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Chapter 2: Problem 12

Describing Transformations Suppose the graph of \(f\) is given. Describe how the graph of each function can be obtained from the graph of \(f\) (a) \(y=f(x+3)+2\) (b) \(y=f(x-7)-3\)

Short Answer

Expert verified
(a) Shift left 3, up 2. (b) Shift right 7, down 3.

Step by step solution

01

Identifying Horizontal Shifts (a)

The term "+3" inside the function argument indicates a horizontal shift. This shift occurs in the opposite direction of the sign within the parentheses. Therefore, the graph of the function will shift 3 units to the left.
02

Identifying Vertical Shifts (a)

The term "+2" outside the function represents a vertical shift. This shift occurs in the same direction as the sign. Therefore, the graph of the function will move 2 units up.
03

Describing Transformation (a)

The graph of the function \(y = f(x+3) + 2\) is obtained by shifting the graph of \(f\) 3 units to the left and 2 units up.
04

Identifying Horizontal Shifts (b)

Similarly, with the term "-7" inside the function argument, it indicates a horizontal shift. This time the graph of the function will shift 7 units to the right.
05

Identifying Vertical Shifts (b)

The term "-3" outside the function causes a vertical shift. The graph of the function will move 3 units down.
06

Describing Transformation (b)

The graph of the function \(y = f(x-7) - 3\) is obtained by shifting the graph of \(f\) 7 units to the right and 3 units down.

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Key Concepts

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

Horizontal Shifts
Horizontal shifts in a function graph occur when you add or subtract a number within the function鈥檚 argument, which is typically inside the parentheses next to 'x'. This number dictates how the graph moves along the horizontal axis.
A positive number will surprisingly shift the graph to the left, while a negative number will shift it to the right.
For example:
  • If you have the function聽\( f(x+3) \), the graph will shift 3 units to the left, even though you are adding to 'x'.
  • Conversely, with the function聽\( f(x-7) \), the negative sign indicates that the graph will shift 7 units to the right, opposite what you might initially think.
This might seem a bit counterintuitive at first, but remember this rule, and you'll master horizontal shifts in no time!
Vertical Shifts
Vertical shifts are usually more intuitive compared to horizontal shifts. They happen when you add or subtract a number directly from the entire function.
This term will always determine how the graph moves upward or downward along the y-axis.
  • If you add a number, such as in聽\( f(x) + 2 \), the entire graph of the function shifts up by that amount. So, here, it would move 2 units up.
  • On the other hand, with a subtraction, as in聽\( f(x) - 3 \), the graph moves downward by that many units, in this case, moving 3 units down.
In essence, vertical shifts are simply moving the function's graph up or down the exact distance specified by the number on the outside of the function.
Graph Transformations
Combining horizontal and vertical shifts results in a full graph transformation. It's like managing a coordinate map and moving it around the plane. For any function, graph transformations involve applying both types of shifts simultaneously.
For instance, for a function聽\( y = f(x+3) + 2 \), perform two movements:
  • Shift the graph of \( f \) 3 units to the left (horizontal shift).
  • Then, move it 2 units upward (vertical shift).
This sequence will give you the new position of the graph.
Similarly, another example, \( y = f(x-7) - 3 \), requires:
  • First, shifting 7 units to the right,
  • and then moving 3 units downward.
By understanding these steps, you'll develop a clearer picture of how a graph can be transformed by manipulating its function directly.

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Most popular questions from this chapter

Suppose that $$h=f \circ g$$ If \(g\) is an even function, is \(h\) necessarily even? If \(g\) is odd, is \(h\) odd? What if \(g\) is odd and \(f\) is odd? What if \(g\) is odd and \(f\) is even?

Graphing Functions Sketch a graph of the function by first making a table of values. $$g(x)=x^{2}+2 x+1$$

DISCUSS: Finding an Inverse "in Your Head" In the margin notes in this section we pointed out that the inverse of a function can be found by simply reversing the operations that make up the function. For instance, in Example 7 we saw that the inverse of $$f(x)=3 x-2 \quad \text { is } \quad f^{-1}(x)=\frac{x+2}{3}$$ because the "reverse" of "Multiply by 3 and subtract 2" is "Add 2 and divide by 3 ". Use the same procedure to find the inverse of the following functions. (a) \(f(x)=\frac{2 x+1}{5}\) (b) \(f(x)=3-\frac{1}{x}\) (c) \(f(x)=\sqrt{x^{3}+2}\) (d) \(f(x)=(2 x-5)^{3}\) Now consider another function: $$f(x)=x^{3}+2 x+6$$ Is it possible to use the same sort of simple reversal of operations to find the inverse of this function? If so, do it. If not, explain what is different about this function that makes this task difficult.

Piecewise Defined Functions In Exercises \(85-88\) we worked with real-world situations modeled by piecewise defined functions. Find other examples of real-world situations that can be modeled by piecewise defined functions, and express the models in function notation.

A home owner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period beginning on a Sunday.
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