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Chapter 3: Problem 22

Describe a sequence of transformations that will transform the graph of the function \(f\) into the graph of the function \(g.\) $$f(x)=x^{2}+x ; \quad g(x)=(x-3)^{2}+(x-3)+2$$

Short Answer

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Question: Describe the transformations required to turn the function \(f(x)=x^{2}+x\) into the function \(g(x)=(x-3)^{2}+(x-3)+2\). Answer: To transform the function \(f(x)\) into \(g(x)\), apply a horizontal shift of 3 units to the right, followed by a vertical shift of 2 units upwards.

Step by step solution

01

Identify the differences between the function equations

We have the functions: $$f(x)=x^{2}+x$$ and $$g(x)=(x-3)^{2}+(x-3)+2$$ The differences we can observe are: 1. \(x\) is replaced by \((x-3)\) in both terms of the function. 2. A constant term \((+2)\) is added to the function.
02

Identify the transformations for each difference

For each of the differences observed in Step 1, we can describe the transformations: 1. Replacing \(x\) by \((x-3)\): This corresponds to a horizontal shift of the function. Since it is replaced by \((x-3)\), the graph shifts to the right by 3 units. 2. Adding a constant term of 2: This corresponds to a vertical shift of the function. Since it is added, the graph shifts upwards by 2 units.
03

Describe the sequence of transformations

To transform the graph of the function \(f\) into the graph of the function \(g\), perform the following transformations in sequence: 1. Apply a horizontal shift of 3 units to the right. 2. Apply a vertical shift of 2 units upwards. After applying these transformations, the graph of the function \(f(x)\) will be transformed into the graph of the function \(g(x)\).

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Key Concepts

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

Horizontal Shift
The concept of a horizontal shift comes into play when you replace the variable in a function with a term involving the variable plus or minus a constant. For example, in the exercise, we go from the function \( f(x) = x^2 + x \) to \( g(x) = (x-3)^2 + (x-3) + 2 \). The replacement of \( x \) with \( (x - 3) \) indicates a horizontal shift.
  • A horizontal shift affects the graph by moving it left or right along the x-axis.
  • If the term inside the function is \( (x - c) \), the graph shifts to the right by \( c \) units. In this case, the graph of \( f(x) \) is shifted 3 units to the right.
  • Conversely, if the term is \( (x + c) \), the graph shifts to the left by \( c \) units.
This transformation does not change the shape of the graph, only its position horizontally.
Vertical Shift
Vertical shifts occur when you add or subtract a constant from the function. In the transformation from \( f(x) = x^2 + x \) to \( g(x) = (x-3)^2 + (x-3) + 2 \), the addition of \(+2\) indicates a vertical shift. Here's what happens during a vertical shift:
  • Adding a positive constant \( c \) moves the graph up by \( c \) units. Here, adding \(+2\) means the graph shifts up by 2 units.
  • Subtracting a constant \( c \) would move the graph down by \( c \) units.
  • This transformation only affects the position vertically, not the shape of the graph.
Vertical shifts are straightforward and keep the graph's orientation unchanged, making it easy to visualize where the graph ends up.
Graph Transformation
Combining horizontal and vertical shifts as seen in the exercise results in a complete graph transformation. A graph transformation alters the appearance or position of a graph in a coordinate plane. Let's see how they work together:
  • First, the horizontal shift moves the original graph right or left, affecting the x-values.
  • Next, the vertical shift moves the graph up or down, affecting the y-values.
  • The sequence of transformations is crucial 鈥� starting with the horizontal shift and followed by the vertical shift as per this exercise.
  • Each transformation is applied independently, ensuring a smooth repositioning of the graph without affecting its core shape.
Overall, graph transformations help visualize how functions change in different contexts by performing simple operations on the basic parent graph. Understanding these shifts makes it easier to predict and graphically analyze any given function.

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Most popular questions from this chapter

Determine the domain of the function according to the usual convention. $$f(t)=\sqrt{-t}$$

Write the rule of a function g whose graph can be obtained from the graph of the function \(f\) by performing the transformations in the order given. \(f(x)=\sqrt{-x} ;\) shift the graph horizontally 3 units to the left, then reflect it in the \(x\) -axis, and shrink it toward the \(x\) -axis by a factor of \(1 / 2\).

Use algebra to find the inverse of the given one-to-one function. $$f(x)=5 x-4$$

Use algebra to find the inverse of the given one-to-one function. $$f(x)=\frac{x^{3}-1}{x^{3}+5}$$

It is possible to write every even natural number uniquely as the product of two natural numbers, one odd and one a power of two. For example: \(46=23 \times 2 \quad 36=9 \times 2^{2} \quad 8=1 \times 2^{3}\) Consider the function whose input is the set of even integers and whose output is the odd number you get in the above process. So if the input is \(36,\) the output is 9. If the input is \(46,\) the output is 23 (a) Write a table of values for inputs 2,4,6,8,10,12 and 14 (b) Find five different inputs that give an output of 3
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