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Chapter 1: Problem 371

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ n(x)=\frac{1}{3}|x-2| $$

Short Answer

Expert verified
The function \( n(x) = \frac{1}{3}|x-2| \) is a right-shifted, vertically compressed version of \( |x| \). Graph has vertex at (2,0).

Step by step solution

01

Identify the Toolkit Function

The given function is \( n(x) = \frac{1}{3}|x-2| \). The toolkit function here is the absolute value function \( f(x) = |x| \). This is a well-known function that produces a V-shaped graph with its vertex at the origin (0,0).
02

Horizontal Shift

The function \( |x-2| \) indicates a horizontal shift of the basic absolute value function. The standard absolute value function \( |x| \) is shifted 2 units to the right, resulting in the graph of \( |x-2| \).
03

Vertical Compression

The factor \( \frac{1}{3} \) outside the absolute value function indicates a vertical compression. This means that the entire graph of \( |x-2| \) is compressed by a factor of \( \frac{1}{3} \), making it less steep than the graph of the basic \( |x| \).
04

Graph the Transformation

Start by graphing the standard absolute value function \( |x| \). Next, shift this graph 2 units to the right to get \( |x-2| \). Then apply the vertical compression by multiplying all y-values by \( \frac{1}{3} \). This results in the final graph of \( n(x) = \frac{1}{3}|x-2| \), which has its vertex at (2,0) and opens upwards, but is less steep due to the compression.

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Key Concepts

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

Horizontal Shift
A horizontal shift involves moving a graph left or right on the coordinate plane.
This is often done by adding or subtracting a constant from the variable inside the function. For the absolute value function, the basic form is typically given by
  • \( f(x) = |x| \)
When you have an expression like \( |x - 2| \), it tells you that the graph will shift horizontally.
The term \(-2\) indicates that the shift will be 2 units to the right.
Every point on the original graph moves to a new location that is 2 units to the right of its original position.
  • Think of it as physically picking up the graph and sliding it across the x-axis.
  • It's important to know that shifts don't affect the shape, only the position.
This step transforms the original vertex from \((0,0)\) to \((2,0)\), creating a shift in the starting point of the graph but not altering its general V-shape.
Vertical Compression
Vertical compression changes the steepness of a graph.
It has the effect of "squeezing" the graph towards the x-axis.In the function \( n(x) = \frac{1}{3}|x-2| \), we see a coefficient \( \frac{1}{3} \) in front of the absolute value.
This indicates a vertical compression by a factor of \( \frac{1}{3} \).
  • Vertical compression reduces the height of the graph, making the V-shape less steep.
  • Every y-value of the graph \(|x-2|\) is multiplied by \( \frac{1}{3} \), effectively reducing them.
This compression narrows in the range of y-values, tightening the graph closer to the x-axis compared to the original absolute value function.
The resulting shape remains upward opening; however, its sides are less inclined than \(|x|\).
Absolute Value Function
The absolute value function is essential in graph transformations as a "toolkit" function.
It is denoted by \( f(x) = |x| \), a simple yet fundamental V-shaped graph.
  • The most distinctive feature of the absolute value function is its sharp vertex.
  • Without any transformations, this vertex is at the origin \((0, 0)\).
Moreover, the graph is symmetric about the y-axis, which makes it a valuable baseline for understanding transformations.
Key characteristics include:
  • The graph always opens upwards.
  • It represents the distance from zero on the number line, which is always non-negative.
Transforming the absolute value function by shifting or compressing results in a nuanced understanding of how individual transformations affect the overall graph.
This helps in visualizing more complex functions and transformations.

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Most popular questions from this chapter

For the following exercises, use each set of functions to find \(f(g(h(x))) .\) Simplify your answers. $$ f(x)=x^{4}+6, \quad g(x)=x-6, \text { and } h(x)=\sqrt{x} $$

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ q(x)=\left(\frac{1}{4} x\right)^{3}+1 $$

For \(f(x)=\frac{1}{x}\) and \(g(x)=\sqrt{x-1},\) write the domain of \((f \circ g)(x)\) in interval notation.

For the following exercises, use each pair of functions to find \(f(g(0))\) and \(g(f(0))\) $$ f(x)=4 x+8, g(x)=7-x^{2} $$

For the following exercises, describe how the formula is a transformation of a toolkit function. Then sketch a graph of the transformation. $$ k(x)=-3 \sqrt{x}-1 $$
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