Problem 37 - Thomas Calculus

Question

Graph the functions in Exercises \(35-54\) $$ y=|x-2| $$

Step-by-Step Solution

Answer

The graph is a V-shape centered at (2, 0), opening upward.

Step 1: Understanding the Absolute Value Function

The given function is \(y = |x - 2|\). This is an absolute value function, which is V-shaped. The expression inside the absolute value, \(x - 2\), shows a horizontal shift to the right by 2 units.

Step 2: Determine the Vertex of the Graph

For absolute value functions \(y = |x - h| + k\), the vertex of the graph is at the point \((h, k)\). In this function, \(h = 2\) and \(k = 0\), so the vertex is at the point \((2, 0)\).

Step 3: Create a Table of Values

Select key x-values around the vertex to determine corresponding y-values. Here are some points: \((0, 2)\), \((1, 1)\), \((2, 0)\), \((3, 1)\), \((4, 2)\). Evaluate \(y\) for these \(x\)-values.

Step 4: Plot the Points on a Coordinate Plane

Using the table from Step 3, plot the points \((0, 2)\), \((1, 1)\), \((2, 0)\), \((3, 1)\), and \((4, 2)\) on a coordinate plane. These points should form a symmetrical V-shape about the vertex \((2, 0)\).

Step 5: Draw the Graph

Connect the plotted points with straight lines. The graph should look like a V with the bottom point at \((2, 0)\). The sides of the V should extend infinitely in the up direction, maintaining symmetry around the vertex.

Key Concepts

Vertex of a Graph

In the context of graphing, the vertex of a graph is a crucial point. For absolute value functions like \(y = |x - h| + k\), it indicates the 'tip' of the V-shaped graph. On a coordinate plane, this vertex gives you the lowest point (or the highest point, if the graph opens downward) of the V. In the specific function \(y = |x - 2|\), the expression inside the absolute value \(x - 2\) reveals a horizontal shift. The general formula tells us that this change moves the vertex to the point \((h, k)\), which, in our case, is \((2, 0)\). This means the graph is shifted 2 units to the right compared to the standard \(y = |x|\) function. The vertex is not just a starting point; it also acts as the axis of symmetry for the entire graph.

Plotting Points

Plotting points is about selecting precise points that represent the graph's behavior. By substituting different x-values into the function, you can calculate corresponding y-values. For the function \(y = |x - 2|\), we chose x-values around the vertex for simplicity and clarity: \((0, 2)\), \((1, 1)\), \((2, 0)\), \((3, 1)\), and \((4, 2)\). Each of these pairs fulfills the equation and helps form the graph's shape. Starting around the vertex ensures a balanced spread of points on both sides, revealing the characteristic V-shape of the function’s graph. It is like drawing a dotted path before solidifying it with the graph's lines. Keep in mind to choose enough points to capture the essence of the curve effectively.

Coordinate Plane

A coordinate plane is where you illustrate the function’s behavior. It consists of the x-axis (horizontal) and y-axis (vertical), creating a grid for plotting points. Each point is identified using a pair of numbers \((x, y)\), known as coordinates. These coordinates tell you exactly where to place a point in the plane. For our function \(y = |x - 2|\), the coordinate plane will host points like \((0, 2)\), \((1, 1)\), \((2, 0)\), etc. As you plot these points, ensure the scales on the axes are consistent to maintain accuracy. The x-coordinate tells how far left or right to move from the origin \((0, 0)\), while the y-coordinate shows how far up or down to move from there. The coordinate plane acts as a canvas, allowing you to visualize mathematical relationships geometrically.

Symmetry in Graphs

Symmetry is a powerful concept in graphing, especially for absolute value functions. It refers to a balanced and identical arrangement of points around a central axis. For the function \(y = |x - 2|\), the graph exhibits symmetry about the vertical line \(x = 2\). This is the same line where the vertex of the graph is located. If you fold the graph along the line \(x = 2\), both sides of the V-shape should perfectly overlap. This symmetry helps simplify graphing because any transformation or shift in the graph retains this mirrored property. An understanding of symmetry ensures that when you plot points on one side, you can easily determine corresponding points on the opposite side, making the graph easier to complete and analyze.