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Magazine Chapter 2: Problem 24
Sketch the graph of the function by first making a table of values. $$H(x)=|x+1|$$
Short Answer
Expert verified
The graph is a 'V' shape with vertex at (-1, 0).
Step by step solution
01
Understand the Function
The given function is \( H(x) = |x+1| \), which is an absolute value function. The expression inside the absolute value affects the shape of the graph.
02
Make a Table of Values
Select a range of x-values, calculate the corresponding y-values (output), and tabulate them. We will choose values around the point where the expression inside the absolute value equals zero (x = -1).- For \( x = -3 \), \( H(-3) = |-3+1| = 2 \)- For \( x = -2 \), \( H(-2) = |-2+1| = 1 \)- For \( x = -1 \), \( H(-1) = |-1+1| = 0 \)- For \( x = 0 \), \( H(0) = |0+1| = 1 \)- For \( x = 1 \), \( H(1) = |1+1| = 2 \)
03
Plot the Points
Using the table from Step 2, plot the points on a coordinate grid. The points are (-3, 2), (-2, 1), (-1, 0), (0, 1), and (1, 2).
04
Connect the Points
Connect the plotted points with straight lines to form a 'V' shape, typical of absolute value functions. The vertex or turning point of the graph is at \((-1, 0)\).
05
Sketch the Graph
Sketch the graph based on the plotted and connected points. It should have a vertex at (-1, 0) and extend symmetrically upward in both directions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Table of Values
A table of values is an essential tool when graphing functions, especially absolute value functions. To create a table of values for the function \( H(x) = |x+1| \), you must select a set of x-values. Choosing values around the critical point where the expression inside the absolute value equals zero is helpful. For \( H(x) = |x+1| \), this critical point is \( x = -1 \). This is crucial because the absolute value will cause a change in the slope of the graph at this point.
- First, choose x-values like -3, -2, -1, 0, and 1, which are around the critical point.
- Next, plug these x-values into the function to find the corresponding y-values: \(-3\), \(-2\), \(-1\), \(0\), \(1\).
- The outputs are determined by taking the absolute value: \(H(-3) = 2\), \(H(-2) = 1\), \(H(-1) = 0\), \(H(0) = 1\), \(H(1) = 2\).
Locating the Vertex of a Function
The vertex of a function, especially in absolute value functions, marks the point where the graph changes direction. For the function \( H(x) = |x+1| \), the vertex is found by determining where the expression inside the absolute value equals zero. This critical point is \( x = -1 \). When you substitute \( -1 \) into the function, the result is:
- \( H(-1) = |-1+1| = 0 \)
- This means the vertex is at the point \((-1, 0)\).
Drawing the Absolute Value Function Graph
An absolute value function graph, such as for \( H(x) = |x+1| \), typically takes on a distinctive 'V' shape. This 'V' stems from how absolute values work鈥攐utputting only non-negative results, causing the graph to change direction at its vertex.
Here鈥檚 how you can draw it using the table of values:
Here鈥檚 how you can draw it using the table of values:
- Plot each point from the table on the coordinate grid: (-3, 2), (-2, 1), (-1, 0), (0, 1), and (1, 2).
- Notice how these points fall into a 'V' formation, indicative of an absolute value function.
- Begin at the vertex, \((-1, 0)\), and connect the dots to form two straight lines extending outward.
- The lines should create a symmetrical pattern about the vertex, highlighting the reflection that occurs at this point.
