Problem 27 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(f(x)=|2 x-2|\)

Step-by-Step Solution

Answer

The graph is a 'V' shape with its vertex at (1, 0).

Step 1: Understand the Function

The function given is a piecewise function due to the absolute value. It can be written as two separate expressions: \[f(x) = \begin{cases} 2x - 2 & \text{if } 2x - 2 \geq 0 \-(2x - 2) & \text{if } 2x - 2 < 0 \end{cases} \]So the function is linear but changes at the point where the expression inside the absolute value equals zero.

Step 2: Find Critical Points

Solve for the value of \(x\) where the expression inside the absolute value switches from non-negative to negative, which is when \(2x - 2 = 0\). This gives: \[2x - 2 = 0\] \[2x = 2\] \[x = 1\] At \(x = 1\), the behavior of the function changes.

Step 3: Create a Table of Values

We'll choose values for \(x\) around the critical point \(x = 1\) and calculate corresponding \(f(x)\) values.For \(x < 1\):- \(x = 0\), \(f(0) = |2(0) - 2| = |-2| = 2\)- \(x = 0.5\), \(f(0.5) = |2(0.5) - 2| = |-1| = 1\)At \(x = 1\):- \(f(1) = |2(1) - 2| = |0| = 0\)For \(x > 1\):- \(x = 1.5\), \(f(1.5) = |2(1.5) - 2| = |1| = 1\)- \(x = 2\), \(f(2) = |2(2) - 2| = |2| = 2\)

Step 4: Sketch the Graph

Plot the points \((0, 2)\), \((0.5, 1)\), \((1, 0)\), \((1.5, 1)\), and \((2, 2)\) on the coordinate plane.Draw two line segments: one descending from \((0, 2)\) to \((1, 0)\) and one ascending from \((1, 0)\) to \((2, 2)\). These two segments form a 'V' shape that opens upwards.

Key Concepts

Piecewise Functions

Piecewise functions are an essential concept to grasp when dealing with absolute value functions, particularly because they allow us to break down complex expressions into simpler, manageable parts. These functions are defined by multiple sub-functions, each applicable to a certain interval of the main function's domain. For an absolute value function like \[f(x) = |2x - 2|\]we can define it using two separate linear expressions based on the condition of the inside expression.

• When the expression inside is non-negative, it remains unchanged: \(f(x) = 2x - 2\).
• When the expression inside is negative, the expression's sign is reversed, converting it to a positive: \(f(x) = -(2x - 2)= -2x + 2\).

This separation based on the condition creates what we call a 'piecewise function'. It is useful because it divides the behavior of the function into distinct segments, making it easier to analyze and graph.

Critical Points

Critical points in piecewise functions, especially those involving absolute values, signify the transitions between different linear behaviors. In our function \[f(x) =|2x - 2|\]we look at where the expression inside the absolute value equals zero. Solving \[2x - 2 = 0\]we find that the critical point occurs at \(x=1\). This point is crucial because it's where the function changes its mathematical expression.
The critical point not only dictates the transition but often represents the vertex of the graph in absolute value functions. For our example, we see a 'V' shape transition at \(x = 1\), highlighting the function's vertex. Identifying such points first helps in sketching the graph accurately later.

Table of Values

Creating a table of values is a fundamental step to help visualize how a function behaves over its domain. For the function\[f(x) = |2x - 2|\]a table of values allows us to observe the outputs of the function as \(x\) varies, especially around the critical point.

• First, you pick values less than the critical point: \(x = 0, 0.5\)
• Note the exact critical point value: \(x = 1\)
• Finally, choose values greater than the critical point: \(x = 1.5, 2\)

By plugging these values into the function, you can find corresponding \(f(x)\) outputs. This detailed tabulation gives specific data points one can plot, providing a structured path for graphing.

Linear Segments

Linear segments form the graphical representation of each piece in a piecewise function. For the given absolute value function\[f(x) = |2x - 2|\]the graphical representation is split into two linear segments meeting at the critical point.

• The first segment runs from the point \((0, 2)\) to the critical point \((1, 0)\). This segment represents when the inside expression is negative and thus takes the negative form of the linear equation.
• The second segment stretches from \((1, 0)\) to \((2, 2)\), representing where the inside expression remains unchanged because it's non-negative.

These segments combined create the well-known 'V' shape characteristic of absolute value functions. Understanding how these linear segments fit together helps in accurately sketching the graph and understanding the function’s behavior across its domain.