Problem 26 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is a 'V' shape turned horizontally, with a diagonal line \(y=-2x\) for \(x<0\) and a horizontal line at \(y=0\) for \(x\geq0\).

Step 1: Identify the function

The function we are analyzing is a piecewise function. It is defined as \(G(x) = |x| - x\) where \(|x|\) is the absolute value of \(x\). This means that the function might behave differently for different values of \(x\).

Step 2: Determine behavior based on x

Examine two cases for the function: when \(x \geq 0\) and when \(x < 0\).- **Case 1:** If \(x \geq 0\), then \(|x| = x\). Therefore, the function becomes \(G(x) = x - x = 0\).- **Case 2:** If \(x < 0\), then \(|x| = -x\). Therefore, the function becomes \(G(x) = -x - x = -2x\).

Step 3: Create a table of values

Now, let's create a table of values to calculate \(G(x)\) for various values of \(x\):\[\begin{array}{c|c} x & G(x) \ \hline -3 & -2(-3) = 6 \ -2 & -2(-2) = 4 \ -1 & -2(-1) = 2 \ 0 & 0 \ 1 & 0 \ 2 & 0 \ 3 & 0 \\end{array}\]

Step 4: Sketch the graph

Using the table of values from Step 3, sketch the graph:1. For \(x < 0\), the function is linear with a slope of 2. Plot the points: \((-3, 6)\), \((-2, 4)\), and \((-1, 2)\).2. For \(x \geq 0\), the function is constant and equal to 0. Plot the points: \((0, 0)\), \((1, 0)\), \((2, 0)\), and \((3, 0)\).3. Draw a line connecting the points \((-3, 6)\) to \((0, 0)\) and then a horizontal line for \(x \geq 0\).

Step 5: Analyze the graph

The graph consists of a line with a negative slope for \(x < 0\) and a constant line at \(y=0\) for \(x \geq 0\). This creates a 'V' shape that has been flipped horizontally.

Key Concepts

Piecewise Functions

Piecewise functions are mathematical objects that have different expressions based on the input value, which splits into different cases. For example, the function in this exercise, defined as \(G(x) = |x| - x\), is a piecewise function because it behaves differently depending on whether \(x\) is less than zero or greater than or equal to zero.

• For \(x \geq 0\), the absolute value \(|x|\) simplifies to \(x\), making \(G(x) = 0\).
• For \(x < 0\), \(|x|\) equals \(-x\), so \(G(x) = -2x\).

Understanding these cases is essential for correctly sketching graphs or solving equations involving piecewise functions. By dissecting the function into intervals, you can represent it as different linear or nonlinear expressions, making it easier to analyze complex behaviors.

Graph Sketching

Graph sketching is an essential skill in understanding how functions behave visually. It brings algebraic expressions to life by showing changes visually. Graph sketches help students gain insights into the general behavior and properties of functions.To sketch the graph of the function \(G(x)=|x|-x\), we followed these steps:1. **Identify the behavior:** We found that \(G(x)\) for \(x \geq 0\) is always zero, forming a horizontal line along the x-axis. Conversely, \(G(x)\) for \(x < 0\) behaves as \(-2x\), creating a sloped line.2. **Plot points:** Using calculated table values, plot specific points such as \((-3, 6)\), \((-2, 4)\), and \((-1, 2)\) for \(x < 0\), and \((0,0)\), \((1,0)\), etc., along the x-axis for \(x \geq 0\).3. **Connect points:** Draw lines through the plotted points to form the graph. The line for \(x < 0\) shows a decreasing linear trend, while for \(x \geq 0\), it's flat at zero.These visual techniques highlight aspects like slopes, intercepts, and overall shapes crucial in analyzing mathematical functions.

Table of Values

Creating a table of values is a fundamental step in exploring and understanding the behavior of a function. It involves evaluating the function's rule over various specified points, then organizes data into a simple format for further analysis.In this exercise, we constructed a table for \(G(x) = |x| - x\), considering the distinct cases for \(x\). For each integer value chosen for \(x\), we computed \(G(x)\) and noted the results:

• When \(x\) was negative, calculations like \(G(-3) = 6\) and \(G(-1) = 2\) were obtained, showing the effect of \(-2x\).
• For non-negative \(x\), \(G(x) = 0\), highlighting its constancy.

The table not only simplifies the graph plotting process but also provides a clear glimpse of the function's dynamic, helping in visualizing and validating our piecewise analysis.