The graph, y=lg x is rotated 90 degrees anticlockwise around the origin. What is the equation of the new graph? Please provide proof.
To rotate the graph of y = log(x) 90 degrees anticlockwise around the origin, we can apply the transformation rules for rotation in the Cartesian coordinate plane.
The rotation of a point (x, y) in the counterclockwise direction around the origin by 90 degrees can be achieved by swapping the x and y coordinates while changing the sign of the new x coordinate.
Let's apply this transformation to the equation y = log(x):
1. Swap the x and y coordinates:
x = log(y)
2. Change the sign of the new x coordinate:
-x = log(y)
So, the equation of the new graph obtained by rotating y = log(x) 90 degrees anticlockwise around the origin is -x = log(y).
To provide further proof, we can graph both equations and observe the transformation:
1. Graph y = log(x):
The graph of y = log(x) is a logarithmic function that passes through the point (1, 0) and approaches the x-axis asymptotically. It is commonly referred to as the logarithmic curve.
2. Graph -x = log(y):
Now, let's graph the equation -x = log(y). To make it easier, we can rewrite it as x = -log(y). By comparing this equation to the original swap and sign change transformation, we can see that they are equivalent.
By comparing the graphs of these two equations, we observe that the graph of -x = log(y) is obtained by rotating the graph of y = log(x) 90 degrees anticlockwise around the origin. Thus, we have proven that the equation of the new graph is -x = log(y).
90° clockwise rotation takes (x,y)->(-y,x)
so, (x,lg x) -> (-lg x,x)
That is, (1/x,e^x) or (x,e^-x)
enter those two functions
log(x)
e^(-x)
at
http://rechneronline.de/function-graphs/