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Chapter 2: Problem 71

Assume that \((a, b)\) is a point on the graph of \(f .\) What is the corresponding point on the graph of each of the following functions? $$ y=f(-x) $$

Short Answer

Expert verified
The corresponding point on the graph of y=f(-x) to the point (a, b) on the graph of f(x) is (-a, b).

Step by step solution

01

Reflection Over the Y-Axis

When negating the argument of a function, such as f(-x), the effect is a reflection over the y-axis. Therefore, the x-value of the corresponding point will be the negative of the original x-value.
02

Apply Reflection to Point

To find the corresponding point on the graph of y=f(-x), we negate the x-coordinate. So, the x-coordinate becomes -a instead of a, while the y-coordinate remains the same b.
03

Find the Corresponding Point

Then, the corresponding point on the graph of y=f(-x) to the point (a, b) on the function y=f(x) is (-a,b).

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Key Concepts

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

Reflection Over the Y-Axis
Imagine flipping a picture horizontally. That's similar to what happens when a function experiences a reflection over the y-axis.
When you adjust a function to become f(-x), every point's x-coordinate on the graph of y = f(x) is flipped to its opposite.
  • If you have a point (a, b) on y = f(x), this transformation alters it to (-a, b) on y = f(-x).
  • This is because only the x-value changes sign. The y-value remains constant.
This effect is purely visual and greatly impacts the symmetry of the graph. Often, reflections are used when solving problems or proving symmetry in functions, making them a crucial graphing technique.
Function Notation
Function notation is like a special shorthand in mathematics that helps describe the relationship between variables clearly. For any function f, you can express it as y = f(x). Here's what that means:
  • The letter "f" represents the function. It's the rule or operation applied to x.
  • The variable "x" is the input or the starting value we give the function.
  • "y" is often the output or result after applying f to x.
  • In the question, f(-x) means applying the function rule f to -x instead of x.
Function notation makes handling complex equations simpler and better organized. It's particularly helpful when talking about transformations or shifts in a function's graph.
Graphing Techniques
Graphing a function accurately is vital to understanding its behavior. Besides simply plotting points, several techniques help make this process easier and more insightful:
  • Reflections: Changes to the graph like f(-x) demonstrate reflections. As described, this results in mirroring the graph about the y-axis.
  • Shifting: Moving graphs up, down, left, or right. For example, y = f(x) + k shifts up if k is positive.
  • Stretching/Compressing: Involves adjusting the graph's "shape." Multiplying f(x) by a coefficient changes the slope, affecting how steep or flat the graph appears.
Graphing techniques do more than just plot a graph; they help in understanding symmetry, patterns, and connections between different functions.

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Most popular questions from this chapter

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The graph shows the amount of money, in billions of dollars, of new student loans from 1993 through 2000 . (graph can't copy) The data shown can be modeled by the function \(f(x)=6.75 \sqrt{x}+12,\) where \(f(x)\) is the amount, in billion of dollars, of new student loans \(x\) years after 1993 . a. Describe how the graph of \(f\) can be obtained using transformations of the square root function \(f(x)=\sqrt{x} .\) Then sketch the graph of \(f\) over the interval \(0 \leq x \leq 9 .\) If applicable, use a graphing utility to verify your hand-drawn graph. b. According to the model, how much was loaned in \(2000 ?\) Round to the nearest tenth of a billion. How well does the model describe the actual data? c. Use the model to find the average rate of change, in billions of dollars per year, between 1993 and 1995 Round to the nearest tenth. d. Use the model to find the average rate of change, in billions of dollars per year, between 1998 and 2000 . Round to the nearest tenth. How does this compare with you answer in part (c)? How is this difference shown by your graph? e. Rewrite the function so that it represents the amount, \(f(x),\) in billions of dollars, of new student loans \(x\) years after 1995

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