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Chapter 1: Problem 28

Graph each equation . Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$y=x^{3}-1$$

Short Answer

Expert verified
The points on graph would be: (-3,-28), (-2,-9), (-1,-2), (0,-1), (1,0), (2,7), (3,26). Joining these points, as x increases, y also increases, thus indicating the graph of the cubic function \(y = x^3 - 1\).

Step by step solution

01

Function Evaluation

Start by substituting the given x-values into the equation. \n For \(x=-3, y= (-3)^{3} - 1 = -28 \). \n For \(x=-2, y= (-2)^{3} - 1 = -9 \). \n For \(x=-1, y= (-1)^{3} - 1 = -2 \). \n For \(x=0, y= (0)^{3} - 1 = -1 \). \n For \(x=1, y= (1)^{3} - 1 = 0 \). \n For \(x=2, y= (2)^{3} - 1 = 7 \). \n For \(x=3, y= (3)^{3} - 1 = 26 \).
02

Point Placement

For every (x,y) pair obtained from Step 1, plot the points on the Cartesian plane. The given points are: \n (-3,-28), (-2,-9), (-1,-2), (0,-1), (1,0), (2,7), (3,26)
03

Curve Sketching

The final step is to draw the shape of the curve. Start at the lowest x-value and join the points in ascending order of x. The curve of a cubic function is smooth, without breaks.

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Key Concepts

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

Function Evaluation
Function evaluation involves substituting specific values into a function to calculate the corresponding outcomes. In the case of the function \(y = x^3 - 1\), the task is to find the outcomes for \(x = -3, -2, -1, 0, 1, 2, 3\).
Here鈥檚 how you can evaluate the function:
  • For \(x = -3\), calculate \((-3)^3 - 1 = -28\).
  • For \(x = -2\), calculate \((-2)^3 - 1 = -9\).
  • For \(x = -1\), calculate \((-1)^3 - 1 = -2\).
  • For \(x = 0\), calculate \((0)^3 - 1 = -1\).
  • For \(x = 1\), calculate \(1^3 - 1 = 0\).
  • For \(x = 2\), calculate \(2^3 - 1 = 7\).
  • For \(x = 3\), calculate \(3^3 - 1 = 26\).
Each evaluation gives a point \((x, y)\) that lies on the curve of the equation. These values tell us how the function behaves at different \(x\) values.
Plotting Points
Once you have a set of \((x, y)\) pairs from evaluating the function, the next step is plotting these points on a graph.
Plotting points involves two steps:
  • Identify the \(x\) value on the horizontal axis.
  • Find the corresponding \(y\) value on the vertical axis and mark a point where these two meet.
For the function \(y = x^3 - 1\), the pairs \((-3, -28), (-2, -9), (-1, -2), (0, -1), (1, 0), (2, 7), (3, 26)\) are plotted to represent the function's behavior visually. Each pair gives a distinct position on the graph, showing how the value of \(y\) changes with \(x\).
Cartesian Plane
The Cartesian plane is a two-dimensional graphing system used for plotting points and graphing equations. It consists of two perpendicular axes:
  • The \(x\)-axis, which is horizontal.
  • The \(y\)-axis, which is vertical.
These axes divide the plane into four quadrants. The intersection where they meet is called the origin, denoted as \((0,0)\).
In graphing \(y = x^3 - 1\), the Cartesian plane helps visualize how the function changes. You plot each \((x, y)\) point based on its coordinates and observe the shape and direction of the curve. Understanding the Cartesian plane is key to effectively interpreting the resulting graph.
Smooth Curve Sketching
After plotting all the points, the next step is to sketch a smooth curve connecting these points. Curve sketching creates a visual representation of the function showing the relationship between \(x\) and \(y\).
Here's how to sketch the curve for \(y = x^3 - 1\):
  • Begin at the point with the lowest \(x\) value.
  • Gradually connect each plotted point from left to right in the order of increasing \(x\).
  • Ensure the curve is smooth. Cubic functions typically have a single bend and do not have sharp corners.
By doing this, you create a graph that is continuous and represents the change in \(y\) as \(x\) moves from negative to positive values. This method enhances understanding of how the function behaves overall.

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