Problem 16 - College Algebra with Modeling and Visualization

Question

Graph \(y=f(x)\) by hand by first plotting points to determine the shape of the graph. $$ f(x)=|2 x-1| $$

Step-by-Step Solution

Answer

Plot \((0, 1), \left(\frac{1}{2}, 0\right), (1, 1), (2, 3)\); shape is a 'V'.

Step 1: Understand Absolute Value

The function given is \(f(x) = |2x - 1|\), which involves an absolute value. The absolute value function creates a 'V' shape on a graph. The 'V' is centered around the point where the expression inside the absolute value is zero.

Step 2: Find the Vertex

To find the vertex, set the expression inside the absolute value to zero: \(2x - 1 = 0 \). Solving for \(x\) gives \(x = \frac{1}{2}\). Thus, the vertex is at \(\left( \frac{1}{2}, 0 \right)\).

Step 3: Choose Points to Plot

Select a few values of \(x\) to evaluate \(f(x)\) and plot points. Choose values on either side of the vertex (e.g., \(x = 0, 1, 2\)).

Step 4: Calculate \(f(x)\) for Each \(x\) Value

Calculate \(f(x)\) for selected \(x\) values:- For \(x = 0\), \(f(x) = |2(0) - 1| = 1\)- For \(x = 1\), \(f(x) = |2(1) - 1| = 1\)- For \(x = 2\), \(f(x) = |2(2) - 1| = 3\)

Step 5: Plot the Points and Sketch the Graph

Plot the calculated points: \((0, 1), \left(\frac{1}{2}, 0\right), (1, 1), (2, 3)\). Sketch a 'V' shape connecting these points, starting from the vertex \(\left(\frac{1}{2}, 0\right)\), going upward to the right and left.

Key Concepts

Graphing Techniques

Graphing is a visual way to represent functions. It helps us understand their behavior just by looking at a picture. To graph an absolute value function like \(f(x) = |2x - 1|\), we employ specific techniques to showcase its unique features.

An absolute value function produces a 'V' shape graph. This happens because absolute value expressions flip any negative results to positive, creating a symmetric reflection across the vertex. Here’s a quick guide on how to graph this function effectively:

• Identify critical points: First, pinpoint the vertex, a focal aspect that changes the graph's direction.
• Symmetry helps: Use its inherent symmetry to find other points around the vertex.
• Connect points smoothly: Draw a smooth 'V' connecting these points.

By using these steps, you can neatly illustrate the full character of the absolute value function on a graph.

Vertex of a Graph

The vertex is a central part of the function graph; it acts much like the hinge of a 'V' shape in an absolute value function. For our function \(f(x) = |2x - 1|\), the vertex not only represents the lowest point but also where the graph changes direction.

To find this point, you'll need to solve the equation inside the absolute value for zero:
- Set \(2x - 1 = 0\). - Solve to get \(x = \frac{1}{2}\).
Once you have the \(x\)-value, simply plug it back in to find the corresponding \(y\)-value. Therefore, the vertex is \(\left( \frac{1}{2}, 0 \right)\).

The importance of the vertex in graphing cannot be overstated. It determines the angle of the 'V' and sets the stage for the rest of the graph. This point gives us the symmetry line, helping us plot other points with minimal calculations while keeping the graph accurate.

Plotting Points

Plotting points is foundational to graphing functions. It involves selecting values, computing corresponding outcomes, and then marking these on a graph.

To graph \(f(x) = |2x - 1|\):

• Choose several \(x\)-values, especially around the vertex.
• For each value, calculate \(f(x)\). For instance, when \(x = 0, 1, 2\), compute:
  - \(f(0) = 1\),
  - \(f(1) = 1\),
  - \(f(2) = 3\).
• Plot these points: \((0, 1)\), \(\left(\frac{1}{2}, 0\right)\), \((1, 1)\), and \((2, 3)\).

Once points are plotted, draw lines to create the 'V' shape around the vertex.
By plotting points carefully, you construct a precise representation of the function. It is these small details that together form an accurate and coherent graph.