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Chapter 0: Problem 38

Let \(f(x)=x^{2}\). Graph the functions \(f(x+1), f(x-1)\), \(f(x+2)\), and \(f(x-2)\). Make a guess about the relationship between the graph of a general function \(f(x)\) and the graph of \(f(g(x))\), where \(g(x)=x+a\) for some constant \(a\). Test your guess on the functions \(f(x)=x^{3}\) and \(f(x)=\sqrt{x}\).

Short Answer

Expert verified
Shifting \( f(x) \) by \( a \) units horizontally yields \( f(x+a) \), a left shift for \( a > 0 \), and a right shift for \( a < 0 \).

Step by step solution

01

Understand the Original Function

The original function is given as \( f(x) = x^2 \). This is a parabola centered at the origin (0,0) with its vertex at this point, and it opens upwards.
02

Analyze Shifts of f(x)

The exercise requires graphing \( f(x+1) \), \( f(x-1) \), \( f(x+2) \), and \( f(x-2) \). These are horizontal shifts of the original function.
03

Graph f(x+1)

Shift the graph of \(f(x) = x^2\) one unit to the left to obtain \( f(x+1) = (x+1)^2 \).
04

Graph f(x-1)

Shift the graph of \(f(x) = x^2\) one unit to the right to obtain \( f(x-1) = (x-1)^2 \).
05

Graph f(x+2)

Shift the graph of \(f(x) = x^2\) two units to the left to obtain \( f(x+2) = (x+2)^2 \).
06

Graph f(x-2)

Shift the graph of \(f(x) = x^2\) two units to the right to obtain \( f(x-2) = (x-2)^2 \).
07

Generalize the Relationship

The general form \( f(g(x)) \), where \( g(x) = x + a \), represents a shift in the graph of the function. If \( a > 0 \), the function shifts to the left by \( a \) units. If \( a < 0 \), the function shifts to the right by \( |a| \) units.
08

Test the Guess with f(x) = x^{3}

For \( f(x) = x^{3} \), consider \( f(x+a) \). For example, \( f(x+1) = (x+1)^{3} \) and \( f(x-1) = (x-1)^{3} \). These represent similar horizontal shifts.
09

Test the Guess with f(x) = \sqrt{x}

For \( f(x) = \sqrt{x} \), consider \( f(x+a) \). For example, \( f(x+1) = \sqrt{x+1} \) and \( f(x-1) = \sqrt{x-1} \) (domain restricted to where it is real-valued). These also represent horizontal shifts.

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Key Concepts

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

horizontal shifts
Horizontal shifts are transformations that move the graph of a function left or right in the coordinate plane. When we have a function of the form \( f(g(x)) \) where \( g(x) = x + a \), this results in a horizontal shift. If \( a \) is positive, the function shifts to the left by \( a \) units, because the inputs must be smaller for \( g(x) \) to achieve the same value. Conversely, if \( a \) is negative, the function shifts to the right by \( |a| \) units, as the inputs must be larger to maintain the same outputs.
graphing functions
Graphing functions involves plotting points or lines in a coordinate plane to visualize the behavior of the function. To start, you can plot individual points by substituting values for \( x \) and solving for \( y \). Connecting these points will show the overall shape. Understanding transformations like shifts, stretches, and reflections helps adjust existing graphs to map new functions without re-plotting all points each time. For example, knowing how to apply horizontal shifts can streamline the process when graphing shifted parabolas, cubic functions, and square root functions.
parabolas
Parabolas are the graphs of quadratic functions, taking the form \( f(x) = ax^2 + bx + c \). They are U-shaped curves that can open upwards or downwards. The graph of \( f(x) = x^2 \) is a standard parabola centered at the origin with vertex (0,0). Horizontal shifts affect where the vertex is located. If the parabola is shifted left or right, this is due to the function being modified to \( f(x + a) \) or \( f(x - a) \). The vertex will move to (-a, 0) or (a, 0) accordingly, changing the function鈥檚 visual placement but retaining its general shape.
cubic functions
Cubic functions, expressed as \( f(x) = x^3 \), generate a graph with an S-like curve. Shifts in cubic functions, similar to quadratic functions, happen horizontally when the input variable is altered. For example, \( f(x+1) = (x+1)^3 \) shifts the graph of \( f(x) = x^3 \) one unit to the left, while \( f(x-1) = (x-1)^3 \) shifts it one unit to the right. These horizontal transformations do not change the overall cubic shape but simply reposition the starting point of the curve.
square root functions
Square root functions have the form \( f(x) = \sqrt{x} \), creating a curve that starts at (0,0) and increases slowly initially before accelerating. Applying a horizontal shift to a square root function, like \( f(x+1) = \sqrt{x+1} \), shifts the entire curve to the left by one unit. Conversely, \( f(x-1) = \sqrt{x-1} \) moves it to the right by one unit. It is important to note the domain restrictions in these functions, such as \( x+a \) needing to be non-negative to maintain real-number outputs, illustrating how transformations can affect not just appearance but also the functional domain.

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Most popular questions from this chapter

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