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Chapter 3: Problem 13

The graph of \(f(x)=\frac{1}{x+6}-3\) is the graph of \(y=\frac{1}{x}\) shifted (left/right) 6 units and (up/down) 3 units.

Short Answer

Expert verified
Shift left 6 units and down 3 units.

Step by step solution

01

Identify the Base Function

Consider the base function, which is given by: \[ y = \frac{1}{x} \]
02

Determine Horizontal Shift

Notice the term in the denominator of the given function: \[ f(x) = \frac{1}{x+6} - 3 \] This represents a horizontal shift. Specifically, the \(x + 6\) term indicates a shift to the left by 6 units.
03

Determine Vertical Shift

Next, examine the constant subtracted outside the fraction: \[ f(x) = \frac{1}{x+6} - 3 \] This results in a vertical shift. The \(-3\) term means the graph is shifted down by 3 units.

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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 changes the position of the graph along the x-axis. In our function, the term \(x + 6\) inside the fraction \(f(x) = \frac{1}{x+6} - 3\) handles this shift. Here, adding 6 to \(x\) means every x-value on the graph of \(y = \frac{1}{x}\) moves to the left by 6 units.

To visualize it:
  • Start by graphing \(y = \frac{1}{x}\).
  • Next, move every point on this graph 6 units to the left because of \(+6\) in the denominator.
Horizontal shifts may seem tricky, but remember that adding a positive number inside the function makes the graph go in the negative x-direction.

So, \(x+6\) means shifting 6 units left, while \(x-6\) means moving 6 units right.
Vertical Shift
The vertical shift changes the position of the graph along the y-axis. Look at the term outside the fraction in the function \(f(x) = \frac{1}{x+6} - 3\).

The \(-3\) term tells us to move the entire graph of \(\frac{1}{x+6} \) down by 3 units.

Here's how you do it:
  • Start with the horizontally shifted graph you got from the \(y = \frac{1}{x}\) base function.
  • Move this entire graph down by 3 units, because of the \(-3\) term.
Shifting vertically is straightforward:

- If you have \(+c\) (like \( \frac{1}{x+6} +3 \)), move the graph up by \(c\) units.
- If you have \(-c\) (like in our function), move the graph down by \(c\) units.

Now, you should see how the combination of horizontal and vertical shifts transforms the graph.
Rational Functions
A rational function is any function that can be written as the quotient of two polynomials. Our original function \(f(x) = \frac{1}{x+6} - 3\) is a classic example of a rational function.

To understand better:
  • The numerator is \(1\): a constant polynomial.
  • The denominator is \(x+6\): a first-degree polynomial.
Rational functions can have:
  • Vertical asymptotes, where the denominator equals zero. For our function, it鈥檚 at \(x = -6\).
  • Horizontal asymptotes, indicating the end behavior of the graph. Our graph approaches \(y = -3\) as \(x\) heads to infinity.
As you plot, remember these features to get a precise graph. Rational functions, like ours here, are a great way to explore shifts and understand a wide range of transformations.

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Most popular questions from this chapter

Use a variation model to solve for the unknown value. The cost to carpet a rectangular room varies jointly as the length of the room and the width of the room. A 10-yd by 15 -yd room costs \(\$ 3870\) to carpet. What is the cost to carpet a room that is 18 yd by 24 yd?

Explain why \(x=-2\) is not a vertical asymptote of the graph of \(f(x)=\frac{x^{2}+7 x+10}{x+2}\).

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