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Chapter 3: Problem 44

Graph each function using the techniques of shifting, compressing, stretching, and/or reflecting. Start with the graph of the basic function (for example, \(y=x^{2}\) ) and show all the steps. Be sure to show at least three key points. Find the domain and the range of each function. $$ f(x)=(x+2)^{3}-3 $$

Short Answer

Expert verified
Shift y=x^3 left by 2 units, then down by 3 units. Domain: \( (-fty, fty) \). Range: \( (-fty, fty) \).

Step by step solution

01

Identify the basic function

The basic function here is \( y = x^3 \). This is a cubic function with a typical 'S' shaped curve that passes through the origin (0,0).
02

Apply horizontal shift

The function \( f(x) = (x+2)^3 - 3 \) involves a horizontal shift. The term \( (x+2)^3 \) indicates a shift to the left by 2 units. So we will move the graph of \( y = x^3 \) leftward by 2 units. Key points on the original graph (like (-1, -1), (0, 0), and (1, 1)) will shift to (-3, -1), (-2, 0), and (-1, 1) respectively.
03

Apply vertical shift

Next, the term \( -3 \) indicates a vertical shift downward by 3 units. We adjust all key points from the previous step by moving them down by 3 units. The new key points are (-3, -4), (-2, -3), and (-1, -2).
04

Graph the transformed function

Plot the points (-3, -4), (-2, -3), and (-1, -2) on a graph and draw the curve that passes through these points. This is the graph of the function \( f(x) = (x+2)^3 - 3 \).
05

Determine the domain and range

The domain of the function \( f(x) \) is all real numbers, \( (-fty, fty) \), because there are no restrictions on the values that \( x \) can take. The range of \( f(x) \) is also all real numbers, \( (-fty, fty) \), as the cubic function extends infinitely in both the positive and negative y-directions.

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Key Concepts

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

Horizontal Shift
A horizontal shift moves the graph of a function left or right along the x-axis. In our example, the function is \( f(x) = (x+2)^3 - 3 \). The term \( (x+2)^3 \) means the graph of the basic function \( y = x^3 \) shifts to the left by 2 units. This happens because we add 2 inside the function's argument, causing each x-coordinate to decrease by 2. So, if you start with key points like (-1, -1), (0, 0), and (1, 1) on \( y = x^3 \), they will move to (-3, -1), (-2, 0), and (-1, 1) on the horizontally shifted graph of \( f(x) \).
Vertical Shift
A vertical shift moves the graph of a function up or down along the y-axis. In \( f(x) = (x+2)^3 - 3 \), the \( -3 \) indicates a vertical shift downward by 3 units. To apply this, you subtract 3 from the y-coordinates of all points on the graph after the horizontal shift. For example, after moving points to (-3, -1), (-2, 0), and (-1, 1), you shift them downward to (-3, -4), (-2, -3), and (-1, -2). This gives you the new position of the key points on the final graph of \( f(x) \).
Cubic Function
A cubic function generally has the form \( y = x^3 \). This type of function creates an 'S' shaped curve that passes through the origin (0,0). It is odd, meaning it is symmetric about the origin. For any cubic function, as x increases or decreases, the y-values will become positive or negative quickly, creating the characteristic 'S' shape. The cubic function can be transformed by shifts, stretches, compressions, and reflections, making it flexible for modeling various real-world scenarios.
Domain and Range
The domain of a function is all the possible x-values it can take, while the range is the set of all possible y-values. For our function \( f(x) = (x+2)^3 - 3 \), the domain is all real numbers, written as \( (-\infty, \infty) \). There are no restrictions on x-values for a cubic function. Similarly, the range is also all real numbers \( (-\infty, \infty) \). This is because the cubic function increases and decreases without bound, covering all possible y-values. So, no matter what value x takes, y can be any real number.

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