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Chapter 0: Problem 22

Graph. (Unless directed otherwise, assume that "Graph" means "Graph by hand.") \(y+1=x^{3}\)

Short Answer

Expert verified
Graph is a cubic curve \( y = x^3 - 1 \) shifted down by 1, passing through points (0, -1), (1, 0), and (-1, -2).

Step by step solution

01

Rearrange the Equation

Start by isolating the variable y to make it easier to identify key points and sketch the graph. The given equation is \( y + 1 = x^3 \). By subtracting 1 from both sides, the equation becomes \( y = x^3 - 1 \). This form shows that the graph is a transformation of the basic cubic function \( y = x^3 \) shifted down by 1 unit.
02

Identify Key Features of the Graph

Next, identify key characteristics of the function \( y = x^3 - 1 \). This includes determining its intercepts and symmetry. For the y-intercept, set \( x = 0 \), giving \( y = 0^3 - 1 = -1 \). So the y-intercept is (0, -1). The function is also odd (since it's symmetrical about the origin) because cube functions are odd.
03

Plot Critical Points

To sketch the graph, select a few x-values and calculate their corresponding y-values to plot on the graph. For instance: when \( x = 1, y = 1^3 - 1 = 0 \), so you have the point (1, 0). Similarly, for \( x = -1, y = (-1)^3 - 1 = -2 \), so the point is (-1, -2). This helps establish the shape of the graph around the intercepts and origins.
04

Sketch the Graph

Using the calculated points, sketch the curve beginning from the origin. Draw the smooth curve passing through the identified points (0, -1), (1, 0), and (-1, -2). Ensure the graph reflects the cubic nature with a point of inflection at the origin and extends towards infinity in both directions following the characteristic shape of a cubic function shifted down by 1 unit.

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Key Concepts

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

Equation rearrangement
Rearranging an equation is a crucial first step when graphing functions. It helps us understand the transformation that the function undergoes. Consider the given equation: \( y + 1 = x^3 \). Our goal is to isolate \( y \) to identify how the cubic function behaves. To do this, subtract 1 from both sides, giving us \( y = x^3 - 1 \).
This rearranged equation indicates a vertical shift. The graph represents the standard cubic function \( y = x^3 \), but it is moved 1 unit down along the y-axis. Through rearrangement, we unlock insights into how the function stretches or shifts, making it easier to predict the graph's appearance.
Key features identification
Identifying key features of a function is essential for sketching an accurate graph. For the equation \( y = x^3 - 1 \), we first determine intercepts and the function's symmetry.
  • **Y-Intercept:** Set \( x = 0 \). This gives \( y = 0^3 - 1 = -1 \), so the y-intercept is at the point \( (0, -1) \).
  • **Symmetry:** Cubic functions like \( x^3 \) are known to be odd functions. They exhibit symmetry around the origin. This characteristic implies that as \( x \) changes sign, \( y \) does also, maintaining the balanced shape of the graph.
Understanding these features allows us to see the overall behavior and symmetry, helping us predict more points accurately.
Plotting critical points
To bring the function graph to life, we plot calculated critical points. Selecting specific values for \( x \) simplifies the sketching process. Let's take a closer look:
  • When \( x = 1 \), we find \( y = 1^3 - 1 = 0 \), leading us to the point (1, 0).
  • Similarly, for \( x = -1 \), \( y = (-1)^3 - 1 = -2 \), resulting in point (-1, -2).
By plotting these critical points, we notice the graph's characteristic cubic S-shape. The graph should smoothly pass through the origin, demonstrating a point of inflection there. These plotted points function as guides, ensuring accuracy when sketching by hand, as the graph extends toward infinity in both positive and negative directions.

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