Problem 7
Question
In \(7-10,\) the domain of each function is the set of real numbers. a. Sketch the graph of each function. b. What is the range of each function? $$ y=|x-1| $$
Step-by-Step Solution
Verified Answer
The graph is a V-shape with vertex at (1,0) and range \([0, \, \infty)\).
1Step 1: Understand the Function
The function given is \( y = |x-1| \). This is an absolute value function, which means it reflects all negative outputs to be positive. This particular function represents a V-shaped graph, shifted 1 unit to the right on the x-axis, due to the \(x-1\) inside the absolute value.
2Step 2: Find Critical Points
To graph the function, let's start by finding the vertex, which is the turning point of the V-shape. Set the inside of the absolute value to zero: \( x-1 = 0 \) gives \( x = 1 \). At this point, the function value \( y = |1-1| = 0 \), so the vertex is at \((1, 0)\).
3Step 3: Choose Additional Points
Select points on either side of the vertex to understand the shape. Choose \( x = 0 \) and \( x = 2 \). For \( x = 0 \), \( y = |0-1| = 1 \). For \( x = 2 \), \( y = |2-1| = 1 \). Therefore, the points \((0, 1)\) and \((2, 1)\) help in sketching the right V-shape.
4Step 4: Sketch the Graph
Draw the vertex at \((1, 0)\) and the points \((0, 1)\) and \((2, 1)\). Sketch the V-shape where the vertex is the lowest point, and it opens upwards and symmetrically.
5Step 5: Determine the Range
The range of the function is determined by the possible values of \( y \). Since the absolute value is always non-negative, the smallest value \( y \) can take is 0, which occurs at the vertex. As \( x \) moves away from 1, \( y \) increases. Therefore, the range of the function is \([0, \, \infty)\).
Key Concepts
Domain and RangeGraphing FunctionsVertex of a Function
Domain and Range
The domain of a function refers to all the possible input values (in this case, values of \( x \)) for which the function is defined. For absolute value functions like \( y = |x-1| \), the domain encompasses all real numbers. This implies you can plug any real number into the function and receive a valid output for \( y \).
The range, on the other hand, represents all possible output values (values of \( y \)) that the function can produce. A characteristic feature of absolute value functions is their ability to return only non-negative values. The graph of \( y = |x-1| \) has a vertex at \((1, 0)\), the lowest point of the function. As a result, the smallest possible \( y \) value is 0. However, there is no upper limit on the values \( y \) can reach, as \( x \) increases or decreases infinitely. Therefore, the range of this absolute value function is \([0, \, \infty)\).
In summary, for \( y = |x-1| \):
The range, on the other hand, represents all possible output values (values of \( y \)) that the function can produce. A characteristic feature of absolute value functions is their ability to return only non-negative values. The graph of \( y = |x-1| \) has a vertex at \((1, 0)\), the lowest point of the function. As a result, the smallest possible \( y \) value is 0. However, there is no upper limit on the values \( y \) can reach, as \( x \) increases or decreases infinitely. Therefore, the range of this absolute value function is \([0, \, \infty)\).
In summary, for \( y = |x-1| \):
- Domain: all real numbers \( (-\infty, \, \infty) \)
- Range: \([0, \, \infty)\)
Graphing Functions
Graphing functions involves plotting their output values for a set of input values, and it provides a visual representation. Consider the function \( y = |x-1| \), which is an absolute value function forming a V-shape. Let's see how you can graph this function easily.
First, identify the vertex, which is the function's turning point. For the function \( y = |x-1| \), find the vertex by setting \( x-1 = 0 \), leading to \( x = 1 \). This means that \( y = |1-1| = 0 \), placing the vertex at point \((1, 0)\).
Next, to sketch the graph effectively, you'll need additional points around the vertex. Choose points like \( x = 0 \) and \( x = 2 \). For \( x = 0 \), \( y = |0-1| = 1 \), and for \( x = 2 \), \( y = |2-1| = 1 \). These points help form the V-shape, symmetric about the vertex.
When you sketch the graph, plot the vertex and these points, and draw the V-shape that opens upwards. The function is symmetric because the absolute value operation negates any negative result for \( x \).
In conclusion, graphing the function \( y = |x-1| \) involves:
First, identify the vertex, which is the function's turning point. For the function \( y = |x-1| \), find the vertex by setting \( x-1 = 0 \), leading to \( x = 1 \). This means that \( y = |1-1| = 0 \), placing the vertex at point \((1, 0)\).
Next, to sketch the graph effectively, you'll need additional points around the vertex. Choose points like \( x = 0 \) and \( x = 2 \). For \( x = 0 \), \( y = |0-1| = 1 \), and for \( x = 2 \), \( y = |2-1| = 1 \). These points help form the V-shape, symmetric about the vertex.
When you sketch the graph, plot the vertex and these points, and draw the V-shape that opens upwards. The function is symmetric because the absolute value operation negates any negative result for \( x \).
In conclusion, graphing the function \( y = |x-1| \) involves:
- Finding the vertex at \((1, 0)\)
- Selecting additional points like \((0, 1)\) and \((2, 1)\)
- Drawing a symmetric V-shape opening upwards
Vertex of a Function
Understanding the vertex of a function is essential for graphing and analyzing its behavior. In the context of absolute value functions such as \( y = |x-1| \), the vertex is the point where the direction of the graph changes. It stands as the pinnacle where the V-shape either peaks or converges.
To find the vertex for the function \( y = |x-1| \), we focus on the expression inside the absolute value sign. By setting \( x-1 = 0 \), we solve for \( x = 1 \). At this point, substitute \( x = 1 \) back into the function to find \( y = |1-1| = 0 \). Therefore, the vertex of the function is at \((1, 0)\).
The vertex isn't just a graph coordinate; it tells us about the function's minimal value, as well. In the graph of \( y = |x-1| \), \( y \) reaches its smallest possible value at this vertex, which is 0. This reveals important insights:
To find the vertex for the function \( y = |x-1| \), we focus on the expression inside the absolute value sign. By setting \( x-1 = 0 \), we solve for \( x = 1 \). At this point, substitute \( x = 1 \) back into the function to find \( y = |1-1| = 0 \). Therefore, the vertex of the function is at \((1, 0)\).
The vertex isn't just a graph coordinate; it tells us about the function's minimal value, as well. In the graph of \( y = |x-1| \), \( y \) reaches its smallest possible value at this vertex, which is 0. This reveals important insights:
- The vertex location dictates where the graph changes direction, crucial for sketching the function accurately.
- The vertex indicates the absolute minimum value of \( y \) in functions resembling \( y = |x-1| \).
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