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Chapter 1: Problem 46

?. Transformations of \(f(x)=\sqrt{x}\) Use shifts and scalings to transform the graph of \(f(x)=\sqrt{x}\) into the graph of \(g .\) Use a graphing utility to check your work. a. \(g(x)=f(x+4)\) b. \(g(x)=2 f(2 x-1)\) c. \(g(x)=\sqrt{x-1}\) d. \(g(x)=3 \sqrt{x-1}-5\)

Short Answer

Expert verified
a. \(g(x)=f(x+4)\) b. \(g(x)=2f(2x-1)\) c. \(g(x)=\sqrt{x-1}\) d. \(g(x)=3\sqrt{x-1}-5\)

Step by step solution

01

a. Transforming \(f(x)=\sqrt{x}\) to \(g(x)=f(x+4)\)

To transform the function \(f(x)\) into the function \(g(x)\), we can perform a horizontal shift of 4 units to the left. This can be represented by the transformation \(g(x) = f(x+4)\). This means, for every point on the graph of \(f(x)\), we move each point 4 units to the left to get the graph of \(g(x)\). Use a graphing utility to check the validity of the transformation.
02

b. Transforming \(f(x)=\sqrt{x}\) to \(g(x)=2f(2x-1)\)

To transform the function \(f(x)\) into the function \(g(x)\), we have to analyze the transformation given. The term \(2f\) means that we need to apply a vertical scaling of 2 to the graph of \(f(x)\). The term \(2x-1\) inside the function means that we need to perform a horizontal scaling of 0.5 and a horizontal shift of 0.5 units to the right. This can be represented by the transformation \(g(x) = 2f(2x-1)\). Use a graphing utility to check the validity of the transformation.
03

c. Transforming \(f(x)=\sqrt{x}\) to \(g(x)=\sqrt{x-1}\)

To transform the function \(f(x)\) into the function \(g(x)\), we only have one transformation to perform: a horizontal shift of 1 unit to the right. This can be represented by the transformation \(g(x) = \sqrt{x-1}\). Use a graphing utility to check the validity of the transformation.
04

d. Transforming \(f(x)=\sqrt{x}\) to \(g(x)=3\sqrt{x-1}-5\)

To transform the function \(f(x)\) into the function \(g(x)\), we need to perform three transformations. The term \(3\sqrt{x-1}\) means that we have a vertical scaling of 3 and a horizontal shift of 1 unit to the right. The term \(-5\) represents a vertical shift of 5 units downward. So, the whole transformation can be represented by \(g(x) = 3\sqrt{x-1} - 5\). Use a graphing utility to check the validity of the transformation.

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Key Concepts

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

Understanding Graph Shifts
Graph shifts are modifications we make to a graph, moving it up, down, left, or right. These shifts are crucial in understanding function transformations. For instance, if we take the function \(f(x) = \sqrt{x}\), a transformation to \(g(x) = f(x+4)\) results in a horizontal shift. This specific move shifts every point on the graph 4 units to the left. It's important to remember:
  • Adding inside the function \((f(x+c))\) shifts left,
  • While subtracting \((f(x-c))\) shifts right.
  • Vertical shifts occur when you adjust outside the function.
Checking these transformations is easier when using a graphing utility, which allows you to visualize how the function shifts on a coordinate plane.
Exploring Horizontal Scaling
Horizontal scaling affects how a graph stretches or compresses along the x-axis. In horizontal scaling, the changes happen inside the function. Let's consider the transformation from \(f(x) = \sqrt{x}\) to \(g(x) = 2f(2x-1)\). This entails horizontal scaling by a factor of 0.5 and a horizontal shift. Here's what this means:
  • The graph is compressed by a factor determined by the coefficient in front of \(x\).
  • If the coefficient is greater than 1, it compresses the graph; if it is between 0 and 1, it stretches it.
  • In this case, the coefficient is 2, so the graph compresses by a factor of 0.5.
Engaging with a graphing utility can help visualize these shifts more clearly.
Understanding Vertical Scaling
Similar to horizontal scaling, vertical scaling alters a graph鈥檚 appearance, but this time along the y-axis. Consider the transformation \(g(x) = 3\sqrt{x-1} - 5\). Vertical scaling happens when you multiply the function itself. Here:
  • We multiply the original function by 3, which stretches the graph vertically by a factor of 3.
  • Vertical compression would occur if we were multiplying by a fraction.
  • This scaling changes the height of the graph but not its horizontal positioning.
These changes affect amplitudes in trigonometric functions or steepness in linear equations. Reviewing with a graphing utility helps you see how these transformations affect the graph visually.
Using a Graphing Utility
A graphing utility is a powerful tool that helps visually confirm the transformations made to a function. With the complexity of transformations like moving from \(f(x) = \sqrt{x}\) to various \(g(x)\) forms, it's helpful to see these changes in real-time.
  • Graphing utilities can display shifts, scaling, or simultaneous transformations accurately.
  • They show exactly how transformations such as horizontal or vertical shifts affect the graph's intercepts and shape.
  • You can plot both the original and transformed functions to compare and understand the transformation easily.
Using a graphing utility, like a graphing calculator or software, is invaluable for students learning how function transformations work.

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Most popular questions from this chapter

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