Problem 46 - Algebra 2

Question

Graph each function. Identify the domain and range. $$ h(x)=|x-3| $$

Step-by-Step Solution

Answer

The graph is V-shaped with vertex at (3,0). Domain: \((-\infty, \infty)\), Range: \([0, \infty)\).

Step 1: Understanding the Function

The function given is a piecewise function represented as the absolute value function \(h(x) = |x-3|\). The absolute value function results in a V-shaped graph where the vertex is at the point that makes the expression inside the absolute value zero.

Step 2: Finding the Vertex

To find the vertex of the function, set the expression inside the absolute value to zero: \(x - 3 = 0\). This gives us \(x = 3\). Hence, the vertex of \(h(x)\) is at the point (3,0).

Step 3: Plotting Key Points

Plot the vertex (3,0) on the coordinate plane. Select additional points on either side of the vertex. For example, evaluate \(h(x)\) at \(x=2\), which gives \(|2-3| = 1\), and at \(x=4\), which also gives \(|4-3| = 1\). Points (2,1) and (4,1) should be plotted.

Step 4: Drawing the Graph

Using the plotted points, draw the V-shaped graph. The left half is the line \(y = 3 - x\) for \(x < 3\), and the right half is the line \(y = x - 3\) for \(x \geq 3\). Ensure the graph is symmetric about the vertex.

Step 5: Determining the Domain

The domain of \(h(x) = |x-3|\) is not restricted by any values of \(x\). Thus, the domain is all real numbers, expressed as \((-\infty, \infty)\).

Step 6: Determining the Range

The output \(h(x)\) represents the distance from 3, hence, it cannot be negative. Thus, \(h(x) \geq 0\). Therefore, the range is \([0, \infty)\).

Key Concepts

Understanding Domain and Range

When working with functions such as the absolute value function, determining the domain and range is crucial. The **domain** of a function is all possible input values (x-values) that the function can accept. For the function \( h(x) = |x-3| \), there's no restriction on the values of \( x \). This means that the domain is all real numbers, which we express as

• Domain: \( (-\infty, \infty) \)

The **range**, on the other hand, refers to all possible output values (y-values) that the function can produce. Since an absolute value function measures distance, it can never be negative. Therefore, the smallest value the function \( h(x) = |x-3| \) can output is \( 0 \), when \( x = 3 \). Consequently, the range of this function is

• Range: \( [0, \infty) \)

Graphing Functions with Absolute Values

Graphing absolute value functions can be straightforward once you understand their unique V-shape. For the absolute value function \( h(x) = |x-3| \), here’s how you can plot its graph:

• Vertex Point: The vertex of this V-shape occurs where the expression inside the absolute value is zero. For \( |x-3| \), set \( x-3 = 0 \) which gives \( x = 3 \). Thus, the vertex is at the point (3, 0).
• Symmetry: Absolute value graphs are symmetric with respect to their vertex line. Whatever happens to the left of the vertex mirrors to the right.
• Select Additional Points: Choose points to the left and right of the vertex to provide structure to your graph. For example, calculate \( h(x) \) at \( x = 2 \) and \( x = 4 \), result in points (2, 1) and (4, 1) respectively.
• Draw the Graph: Use these points, along with the vertex, to draw the V-shape. Check the pattern \( y = 3 - x \) on the left and \( y = x - 3 \) on the right.

By connecting your plotted points, you generate the entire graph of the function.

Vertex of a Function Explained

The vertex of a function, especially in absolute value functions, plays a significant role in its graph. For \( h(x) = |x-3| \), the vertex is a critical point that dictates the shape and symmetry of the graph.To find the vertex, we need to find the input \( x \) that makes the expression inside the absolute value equal to zero. This requires solving the equation \( x - 3 = 0 \). Solving this, we find:

• \( x = 3 \)

Thus, the vertex of the function is at the point (3, 0). At this point, the graph changes direction, forming the base of the "V" shape. The vertex is also important for understanding the function's translation and transformation:

• It tells you the horizontal shift; here, the graph of \( |x| \) is shifted 3 units to the right.
• It determines symmetry; the function is symmetric around this vertex line.

Hence, recognizing the vertex helps predict the behavior and appearance of the absolute value function graph.