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Magazine Chapter 1: Problem 37
Graph the functions. $$y=|x-2|$$
Short Answer
Expert verified
The graph of \(y=|x-2|\) is a V-shape with the vertex at (2,0).
Step by step solution
01
Understand the Function
The function given is \( y = |x-2| \). This is an absolute value function, which means it creates a V-shaped graph with a vertex, or point of inflection. Absolute value functions reflect all negative output values to be positive.
02
Identify the Vertex of the Graph
The vertex of the function \( y = |x-a| \) is located at \( (a,0) \). In the given function \( y = |x-2| \), the vertex is at \( (2,0) \). This point is where the graph changes direction.
03
Determine the Shape and Direction
The graph of \( y = |x-2| \) is symmetrical about the vertical line \( x = 2 \). To the right of the vertex, the graph will increase linearly, while to the left, it will decrease linearly, both with a slope of 1 (i.e., 45 degrees relative to the x-axis).
04
Plot the Vertex and Some Points
Plot the vertex \( (2,0) \) on the coordinate plane. Choose some points to the left and right of 2, such as \( x = 1 \) and \( x = 3 \). When \( x = 1 \), \( y = |1-2| = 1 \), and when \( x = 3 \), \( y = |3-2| = 1 \). Similarly, when \( x = 0 \), \( y = 2 \), and when \( x = 4 \), \( y = 2 \).
05
Connect the Points with a V-Shaped Line
Connect the points \( (0,2), (1,1), (2,0), (3,1), (4,2) \) with straight lines to form a V-shape. This represents the graph of \( y = |x-2| \), with the vertex at the lowest point.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Functions
Graphing functions involves plotting points on a coordinate plane to visualize a mathematical equation. In the case of an absolute value function like \( y = |x-2| \), it produces a V-shaped graph. The basic steps to graph a function are identifying important points like the vertex and then plotting additional points to get a sense of the graph's shape.
Understanding how to graph various functions, especially those involving absolute values, enhances your ability to visualize mathematical relationships.
- Start by determining the vertex, which is the starting point of your graph.
- Next, choose points to the left and right of the vertex and calculate their corresponding \( y \)-values. These points will help you draw the rest of the graph.
- Finally, connect these points, ensuring you follow the prescribed shape dictated by the function type. For \( y = |x-2| \), this shape is a V, ensuring each arm of the V corresponds exactly to the calculated points.
Understanding how to graph various functions, especially those involving absolute values, enhances your ability to visualize mathematical relationships.
Vertex
The vertex is a significant point in graphing, especially for absolute value functions like \( y = |x-2| \). It serves as the pivot or turning point of the graph. For the equation \( y = |x-a| \), the vertex is located at \( (a, 0) \).
In the equation \( y = |x-2| \), the vertex is \( (2, 0) \). This point is crucial for drawing the graph as it marks where the direction of the graph changes.
In the equation \( y = |x-2| \), the vertex is \( (2, 0) \). This point is crucial for drawing the graph as it marks where the direction of the graph changes.
- To find the vertex, solve \( x = a \). It is straightforward since it directly impacts where the absolute value part equals zero, causing a change in direction.
- The vertex acts as a guidepost. Once pinpointed on the graph, it makes identifying the rest of the graph simpler by indicating how the function increases or decreases from that point.
Symmetrical Graph
A symmetrical graph has a balance on either side of a central line. For absolute value functions like \( y = |x-2| \), this symmetry is about a vertical line passing through the vertex. Here, it's the line \( x = 2 \), which means each side of the graph mirrors the other.
- Symmetry simplifies graphing since you only need to calculate points on one side of the vertex and mirror them across the line of symmetry.
- In equations like \( y = |x-2| \), symmetry ensures that as you move away from the vertex, the \( y \)-values are the same for equidistant \( x \)-values on either side.
- This symmetry leads to a uniform V-shaped graph, where both arms rise at the same angle, usually a \( 45^{\circ} \), representing a slope of 1.
