Problem 60 - Algebra 2

Question

Evaluate each function for five values of \(x .\) Then graph each function. \(f(x)=\frac{1}{3}\left|\frac{1}{3} x-3\right|\)

Step-by-Step Solution

Answer

Results for the five values are: for x = -6, f(x) = 5/3; for x = -3, f(x) = 4/3; for x = 0, f(x) = 1; for x = 3, f(x) = 2/3; for x = 6, f(x) = 1/3. The graph is a 'V' shape centered around the point (9,0).

Step 1: Choose Values for x

Choose five values for x to plug into the function. It's helpful to choose a range of values including negative numbers, zero, and positive numbers. For example, let's choose \(x = -6, -3, 0, 3, \text{and } 6\).

Step 2: Plug in the Values

For each chosen value of x, substitute it into the function and simplify. For \(f(x)=\frac{1}{3}\left|\frac{1}{3} x-3\right|\), plug in each value and calculate the result.

Step 3: Calculate the Function's Value for Each x

Calculate \(f(x)\) using the chosen values.1. For \(x = -6\), \(f(-6)=\frac{1}{3}\left|\frac{1}{3}(-6)-3\right|=\frac{1}{3}\left|-2-3\right|=\frac{1}{3}(5)=\frac{5}{3}\).2. For \(x = -3\), \(f(-3)=\frac{1}{3}\left|\frac{1}{3}(-3)-3\right|=\frac{1}{3}\left|-1-3\right|=\frac{1}{3}(4)=\frac{4}{3}\).3. For \(x = 0\), \(f(0)=\frac{1}{3}\left|\frac{1}{3}(0)-3\right|=\frac{1}{3}\left|-3\right|=\frac{1}{3}(3)=1\).4. For \(x = 3\), \(f(3)=\frac{1}{3}\left|\frac{1}{3}(3)-3\right|=\frac{1}{3}\left|1-3\right|=\frac{1}{3}(2)=\frac{2}{3}\).5. For \(x = 6\), \(f(6)=\frac{1}{3}\left|\frac{1}{3}(6)-3\right|=\frac{1}{3}\left|2-3\right|=\frac{1}{3}(1)=\frac{1}{3}\).

Step 4: Plot the Points on the Graph

On a coordinate plane, plot the points \( (x, f(x)) \) found in the previous step: \( (-6,\frac{5}{3}), (-3,\frac{4}{3}), (0,1), (3,\frac{2}{3}), (6,\frac{1}{3}) \).

Step 5: Connect the Points

Since the function is a transformation of an absolute value function and is continuous, connect the points with straight lines forming a 'V' shape that opens upwards.

Key Concepts

Evaluating Functions

Evaluating a function involves finding the output values when specific inputs are given. Just like plugging in coordinates to a GPS to get your route, inputting numbers into a function such as \(f(x)=\frac{1}{3}\left|\frac{1}{3} x-3\right|\) gives us a set of results we can plot on a graph.
When performing this evaluation, it's akin to testing how the function behaves under certain conditions; choosing a range of values for x is crucial, as it allows a clearer overview of the function's pattern. A selection often includes negatives, zero, and positives. Here’s a breakdown of what this looks like in practice:

• Input: You select values such as -6, -3, 0, 3, and 6.
• Substitution: Replace the x in the function with the chosen values.
• Simplification: Resolve the equation to find corresponding y values for each x.

By methodically working through inputs and calculating outputs, a comprehensive picture of the function's behavior is developed, essential for creating an accurate graph.

Absolute Value Transformations

The concept of absolute value transformations can turn a simple 'V' shape into a diverse collection of figures on a graph. When dealing with a function like \(f(x)=\frac{1}{3}\left|\frac{1}{3} x-3\right|\), we're observing how this mathematical 'sculpting' takes place.
Absolute value functions are distinctive with their sharp vertex pointing either up or down. Think of this point as the turning spot or pivot of the graph. Transformations can shift this graph horizontally and vertically, and even change its width and orientation. In this specific function, the coefficient \(\frac{1}{3}\) outside of the absolute value contracts the graph vertically, making it 'flatter' than the standard \(|x|\). Meanwhile, the operation inside the absolute value, \(-3\), shifts the entire graph to the right by 3 units. These transformations are key in shaping the graph's final look, transitioning from a basic to a more complex form.

Piecewise Linear Graphs

Piecewise linear graphs are essentially connect-the-dots for mathematicians, where the 'dots' are specific points on a graph and the connections are straight lines. Functions like \(f(x)=\frac{1}{3}\left|\frac{1}{3} x-3\right|\) exhibit linear behavior in pieces, with each segment having its own linear equation.
Visualize it as a path formed step by step or segment by segment. Why are they treated piece by piece? Because the absolute value function creates a distinct break at the point where x causes the value inside the absolute value to switch from positive to negative or vice versa. The graph's appearance may change drastically across the domain, but its essence, built from these linear pieces, is always predictable and straightforward. This segmented approach renders an accurate depiction of how the function operates across different intervals of x, providing a clear view of the function's behavior.