Problem 24 - Algebra and Trigonometry

Question

Sketch the graph of the function by first making a table of values. \(H(x)=|x+1|\)

Step-by-Step Solution

Answer

The graph of \(H(x) = |x+1|\) is a "V" shape with vertex at \((-1, 0)\).

Step 1: Understanding the Function

The function given is \(H(x) = |x+1|\). This is an absolute value function that shifts the basic absolute value graph \(y = |x|\) to the left by 1 unit. Our goal is to create a table of values and sketch the graph based on this table.

Step 2: Setting Up a Table of Values

Choose a range of \(x\) values around the vertex of the absolute value function, which is at \(x=-1\) here. We'll use the points: \(-3, -2, -1, 0, 1, 2\). Calculate \(H(x)\) for each \(x\) value and fill in the table.

Step 3: Calculating the Values

For each \(x\) chosen, compute \(H(x) = |x+1|\).- When \(x = -3\), \(H(x) = |-3 + 1| = 2\).- When \(x = -2\), \(H(x) = |-2 + 1| = 1\).- When \(x = -1\), \(H(x) = |-1 + 1| = 0\).- When \(x = 0\), \(H(x) = |0 + 1| = 1\).- When \(x = 1\), \(H(x) = |1 + 1| = 2\).- When \(x = 2\), \(H(x) = |2 + 1| = 3\).

Step 4: Completing the Table

Now the table of values is:\[\begin{array}{c|c} x & H(x) \ \hline -3 & 2 \ -2 & 1 \ -1 & 0 \ 0 & 1 \ 1 & 2 \ 2 & 3 \\end{array}\]

Step 5: Sketching the Graph

Plot the points from the table onto a Cartesian coordinate system. The graph should form a "V" shape, with the vertex at the point \((-1, 0)\). Connect the points with straight lines to complete the absolute value graph.

Key Concepts

Graphing Absolute Value

Graphing an absolute value function involves a few key ideas. The graph of an absolute value function often takes a distinct "V" shape. In its basic form, the absolute value function is given by \( y = |x| \). This creates a graph with its vertex at the origin \((0, 0)\). The graph is symmetric around the y-axis, and each leg of the "V" extends out diagonally.To graph \( H(x) = |x+1| \), we need to understand how this function modifies the basic absolute value graph. The term \(+1\) inside the absolute value symbol \(|x+1|\) affects the graph's horizontal position. Changes within the absolute value symbols will result in shifts, which we discuss next. This function highlights how absolute value functions can represent various transformations and translations, making them versatile in graphing.

Shifting Graphs

Shifting a graph involves moving it around on the coordinate plane without altering its shape. For the function \( H(x) = |x+1| \), we are dealing with a horizontal shift. Specifically, the \(+1\) inside the absolute value indicates a shift to the left.Here’s why: think of the transformation \( x \rightarrow x + 1 \). To make this zero (as in the bottom of the "V"), \( x \) would have to be \(-1\). So, the graph that normally centers about zero shifts left by one unit, making the vertex at \((-1, 0)\).

• Horizontal Shift: Left 1 unit
• The shape ("V" structure) remains the same.

Graph shifts are fundamental in modifying a function's location to match a problem's constraints or tasks.

Table of Values

Creating a table of values is an essential step in graphing functions. It helps in visualizing the function by pinpointing exact coordinate points. For \( H(x) = |x+1| \), we choose values around the vertex, which we've identified as \((-1, 0)\).To construct a table:

• Select \( x \) values around the vertex, such as \(-3, -2, -1, 0, 1,\) and \( 2 \).
• Calculate \( H(x) \) for each chosen \( x \) value.
• Plot these \( (x, H(x)) \) pairs to sketch the graph.

For example, when \( x = -2\), \( H(x) = |-2+1| = 1 \). The table provides a clear depiction of the absolute value function's behavior across various points, indicating how the function opens outward from the vertex.

Vertex of a Function

The vertex of an absolute value function is a pivotal point, often representing its minimum or maximum, depending on the direction the "V" opens. In our case, the function \( H(x) = |x+1| \) has a vertex at \((-1, 0)\).This point is crucial because:

• The graph changes direction at the vertex.
• For this upward opening function, \((-1, 0)\) is the minimum value.
• The vertex serves as a reference point for shifts and transformations.

Understanding the vertex's role can simplify graphing by providing a clear location from which to plot additional points and ensure symmetry. The vertex lends absolute value functions their characteristic and recognizable "V" shape.