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Chapter 2: Problem 52

Graph the functions on the same screen using the given viewing rectangle. How is each graph related to the graph in part (a)? Viewing rectangle \([-6,6]\) by \([-4,4]\) (a) \(y=\frac{1}{\sqrt{x}}\) (b) \(y=\frac{1}{\sqrt{x+3}}\) (c) \(y=\frac{1}{2 \sqrt{x+3}}\) (d) \(y=\frac{1}{2 \sqrt{x+3}}-3\)

Short Answer

Expert verified
Part (b) shifts (a) left by 3 units. Part (c) compresses (b) vertically by 1/2. Part (d) shifts (c) down by 3 units.

Step by step solution

01

Understand the Base Function

First, examine the base function in part (a), which is \( y = \frac{1}{\sqrt{x}} \). This function is only defined for \( x > 0 \) due to the square root. The graph is a decreasing curve in the first quadrant as \( x \) increases.
02

Graph the Base Function

Plot \( y = \frac{1}{\sqrt{x}} \) on the given viewing rectangle \([-6,6]\) by \([-4,4]\). Since the function is undefined for \( x \leq 0 \), focus only on the positive values of \( x \). The graph starts at positive infinity when \( x \) is near zero and approaches zero as \( x \) increases.
03

Modify for Part (b)

Graph \( y = \frac{1}{\sqrt{x+3}} \). This function results from shifting the graph of \( y = \frac{1}{\sqrt{x}} \) left by 3 units. This means we take the previous curve and move every point three units to the left on the x-axis.
04

Modify for Part (c)

Graph \( y = \frac{1}{2\sqrt{x+3}} \). This variation involves a vertical compression by a factor of 1/2 on the graph of \( y = \frac{1}{\sqrt{x+3}} \). This means we multiply all the y-values of the shifted graph in part (b) by 1/2, which makes the curve closer to the x-axis.
05

Modify for Part (d)

Graph \( y = \frac{1}{2\sqrt{x+3}} - 3 \). This involves a vertical shift downwards by 3 units applied to the graph from part (c). This means we subtract 3 from every y-coordinate of the graph in part (c), which moves the entire curve down by three units.

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Key Concepts

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

Function Shifts
Function shifts refer to the movement of a graph along the axes, either horizontally or vertically. For example, if we consider the function given in part (b), \( y = \frac{1}{\sqrt{x+3}} \), we see a horizontal shift compared to part (a), \( y = \frac{1}{\sqrt{x}} \). This graph has been shifted 3 units to the left because of the \(+3\) inside the square root.
This happens because changes within the function's argument, such as \(+3\), affect its position on the x-axis. It's like telling each point from the original graph to "move left by 3 steps."
Function shifts are intuitive once you visualize the points moving parallel to the axis you're shifting along. It is crucial to remember that:
  • Adding a number inside the function shifts it to the left.
  • Subtracting a number shifts it to the right.
Vertical Compression
Vertical compression occurs when a graph is "squeezed" toward the x-axis vertically, adjusting how "tall" the graph appears. In part (c), the function \( y = \frac{1}{2\sqrt{x+3}} \) is a vertically compressed version of \( y = \frac{1}{\sqrt{x+3}} \).
Here a factor of \(\frac{1}{2}\) is applied to the function, meaning every y-value gets halved, thus lowering each point on the graph closer to the x-axis.
Imagine taking a piece of flexible wire representing the graph and pushing it down towards the x-axis without bending it horizontally. That's what a vertical compression does:
  • Multiplying the function by a fraction between 0 and 1 shrinks it vertically.
  • If the factor was greater than 1, it would stretch the graph away from the x-axis, not compress it.
Vertical Displacement
Vertical displacement shifts the entire graph up or down along the y-axis. In part (d), the function \( y = \frac{1}{2\sqrt{x+3}} - 3 \) is derived by shifting the graph of \( y = \frac{1}{2\sqrt{x+3}} \) downward by 3 units.
This change in the y-values moves the curve further into the negative part of the y-axis. Think of this as taking a picture and sliding it down on a wall: everything shifts uniformly.
Understand the basics of vertical displacement:
  • Subtracting a number shifts the graph down.
  • Adding a number shifts the graph up.

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Most popular questions from this chapter

The amount of a commodity sold is called the demand for the commodity. The demand \(D\) for a certain commodity is a function of the price given by $$D(p)=-3 p+150$$ (a) Find \(D^{-1} .\) What does \(D^{-1}\) represent? (b) Find \(D^{-1}(30) .\) What does your answer represent?

Express the function in the form \(f \circ g\) $$H(x)=\left|1-x^{3}\right|$$

An airplane is flying at a speed of \(350 \mathrm{mi} / \mathrm{h}\) at an altitude of one mile. The plane passes directly above a radar station at time \(t=0\) (a) Express the distance \(s\) (in miles) between the plane and the radar station as a function of the horizontal distance \(d\) (in miles) that the plane has flown. (b) Express \(d\) as a function of the time \(t\) (in hours) that the plane has flown. (c) Use composition to express \(s\) as a function of \(t\) (IMAGE CAN'T COPY)

Sales Growth The annual sales of a certain company can be modeled by the function \(f(t)=4+0.01 t^{2},\) where \(t\) represents years since 1990 and \(f(t)\) is measured in millions of dollars. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=f(t) ?\) (b) Suppose you want \(t\) to represent years since 2000 instead of \(1990 .\) What transformation would you have to apply to the function \(y=f(t)\) to accomplish this? Write the new function \(y=g(t)\) that results from this transformation.

A savings account earns \(5 \%\) interest compounded annually. If you invest \(x\) dollars in such an account, then the amount \(A(x)\) of the investment after one year is the initial investment plus \(5 \% ;\) that is, \(A(x)=x+0.05 x=1.05 x .\) Find $$\begin{array}{l}A \circ A \\\A \circ A \circ A \\\A \circ A \circ A \circ A\end{array}$$ What do these compositions represent? Find a formula for what you get when you compose \(n\) copies of \(A\)
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