Reflecting the graph calculus problem f(x) = 5^x?
reflecting the graph about the x-axis and the y-axis: y=___________
reflecting the graph about the line y = 3. y=__________
To reflect the graph of the function f(x) = 5^x about the x-axis, we can multiply the function by -1. Therefore, the equation of the reflected graph about the x-axis is y = -5^x.
To reflect the graph of the function f(x) = 5^x about the y-axis, we can substitute x with -x in the function. Therefore, the equation of the reflected graph about the y-axis is y = 5^(-x).
To reflect the graph of the function f(x) = 5^x about the line y = 3, we need to first find the distance between the graph and the line at any given point. Then, we can find the corresponding reflection point by subtracting the distance from the line or adding the distance to the line, depending on the position of the point.
Let's consider a point (x, y) on the graph. The distance between the point and the line y = 3 is |y - 3|. Therefore, the reflected point will have a distance of |y - 3| from the line y = 3.
If the point is above the line y = 3 (y > 3), we subtract the distance from the line to get the reflected point. So the equation of the reflected graph about y = 3 is y = 3 - |5^x - 3|.
If the point is below the line y = 3 (y < 3), we add the distance to the line to get the reflected point. So the equation of the reflected graph about y = 3 is y = 3 + |5^x - 3|.
Note that the absolute value function |5^x - 3| ensures that the reflection is symmetric with respect to the line y = 3.