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Chapter 3: Problem 21

Write the function whose graph is the graph of \(y=x^{3},\) but is: Shifted up 4 units

Short Answer

Expert verified
The function is \( y = x^{3} + 4 \).

Step by step solution

01

- Understand the Basic Graph

Start with the given function, which is the cubic function: \( y = x^{3} \). This is the basic graph that you will modify.
02

- Apply the Shift

Since the graph needs to be shifted up by 4 units, add 4 to the function \( y = x^{3} \). This results in \( y = x^{3} + 4 \).
03

- Write the Final Function

The function after applying the vertical shift is \( y = x^{3} + 4 \). This is the final function whose graph is shifted up by 4 units compared to the original cubic function.

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Key Concepts

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

Cubic Function
A cubic function is a type of polynomial function where the highest degree is three. The general form of a cubic function is \( y = ax^3 + bx^2 + cx + d \), Here,
  • \( a \) , \( b \) , \( c \) , and \( d \) are coefficients
  • \( x \) is the variable.
  • The simplest form of a cubic function is \( y = x^3 \), which has a unique S-shaped curve through the origin.
  • Cubic functions can have different shapes but they are always curvy and will either increase or decrease as you move along the x-axis.
Cubic functions are an important part of algebra and calculus, helping in understanding complex polynomial behaviors.
Vertical Shift
A vertical shift moves the graph of a function up or down on the coordinate plane. This is achieved by adding or subtracting a constant value from the function.To apply a vertical shift:
  • Add a constant value to move the graph up.
  • Subtract a constant value to move the graph down.
For example, in the given exercise, we shift the cubic function \( y = x^3 \) up by 4 units. We do this by adding 4 to the original function: \( y = x^3 + 4 \).After the shift, the graph is still a cubic curve, but each point on the original graph is now 4 units higher. This is called a vertical shift.
Graph Transformations
Graph transformations involve changing the appearance or position of a graph without altering its shape or orientation. These transformations include:
  • Translations (shifting graphs horizontally or vertically): Adding/subtracting constants.
  • Reflections (flipping graphs across axes).
  • Stretching and compressing (scaling graphs vertically or horizontally).
In the case of the vertical shift for the cubic function, we performed a translation by moving the graph up. To understand the transformations clearly, it's important to practice with different functions and types of shifts or changes. Graphs can be combined and modified in multiple ways to suit various applications in mathematics.

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