Questions: Compare the graph of the function, f(x)=x, with the graph of the function, g(x)=x/3. For each point, (a, b), on f(x), there is a corresponding point, (3a, b), on g(x). The graph of g(x) is a horizontal stretch of magnitude 3 applied to the graph of f(x). For each point, (a, b), on f(x), there is a corresponding point, (1/3a, b), on g(x). The graph of g(x) is a horizontal shrink of magnitude 3 applied to the graph of f(x). For each point, (a, b), on f(x), there is a corresponding point, (a, 1/3b), on g(x). The graph of g(x) is a vertical shrink of magnitude 3 applied to the graph of f(x).

Compare the graph of the function, f(x)=x, with the graph of the function, g(x)=x/3.

For each point, (a, b), on f(x), there is a corresponding point, (3a, b), on g(x). The graph of g(x) is a horizontal stretch of magnitude 3 applied to the graph of f(x).

For each point, (a, b), on f(x), there is a corresponding point, (1/3a, b), on g(x). The graph of g(x) is a horizontal shrink of magnitude 3 applied to the graph of f(x).

For each point, (a, b), on f(x), there is a corresponding point, (a, 1/3b), on g(x). The graph of g(x) is a vertical shrink of magnitude 3 applied to the graph of f(x).

Transcript text: Compare the graph of the function, $f(x)=|x|$, with the graph of the function, $g(x)=\left|\frac{x}{3}\right|$. For each point, $(a, b)$, on $f(x)$, there is a a corresponding point, $(3 a, b)$, on $g(x)$. The graph of $g(x)$ is a horizontal stretch of magnitude 3 applied to the graph of $f(x)$. For each point, $(a, b)$, on $f(x)$, there is a b corresponding point, $\left(\frac{1}{3} a, b\right)$, on $g(x)$. The graph of $g(x)$ is a horizontal shrink of magnitude 3 applied to the graph of $f(x)$. For each point, $(a, b)$, on $f(x)$, there is a corresponding point, $\left(a, \frac{1}{3} b\right)$, on $g(x)$. The graph of $g(x)$ is a vertical shrink of magnitude 3 applied to the graph of $f(x)$.

Solution

Solution Steps

To compare the graphs of $ f(x) = |x| $ and $ g(x) = \left|\frac{x}{3}\right| $, we need to analyze how the transformation affects the points on the graph. Specifically, we need to determine how the x-coordinates and y-coordinates of points on $ f(x) $ are transformed to points on $ g(x) $.

For each point $(a, b)$ on $ f(x) $, the corresponding point on $ g(x) $ can be found by substituting $ x $ with $ \frac{x}{3} $ in $ f(x) $.
This transformation results in a horizontal stretch by a factor of 3.

Step 1: Define the Functions

We start by defining the functions $ f(x) = |x| $ and $ g(x) = \left|\frac{x}{3}\right| $. These functions represent the absolute value of $ x $ and the absolute value of $ \frac{x}{3} $, respectively.

Step 2: Generate $ x $ Values

We generate a range of $ x $ values from $-10$ to $10$ to evaluate the functions. This range is chosen to provide a comprehensive view of the behavior of both functions.

Step 3: Calculate $ y $ Values

We calculate the corresponding $ y $ values for both functions:

For $ f(x) $, $ y_f = |x| $
For $ g(x) $, $ y_g = \left|\frac{x}{3}\right| $

Step 4: Analyze the Transformation

To compare the graphs of $ f(x) $ and $ g(x) $, we observe the transformation:

For each point $(a, b)$ on $ f(x) $, the corresponding point on $ g(x) $ is $(3a, b)$.
This indicates that the graph of $ g(x) $ is a horizontal stretch of magnitude 3 applied to the graph of $ f(x) $.

Final Answer

$\boxed{\left(\frac{1}{3} a, b\right)}$

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