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)?
To graph the equation y = -log5(-x), we will start by looking at the original equation y = log5(x) so that we can understand the transformations.
The graph of y = log5(x) is commonly referred to as the logarithmic function with base 5. It has a vertical asymptote at x = 0, meaning the graph approaches but never touches the x-axis. The graph passes through the point (1,0) because log5(1) = 0.
Now, let's consider the transformations applied to the original logarithmic function.
1) Reflection: The negative sign (-) in front of the log function in y = -log5(x) indicates a reflection about the x-axis. This means that any point that was originally above the x-axis will now be below the x-axis, and vice versa.
2) Horizontal reflection: The negative sign (-) before the x-value in -log5(-x) causes the function to be horizontally reflected. This means that any point that was originally to the right of the y-axis will now be to the left of the y-axis, and vice versa.
To graph y = -log5(-x), we can start by plotting a few key points:
- Choose a few x-values, such as -1, -10, 1, and 10.
- Use these x-values to find the corresponding y-values by substituting into the equation y = -log5(-x).
- Plot the points (-1, y1), (-10, y2), (1, y3), and (10, y4) on the graph.
To find the inverse function, we need to interchange the x and y variables and solve for y.
In this case, the inverse function of y = -log5(-x) is found by interchanging x and y:
x = -log5(-y)
Next, we can solve for y:
5^(-x) = -y
y = -5^(-x)
Therefore, the inverse function of y = -log5(-x) is x = -5^(-y).