91Ó°ÊÓ

Vertical compression is a transformation that "squeezes" the graph of a function towards the x-axis. Imagine you have a rubber band stretched vertically. If you compress it, the height decreases. This is similar to what happens with the function's graph.
When a function is vertically compressed by a factor, each y-value of the original function is multiplied by that factor. For instance, applying a compression factor of \( \frac{1}{3} \) to the function \( f(x) = x \), gives us the new function \( g(x) = \frac{1}{3}x \). This means every y-value is now one third of its original value.
Vertical compression affects the steepness of the line. Here, the steeper the line, the less it compresses. However, the x-values remain unchanged in this transformation, only the height (y-values) changes.

A horizontal shift moves the graph of a function left or right along the x-axis. If you think of it as sliding the graph without changing its shape, you're on the right track.
To perform a horizontal shift, you adjust the x-variable in the function's equation. For a shift to the right by 1, replace \(x\) with \(x - 1\) in the function. For example, changing \(f(x) = \frac{1}{3}x\) to \(g(x) = \frac{1}{3}(x-1)\).
This means that for every x-value, we now look 1 unit to the right compared to the original function. Shifting doesn't affect the slope but alters where the function's peak (or any specific point) is located along the x-axis.

A vertical shift is a transformation that moves a graph up or down along the y-axis. Imagine taking a paper with a graph drawn on it and sliding it upwards or downwards without altering its shape.
To perform a vertical shift, a constant is added to (or subtracted from) the function. For instance, adding 3 to \(f(x) = \frac{1}{3}(x-1)\) results in \(g(x) = \frac{1}{3}(x-1) + 3\). This transformation shifts all y-values up by 3 units.
This change impacts the y-intercept, as well. The whole graph moves as one unit without affecting the slope. It's like moving a picture frame without tilting it.

Slope is a measure of how steep a line is, often described as "rise over run." It indicates the change in the y-values for each unit change in the x-values.
For the linear function \( g(x) = \frac{1}{3}x + \frac{8}{3} \), the coefficient of \(x\), \( \frac{1}{3} \), represents the slope. This tells us that for every unit increase in x, the y-value increases by \( \frac{1}{3} \).
The slope helps determine the angle at which the line inclines or declines. A positive slope, like \( \frac{1}{3} \), means the line ascends. Understanding slope is crucial for analyzing a function's behavior as it shows how outputs change with varying inputs.

The y-intercept is the point where a line crosses the y-axis. This happens where the x-value is zero.
In the equation \( g(x) = \frac{1}{3}x + \frac{8}{3} \), the term \( \frac{8}{3} \) is the y-intercept. This means when x is 0, the value of g(x) is \( \frac{8}{3} \).
Visually, it's the starting point of the function on a graph. Knowing the y-intercept is helpful because it gives a direct cue about where a graph begins when plotted. It's like setting a baseline from which the line extends up or down based on the slope.