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Chapter 0: Problem 20

Sketch the graph of the equation by translating, reflecting, compressing, and stretching the graph of \(y=x^{2}, y=\sqrt{x}\), \(y=1 / x, y=|x|,\) or \(y=\sqrt[3]{x}\) appropriately. Then use a graphing utility to confirm that your sketch is correct. $$ y=\sqrt{x^{2}-4 x+4} $$

Short Answer

Expert verified
The graph is \( y = |x-2| \), a V-shape shifted 2 units to the right.

Step by step solution

01

Recognize the Function Type

The given function, \( y=\sqrt{x^2-4x+4} \), can be analyzed by recognizing it as a transformed absolute value function. Rewrite it as \( y = \sqrt{(x-2)^2} = |x-2| \). This shows the function is a transformed version of \( y = |x| \).
02

Identify Basic Transformations

The function \( y = \sqrt{(x-2)^2} \) involves a horizontal shift. The term \((x-2)\) implies a shift to the right by 2 units of the basic absolute value function \( y = |x| \). There are no vertical shifts, reflections, stretches, or compressions involved.
03

Sketch the Graph

Start with the graph of \( y = |x| \), which is a V-shaped graph opening upwards. With the transformation identified, shift this graph 2 units to the right. The vertex of the graph moves from the origin (0,0) to the point (2,0).
04

Check with Graphing Utility

Use a graphing utility to plot \( y = \sqrt{x^2 - 4x + 4} \) or equivalently \( y = |x-2| \). Verify that the graph is a V-shape, with the vertex at (2,0), confirming our sketch is correct.

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Key Concepts

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

Absolute Value Function
In the original exercise, we transform the expression \( y = \sqrt{x^2 - 4x + 4} \) into \( y = |x-2| \). This transformation demonstrates that we're dealing with an absolute value function. An absolute value function can be recognized by the notation \( |x| \), which represents the distance of \( x \) from zero on a number line. Basically, this function creates a V-shaped graph that is symmetric around the y-axis.

The absolute value function outputs non-negative values because it measures distance. This means that no matter the input, the graph will not dip below the x-axis.
  • The basic graph of an absolute value function, \( y = |x| \), has its vertex at the origin (0,0).
  • The two legs of the graph open upwards at identical angles to the x-axis from the origin, forming a V shape.
Horizontal Shift
The original function \( y = \sqrt{x^2 - 4x + 4} \) can be rewritten as \( y = |x-2| \), which illustrates a horizontal shift of the basic absolute value graph. This occurs due to the expression \((x-2)\). When we see \(x - h\) inside the absolute value, it represents a shift to the right by \(h\) units.

In this case, the transformation has shifted the graph 2 units to the right. This kind of manipulation doesn't change the shape of the graph, just its position along the x-axis. It impacts the vertex of the graph, moving it from its standard position at (0,0) to its new position at (2,0).
  • Horizontal shifts occur when a constant is added or subtracted inside the function.
  • A shift to the right is represented by a negative value inside the function \((x-h)\).
  • Conversely, a shift to the left would be shown by a positive value \((x+h)\).
Basic Transformations
Basic transformations refer to the various ways we can manipulate a parent function. These include shifting, reflecting, stretching, and compressing the graph. For the exercise, the main focus was on horizontal shifting, as identified by transforming \( y = \sqrt{x^2 - 4x + 4} \) into the absolute value function \( y = |x-2| \).

However, transformations can take many forms:
  • Reflections: Flipping the graph over the x-axis or y-axis.
  • Vertical Shifts: Moving the graph up or down without affecting its horizontal position.
  • Horizontal Shifts: Moving the graph left or right on the x-axis.
  • Stretches and Compressions: Changing the steepness or wideness of the graph.
With our specific function, only horizontal shifting has occurred. A good understanding of these transformations makes it easier to predict how a graph will alter when different changes are applied.
Sketching Graphs
Sketching graphs involves visually representing mathematical functions on a coordinate plane. For this exercise, understanding how to graph the absolute value function and applying horizontal shifts is crucial.

To sketch the transformed graph \( y = |x-2| \):
  • Start by sketching the parent graph, \( y = |x| \).
  • Identify the location of the vertex, which in the absolute value graph is the lowest point.
  • Shift this vertex 2 units to the right, moving it from (0,0) to (2,0).
  • Draw the two lines of the V-shape from this new vertex; both lines should rise at a 45-degree angle from the x-axis.
Using these steps will help create an accurate representation of the function on a graph. Additionally, using a graphing utility can confirm the sketch's accuracy by comparing it to the plotted graph. This practice helps solidify understanding and ensure the graph's precision.

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Most popular questions from this chapter

A variable \(y\) is said to be inversely proportional to the square of a variable \(x\) if \(y\) is related to \(x\) by an equation of the form \(y=k / x^{2},\) where \(k\) is a nonzero constant, called the constant of proportionality. This terminology is used in these exercises. According to Coulomb's \(\operatorname{law},\) the force \(F\) of attraction between positive and negative point charges is inversely proportional to the square of the distance \(x\) between them. (a) Assuming that the force of attraction between two point charges is 0.0005 newton when the distance between them is 0.3 meter, find the constant of proportionality (with proper units). (b) Find the force of attraction between the point charges when they are 3 meters apart. (c) Make a graph of force versus distance for the two charges. (d) What happens to the force as the particles get closer and closer together? What happens as they get farther and farther apart?

Determine whether the statement is true or false. Explain your answer. The natural logarithm function is the logarithmic function with base \(e .\)

Expand the logarithm in terms of sums, differences, and multiples of simpler logarithms. $$ \text { (a) } \log \frac{\sqrt[3]{x+2}}{\cos 5 x} \quad \text { (b) } \ln \sqrt{\frac{x^{2}+1}{x^{3}+5}} $$

Find a formula for \(f^{-1}(x),\) and state the domain of the function \(f^{-1}\). $$ f(x)=\sqrt{x+3} $$

Let \(f(x)=x^{2}, x>1,\) and \(g(x)=\sqrt{x}\) (a) Show that \(f(g(x))=x, x>1,\) and \(g(f(x))=x\) \(x>1\) (b) Show that \(f\) and \(g\) are not inverses by showing that the graphs of \(y=f(x)\) and \(y=g(x)\) are not reflections of one another about \(y=x\) (c) Do parts (a) and (b) contradict one another? \(\mathrm{Ex}-\) plain.
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