The graph of a logarithmic function is given. Match each...

The graph of a logarithmic function is given. Match each graph to one of the following functions:
(a) $y=\log _{3} x$
(b) $y=\log _{3}(-x)$
(c) $y=-\log _{3} x$
(d) $y=-\log _{3}(-x)$
(e) $y=\log _{3} x-1$
(g) $y=\log$
(h) $y=1-\log _{3} x$
(FIGURE CAN NOT COPY)

Key Concepts

Translations

Translations move a function's graph horizontally or vertically without changing its shape. A vertical translation results from adding or subtracting a constant to the function output, shifting the graph up or down. Horizontal translations occur when a constant is added to or subtracted from the input variable x, moving the graph left or right. Recognizing translations is essential when matching a graph to a specific form of a logarithmic function.

Reflections

Reflections change the orientation of a function's graph. A reflection across the y-axis, achieved by replacing x with -x in the function, flips the graph horizontally. Similarly, reflecting across the x-axis, done by multiplying the output by -1, inverts the graph vertically. These reflections alter the observable properties such as intercepts while preserving the basic shape of the logarithmic function.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and typically have a vertical asymptote, a domain of positive numbers (unless modified), and a range spanning all real numbers. Their graphs increase or decrease depending on the base, and understanding their basic shape is crucial to analyzing transformations applied to them.

Function Transformations

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a basic function. For logarithmic functions, these transformations can alter the graph’s position and orientation without changing its overall behavior, such as its asymptote or increasing/decreasing nature.

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