Problem 15 - Calculus with Concepts in Calculus

Question

Sketch the graph of the function. $$ g(x)=|x-2| $$

Step-by-Step Solution

Answer

The graph is a V-shape with vertex at (2, 0).

Step 1: Understand Absolute Value Function

Start by understanding the structure of the function: \( g(x) = |x-2| \). This function represents an absolute value function, which means it will create a V-shaped graph. The expression inside the absolute value \( x-2 \) affects the horizontal translation of the graph.

Step 2: Determine the Vertex

The vertex of the graph \( g(x) = |x-2| \) is found by setting the inside of the absolute value to zero: \( x-2=0 \). Solving this gives \( x=2 \). So the vertex is at the point \( (2, 0) \).

Step 3: Table of Values

Create a table of values to help sketch the graph. Choose points around the vertex, for example, \( x = 0, 1, 2, 3, 4 \), and calculate the corresponding \( g(x) \) values:- \( x = 0 \), \( g(0) = |0-2| = 2 \)- \( x = 1 \), \( g(1) = |1-2| = 1 \)- \( x = 2 \), \( g(2) = |2-2| = 0 \)- \( x = 3 \), \( g(3) = |3-2| = 1 \)- \( x = 4 \), \( g(4) = |4-2| = 2 \).

Step 4: Plot Points and Sketch the Graph

Use the table of values to plot the points on a graph: \((0, 2), (1, 1), (2, 0), (3, 1), (4, 2)\). Connect these points with straight lines to form a "V" shape. The vertex \((2, 0)\) is the lowest point, and the arms of the "V" will extend upwards and outwards.

Key Concepts

Vertex of a Graph

The vertex is a fundamental part of understanding a graph, especially for an absolute value function. For the function \( g(x) = |x-2| \), the vertex is where the graph changes direction, creating the distinct V-shape reflective of absolute value functions. Finding the vertex involves setting the inside of the absolute value expression to zero. Here, it means solving \( x-2=0 \) which gives us \( x=2 \). That leads to the vertex at the point \((2, 0)\). This point is crucial because it determines the "tip" of the V shape on the graph. Knowing where the vertex lies helps you predict how the rest of the graph will be positioned and formed. The entire graph is symmetrical around this vertex point, making it a very powerful tool for sketching accurate graphs.

Horizontal Translation

Horizontal translation is a handy concept when dealing with translating graphs from their parent functions. In the absolute value function \( g(x) = |x-2| \), the term \( x-2 \) signifies a horizontal shift. Instead of the graph starting at the origin where \( x=0 \), it is translated 2 units to the right. This translation aligns with locating the vertex, which helps in visualizing how the graph behaves relative to its base form \( |x| \).To understand translations intuitively:

• If the expression is \( x-h \), it shifts the graph \( h \) units to the right.
• If the expression were \( x+h \), it would shift\( h \) units to the left.

By comprehending horizontal translations, students can effectively predict the position of graphs, aiding in both plotting and understanding multiple similar functions in succession.

Table of Values

Using a table of values is an effective way to sketch graphs, particularly when starting with new functions. For \( g(x) = |x-2| \), creating a table lets us see how inputs (or \( x \) values) correspond to outputs \( g(x) \) which translates into points on the graph.Here's how the table looks:

• \( x = 0, g(x) = 2 \)
• \( x = 1, g(x) = 1 \)
• \( x = 2, g(x) = 0 \)
• \( x = 3, g(x) = 1 \)
• \( x = 4, g(x) = 2 \)

As you can see, as \( x \) moves away from the vertex \( (2,0) \), \( g(x) \) increases in a mirror pattern around the vertex. Plotting these points solidly grounds your understanding of the function's map. Once you've plotted the points on a paper or software, connecting them shows the V shape clearly. This structured approach ensures accuracy and helps visualize concepts fully rather than intuition alone.