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Chapter 2: Problem 28

A function \(f\) is given, and the indicated transformations are applied to its graph (in the given order). Write the equation for the final transformed graph. \(f(x)=x^{3} ;\) shift downward 1 unit and shift 4 units to the left

Short Answer

Expert verified
The transformed function is \( f(x) = (x+4)^3 - 1 \).

Step by step solution

01

Understand the Function

The original function is given as \( f(x) = x^3 \). The graph of this function is a standard cubic curve.
02

Apply the Vertical Shift

The first transformation is a downward shift by 1 unit. This means we subtract 1 from the function: \( f(x) = x^3 - 1 \).
03

Apply the Horizontal Shift

The second transformation is a horizontal shift to the left by 4 units. This requires substituting \( x+4 \) for \( x \) in the function: \( f(x) = (x+4)^3 - 1 \).
04

Write the Final Transformed Equation

After applying both transformations, the final equation of the transformed function is \( f(x) = (x+4)^3 - 1 \). This accounts for both the shift downward 1 unit and the shift 4 units to the left.

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Key Concepts

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

Vertical Shift
When considering vertical shifts in functions, it's crucial to grasp that these transformations involve moving the graph up or down the coordinate plane. A vertical shift occurs when a constant is added or subtracted from the original function. In the context of our cubic function, which has the form of \(f(x) = x^3\), a vertical shift is achieved by adjusting its output values.
  • If you add a constant, the graph will shift upwards by that many units.
  • If you subtract a constant, like we did with the transformation \(f(x) = x^3 - 1\), the graph will shift downwards. Here, the graph moves 1 unit down.
Vertical shifts are helpful for modeling situations where an entire function has to be adjusted vertically, allowing for changes in the baseline or origin of the data without altering the overall shape of the graph.
Horizontal Shift
Horizontal shifts involve moving the graph of a function left or right. This requires altering the input of the function, rather than its output. In formula terms, a horizontal shift involves adding or subtracting a constant inside the function's variable. Specifically for our cubic function \(f(x) = x^3\), to shift the graph 4 units to the left, you replace \(x\) with \(x+4\). This alteration gives us the transformed function \(f(x) = (x+4)^3 - 1\).
  • Shifting left involves adding a constant to \(x\) within the function: \(f(x)\) becomes \(f(x+c)\).
  • Shifting right would involve subtracting a constant from \(x\), translating the function to \(f(x-c)\).
Understanding horizontal shifts is particularly useful when analyzing how function behaviors change with different starting points, or when aligning multiple functions on a graph.
Cubic Function
A cubic function is a type of polynomial function of degree three. The general form is \(f(x) = ax^3 + bx^2 + cx + d\), but in its simplest form, like in our problem, it is \(f(x) = x^3\). Cubic functions are known for their distinct S-shaped curve, sometimes referred to as an inflection point. Here are some characteristics of cubic functions:
  • They can have one real root or three real roots, depending on the discriminant.
  • The curve can start in one direction and reverse, crossing the x-axis possibly more than once.
  • Cubic functions are powerful for modeling scenarios where transformations and complex changes are involved because they can capture upwards and downwards trends within the same function.
In many math problems, like ours, cubic functions are transformed through vertical and horizontal shifts to better fit certain data or conditions, making them versatile tools for analysis.

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Most popular questions from this chapter

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