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Chapter 12: Problem 32

Sketch the graph of \(f(x) .\) Then, graph \(g(x)\) on the same axes using the transformation techniques. $$\begin{aligned}&f(x)=|x+4|\\\&g(x)=-|x+4|\end{aligned}$$

Short Answer

Expert verified
To sketch the graph of f(x) = |x + 4|, we draw a V-shape with a vertex at (-4,0) and slopes of +1 and -1 for the sides. To graph g(x) = -|x + 4| on the same axes, we reflect f(x) across the x-axis, resulting in a downward-opening V-shape with vertex at (-4,0) and slopes of -1 and +1 for the sides. This shows the transformation from f(x) to g(x) by reflecting f(x) across the x-axis.

Step by step solution

01

Sketch the graph of f(x)

First, we need to sketch the graph of f(x) = |x + 4|. This is an absolute value function, which produces a V-shaped graph. To draw it, we should determine: a) The vertex: find the value inside the absolute value, which is x = -4. The vertex is at the point where the graph has its minimum (or maximum). To find the y-value of this vertex point, plug x = -4 into the f(x) equation: f(-4) = |(-4) + 4| = 0. Therefore, the vertex is (-4,0). b) The slope: for an absolute value function, the graph has two linear pieces on either side of the vertex. Since this is the absolute value of a linear function (x+4), the slopes are +1 and -1 for the respective sides. Now we can sketch the graph of f(x): Draw a V-shape with a vertex at the point (-4,0) and slopes of +1 and -1 for the sides.
02

Apply transformation for g(x)

Next, we need to graph g(x) = -|x + 4| on the same axes as f(x). To do this, we can use transformation techniques. Notice that g(x) is simply f(x) multiplied by -1. This means that g(x) is a reflection of f(x) across the x-axis. In other words, we can flip the graph of f(x) over the x-axis to obtain the graph of g(x).
03

Sketch the graph of g(x)

Now, we sketch the graph of g(x) by reflecting f(x) across the x-axis. This means the V-shape will open downward instead of upward. The vertex remains at the same point (-4,0), but the slopes will be the opposite of f(x): -1 and +1 for the respective sides. Draw the graph of g(x) as a downward opening V-shape with the vertex at the point (-4,0) and slopes of -1 and +1 for the sides. Now, we have successfully sketched the graphs of f(x) and g(x) on the same axes, showing the transformation of f(x) to g(x) by reflecting it across the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Function
The absolute value function is a mathematical concept that transforms both positive and negative inputs into non-negative outputs. When you graph an absolute value function, like \( f(x) = |x+4| \), you get a characteristic V-shaped graph. This V-shape is due to the nature of the absolute value, which reflects any input about the x-axis to ensure that the output is always positive or zero.
To graph these functions:
  • Identify the expression inside the absolute value, which determines the graph's horizontal shift. For \( |x+4| \), the horizontal shift is 4 units to the left since it's \( x + 4 \).
  • The graph displays two linear pieces meeting at one point, known as the vertex.
  • The vertex of the graph is found by solving \( x+4=0 \), which leads to the vertex at \((-4, 0)\).
Reflection Across the X-axis
Reflection across the x-axis is an essential transformation technique used in graphing. It flips the graph over the x-axis, converting the positive outputs to negative and vice versa.
When you reflect the graph of \( f(x)= |x+4| \) across the x-axis to get \( g(x) = -|x+4| \), every point on the graph of \( f(x) \) has its y-coordinate negated. In simpler terms, the upside-opening V of \( f(x) \) now opens downwards for \( g(x) \). The vertex stays the same at \( (-4, 0) \), but the orientation of the graph changes.
To effectively apply this transformation:
  • Recognize the operation here: multiplying the function by -1.
  • Visualize this as flipping each output of the function over the x-axis.
  • The slopes of the linear pieces connected at the vertex also become negative if they were positive and vice versa.
V-shaped Graph
A V-shaped graph is a hallmark of absolute value functions. The shape comes from the combination of two linear segments intersecting at the graph's vertex.
In a V-shaped graph, like that of \( f(x) = |x+4| \):
  • One side of the V extends upward with a positive slope, while the other side descends with a negative slope.
  • The vertex of the graph, in our case \((-4, 0)\), is at the bottom (or top, if reflected) of the V, depending on whether the graph opens upwards or downwards.
The angles of the V are typically considered sharp due to the sudden change in direction at the vertex. These graphs are symmetric along the line passing through the vertex.
Vertex of a Parabola
In the context of graph transformations, understanding the vertex of a parabola is crucial, especially for absolute value functions. Although an absolute value graph forms a V-shape, it's helpful to learn about parabolas for further transformations and calculus concepts.
The vertex represents the turning point of the graph:
  • In absolute value functions, it is the point of symmetry and typically where the graph attains its minimum or maximum value, depending on the direction it opens.
  • For \( f(x) = |x+4| \), the vertex \((-4, 0)\) is where the absolute value piecewise definition changes from one linear segment to the other.
Grasping the concept of a vertex helps in making interpretations about the graph's behavior, such as determining intervals of increase and decrease.

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Most popular questions from this chapter

Determine the domain of each function. For labor only, the Arctic Air-Conditioning Company charges \(\$ 40\) to come to the customer's home plus \(\$ 50\) per hour. These labor charges can be described by the function \(L(h)=50 h+40,\) where \(h\) is the time, in hours, and \(L\) is the cost of labor, in dollars. A. Find \(L(1)\) and explain what this means in the context of the problem. B. Find \(L(1.5)\) and explain what this means in the context of the problem. C. Find \(h\) so that \(L(h)=165,\) and explain what this means in the context of the problem.

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