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Chapter 0: Problem 18

Sketch the graph of \(g(x)=(x+1)^{3}-3\) using translations.

Short Answer

Expert verified
The graph of \(g(x) = (x+1)^3 - 3\) is a leftward shift of 1 unit and a downward shift of 3 units of the cubic function \(y = x^3\).

Step by step solution

01

Identify the Parent Function

The parent function is the base function before any transformations are applied. For the given function \(g(x) = (x+1)^3 - 3\), the parent function is \(f(x) = x^3\), which is a cubic function with a characteristic S-shaped curve centered at the origin.
02

Determine Horizontal Translation

The term \(x+1\) inside the cube indicates a horizontal translation. Because the term is \((x+1)\), we shift the parent function to the left by 1 unit. This is a result of setting \(x+1=0\). Solving gives \(x=-1\), which means the origin of the parent function moves to \(-1,0\).
03

Determine Vertical Translation

The \(-3\) outside the cube indicates a vertical translation. We move the entire graph down by 3 units. This adjusts the new point of \(-1,0\) to \(-1,-3\).
04

Sketch the Transformed Graph

Start with the curve of the basic cubic function \(y=x^3\). Shift this curve 1 unit to the left and then 3 units down. The inflection point (the point where the curve changes direction) originally at \(0,0\) is now at \(-1,-3\). The shape of the graph remains unchanged, but its position on the coordinate plane is altered based on the translations.

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Key Concepts

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

Cubic Functions
A cubic function is a type of polynomial function and it is characterized by the highest degree of three. The general form of a cubic function is \(f(x) = ax^3 + bx^2 + cx + d\), where \(a, b, c,\) and \(d\) are constants and \(a eq 0\).

The graph of a basic cubic function \(y = x^3\) features an S-shaped curve. This curve smoothly transitions from the lower left to the upper right, crossing through the origin point (0, 0).
  • It has one inflection point, which in the case of \(y = x^3\), occurs at the origin where the curve changes concavity.
  • The domain and range of a cubic function are all real numbers.
  • The symmetry around the origin means it is an odd function, meaning \(f(-x) = -f(x)\).
Understanding these properties provides a fundamental basis for applying transformations and predicting how the graph might change with modifications.
Horizontal Translation
Horizontal translation involves shifting a function left or right along the x-axis. This is determined by changes within the function's argument or equation.

For the function \(g(x) = (x + 1)^3\), the presence of \(x+1\) suggests a horizontal translation. Here's how it works:
  • The term \(x+1\) indicates a shift to the left by 1 unit. This occurs because setting \(x+1=0\) resolves to \(x=-1\).
  • If it were \(x-1\), the graph would shift to the right by 1 unit.
This transformation does not affect the shape of the graph but merely alters its position on the coordinate plane. By translating the graph of the parent cubic function \(y = x^3\), the inflection point and other key properties are shifted to their new locations.
Vertical Translation
A vertical translation moves the graph up or down along the y-axis. It results from adding or subtracting a constant to the function.

For example, in the function \(g(x) = (x+1)^3 - 3\), the subtraction of 3 indicates a downward shift.
  • This moves every point on the original graph of \((x+1)^3\) down by 3 units.
  • Vertical translation affects the y-values, changing the position of every point while maintaining the graph's shape.
In this particular case, although the graph reflects a translation downward by 3 units, the form of the S-shape in the cubic graph does not change. Its symmetry and inflection point are preserved in this new context, now positioned from \((-1, 0)\) to \((-1, -3)\).

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Most popular questions from this chapter

Write \(p(x)=1 / \sqrt{x^{2}+1}\) as a composite of four functions.

Use a computer or a graphing calculator in Problems \(37-40 .\) Let \(f(x)=x^{2}-3 x .\) Using the same axes, draw the graphs of \(y=f(x), y=f(x-0.5)-0.6,\) and \(y=f(1.5 x),\) all on the domain [-2,5] .

Let \(f(x)=\frac{a x+b}{c x-a}\). Show that \(f(f(x))=x,\) provided \(a^{2}+b c \neq 0\) and \(x \neq a / c\)

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{2}+y^{2}=4\)

Sketch the graph of each of the following, making use of translations. (a) \(y=\frac{1}{4} x^{2}\) (b) \(y=\frac{1}{4}(x+2)^{2}\) (c) \(y=-1+\frac{1}{4}(x+2)^{2}\)
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