a horizontal line).
3. Plot the points (-4, 6), (-2, 6) on the left side of the graph.
4. Plot the points (-1, 3/2), (-5, 3) on the right side of the graph.
5. Connect the points on each side of the graph, keeping in mind the behavior around the vertical asymptote. As x approaches -3 from the left, the function tends to positive infinity, and as x approaches -3 from the right, the function tends to negative infinity.
The resulting graph should display a vertical asymptote at x = -3, a horizontal asymptote at y = 0, and the plotted points on each side.