Question

The point (1,13)
lies on the line f(x)=13^x
. The function g(x)=log13 x
includes a point of a reflection across the line y=x
that lies on the line g(x)=log13 x
. In which quadrant of the coordinate plane is this point located?(1 point)
Responses

Quadrant I
Quadrant I

Quadrant IV
Quadrant IV

Quadrant III
Quadrant III

Quadrant II

Answer 1

To find the point that reflects across the line \( y = x \), we can start with the function \( g(x) = \log_{13}(x) \).

The point \( (1, 13) \) lies on the function \( f(x) = 13^x \). To find the corresponding point on \( g(x) \), we will check where \( g(x) \) is evaluated at \( x = 13 \):

\[
g(13) = \log_{13}(13) = 1
\]

This gives us the point \( (13, 1) \) on the function \( g(x) \).

Now, to find the reflection of the point \( (13, 1) \) across the line \( y = x \), we switch the x and y coordinates. This gives us the reflected point:

\[
(1, 13)
\]

The reflected point \( (1, 13) \) is located in the second quadrant (where x is positive and y is greater than or equal to 0, which would be Quadrant II).

Thus, the answer is:

**Quadrant II**