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Magazine Chapter 0: Problem 10
Graph each pair of equations on one set of axes. $$y=x^{3} \text { and } y=x^{3}+1$$
Short Answer
Expert verified
Graph \( y = x^3 \), then shift it up by 1 unit to get \( y = x^3 + 1 \).
Step by step solution
01
Understanding the Equations
We are given two equations to graph: 1. \( y = x^3 \)2. \( y = x^3 + 1 \)The first equation is a basic cubic function, while the second equation is the same cubic function shifted up by 1 unit.
02
Graphing the First Equation
Start by plotting the points for \( y = x^3 \). Choose several values of \( x \) to calculate the corresponding \( y \)-values:- When \( x = -2 \), \( y = (-2)^3 = -8 \)- When \( x = -1 \), \( y = (-1)^3 = -1 \)- When \( x = 0 \), \( y = 0^3 = 0 \)- When \( x = 1 \), \( y = 1^3 = 1 \)- When \( x = 2 \), \( y = 2^3 = 8 \)Now, plot these points and draw a smooth curve through them to represent the cubic function.
03
Graphing the Second Equation
Next, graph \( y = x^3 + 1 \). This is the same as \( y = x^3 \) but shifted upward by 1 unit. Use the same \( x \)-values:- When \( x = -2 \), \( y = (-2)^3 + 1 = -8 + 1 = -7 \)- When \( x = -1 \), \( y = (-1)^3 + 1 = -1 + 1 = 0 \)- When \( x = 0 \), \( y = 0^3 + 1 = 0 + 1 = 1 \)- When \( x = 1 \), \( y = 1^3 + 1 = 1 + 1 = 2 \)- When \( x = 2 \), \( y = 2^3 + 1 = 8 + 1 = 9 \)Plot these new points, and draw a smooth curve through them. Notice that each point is one unit above the corresponding point on the graph of \( y = x^3 \).
04
Final Graph Interpretation
After plotting both graphs on the same set of axes, observe that the graph of \( y = x^3 + 1 \) is simply the graph of \( y = x^3 \) shifted up by 1 unit. This shows that the vertical shift transformation has been applied correctly.
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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 with a degree of three. Its general form is given by the equation: \( y = ax^3 + bx^2 + cx + d \). In the given exercise, we are dealing with the simplest form of a cubic function: \( y = x^3 \). This function has several key characteristics:
- The graph of a cubic function typically has an 'S' shape.
- It can intersect the x-axis at up to three points, depending on its roots.
- There is one inflection point where the graph changes curvature.
Graph Shifting
Graph shifting is a transformation that alters the position of the graph on the coordinate plane without changing its shape. In this exercise, we examine a vertical shift, where every point on the graph moves up or down by the same amount. Consider the two functions in the exercise:
For \( y = x^3 + 1 \), every point on the graph of \( y = x^3 \) is moved up by 1 unit. This transformation does not affect the x-coordinates but increases every y-coordinate by 1. The result is a graph that looks identical to the original, just higher on the y-axis.
- \( y = x^3 \)
- \( y = x^3 + 1 \)
For \( y = x^3 + 1 \), every point on the graph of \( y = x^3 \) is moved up by 1 unit. This transformation does not affect the x-coordinates but increases every y-coordinate by 1. The result is a graph that looks identical to the original, just higher on the y-axis.
Plotting Points
Plotting points is a fundamental technique used to draw the graph of any function. By selecting several x-values and computing their corresponding y-values, one can plot these coordinates on a graph to see the shape of the function. In our exercise, we start with the function \( y = x^3 \).To plot this:
- Choose x-values (e.g., -2, -1, 0, 1, 2)
- Calculate the y-values based on the equation \( y = x^3 \) for these x-values.
- Plot the points: (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8).
Vertical Shift
A vertical shift changes the position of a graph up or down. In the exercise, we see this with the function \( y = x^3 \) and its shifted version \( y = x^3 + 1 \).The equation \( y = x^3 + 1 \) represents a vertical shift of the original function \( y = x^3 \) upward by 1 unit. This means that for every x-value:
- If \( y = x^3 \) gives a y-coordinate, \(y=x^3 + 1 \) will be that y-coordinate plus 1.
- Thus, points like (-2, -8) become (-2, -7), and (0, 0) becomes (0, 1).
