How do you graph y= -log (base 5)(-x)?
The written question is: Sketch above equation by first describing the transformations performed on the equation y= log (base 5)(x)
And what is the inverse function? x=log (base 5)(y)?
Write the equation for this situation: the cost, c, in dollars of a car rental is equal to 10 dollars plus 1/2 the miles driven, m.
Then graph the equation and use the graph to determine the cost of the car rental if the number of miles driven is 12. Please show all of your work. Submit your graph through the dropbox. (Points : 5)
To graph the equation y = -log⁵(-x), we can follow these steps:
Step 1: Understand the parent function.
First, it is important to understand the parent function, which is y = log⁵(x). The base-5 logarithm functions have the following properties:
- The domain is all positive real numbers, (x > 0).
- The range is all real numbers, (y ∈ R).
- The graph passes through the point (1, 0).
- It is an increasing function, meaning as x increases, y also increases.
Step 2: Apply the transformations.
The equation y = -log⁵(-x) represents a transformation of the parent function. Here are the transformations applied:
a) Reflection about the y-axis: The negative sign in front of the logarithm causes the graph to reflect about the y-axis.
b) Horizontal compression/stretch: The negative sign inside the logarithm function indicates a horizontal compression by a factor of |-1| = 1. In this case, there is no horizontal compression.
c) Horizontal translation: The negative sign inside the logarithm function indicates a horizontal translation by 1 unit to the right.
d) Vertical stretch/compression: There is no vertical stretch or compression.
e) Vertical translation: There is no vertical translation.
Step 3: Sketch the graph.
Based on the transformations, we can now sketch the graph of y = -log⁵(-x), using the graph of y = log⁵(x) as reference.
Start by plotting key points on the transformed graph:
- When x = 1, y = -log⁵(-1) ≈ 0. This point corresponds to the original graph's point at (1, 0).
- Choose a few more x-values (positive and negative) and evaluate y to get additional points.
- Connect the points smoothly, keeping in mind the increasing nature of the logarithmic function.
Now, let's discuss the inverse function.
The inverse function of y = log⁵(x) can be found by swapping the x and y variables in the equation and solving for y:
x = log⁵(y)
To isolate y, rewrite the equation in exponential form:
5^x = y
So, the inverse function is x = log⁵(y) is equivalent to y = 5^x.
In conclusion, the graph of y = -log⁵(-x) can be obtained by applying transformations to the graph of y = log⁵(x), and the inverse function is x = log⁵(y), which can be expressed as y = 5^x.