Problem 27 - Algebra and Trigonometry

Question

Graph each equation in Exercises \(13-28 .\) Let \(x=-3,-2,-1,0\) \(1,2,\) and 3. $$ y=x^{3} $$

Step-by-Step Solution

Answer

The points obtained were \((-3,-27), (-2,-8), (-1,-1), (0,0), (1,1), (2,8), (3,27)\) and after plotting these points, a cubic curve is obtained for \(y = x^{3}\).

Step 1: Calculation for Values of Y

Using each value of x provided, the corresponding value of y is calculated by cubing x using the given equation. For example, when \(x=-3\), \(y=(-3)^{3}=-27\). Repeat this process for each given value of x.

Step 2: Points Plotting

The calculated points are as follows: \((-3,-27), (-2,-8), (-1,-1), (0,0), (1,1), (2,8), (3,27)\). Next, these points are plotted on a graph.

Step 3: Graph Plotting

Now, plot all these points in x-y plane. After plotting, connect the points to draw the cubic curve.

Key Concepts

Cubing Numbers

Cubing a number means taking a number and multiplying it by itself twice. This is done to get the cube of the number, which is like raising it to the power of three. For example, if you take the number 2, cubing it involves calculating:

• First, multiply 2 by itself: 2 x 2 = 4
• Then, multiply that result by 2 again: 4 x 2 = 8

Thus, the cube of 2 is 8, written as 2³ = 8. Similarly, if you cube -3, it would be:

• -3 x -3 = 9 (two negatives make a positive)
• 9 x -3 = -27 (multiply positive by negative)

So, -3³ = -27. Understanding how to cube numbers helps in graphing cubic functions.

Plotting Points

Plotting points is essential for representing the relationship between variables in a function. Each pair of \(x, y\) coordinates represents an individual point on a graph. To plot the points, you follow these simple steps:

• Begin at the origin, where the x-axis and y-axis intersect, which is (0,0).
• From the origin, move horizontally to the x-value of the point.
• Then, move vertically to reach the y-value.

Consider the point (1,1):

• Start at the origin (0,0).
• Move 1 unit to the right to x=1.
• Move 1 unit up to y=1.

Place a dot where you land. Plotting each point accurately will ensure the resulting graph reflects the behavior of the underlying function.

Cubic Functions

A cubic function is a type of polynomial function that involves a variable raised to the power of three. These functions are expressed in the form: \[ y = ax^3 + bx^2 + cx + d \] where \(a, b, c,\) and \(d\) are constants and \(aeq 0\). For the function \(y = x^3\), only the term \(ax^3\) is present, simplifying it to \(y = x^3\) with \(a = 1\), and \(b, c, d = 0\).Cubic functions often feature interesting behavior:

• They can have up to three real roots or solutions.
• The graph is typically an S-shaped curve called an inflection point.

The S-shape differs slightly if the leading coefficient \(a\) is negative. Knowing how a cubic function behaves helps visualize and graph its specific curve.

Algebraic Graphing

Algebraic graphing involves plotting points derived from algebraic equations to visually represent the relationship between variables. To graph algebraically, you:

• Identify the function (e.g., \(y = x^3\)).
• Calculate y-values for each specified x (like in steps where \(-3\) to \(3\) were used).
• Obtain coordinates such as \((-3,-27)\) and \((3,27)\).

With these steps complete, plotting the calculated points on the graph is straightforward.

• Once points are plotted, connect them smoothly to reveal the cubic curve.
• The curve visually shows how y changes with x, highlighting key features (like inflection points).

Effective algebraic graphing helps in understanding relationships and predicting values.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which you can plot points, lines, and curves to represent mathematical equations. It consists of two perpendicular lines:

• The horizontal line is known as the x-axis (represents input or independent variable).
• The vertical line is known as the y-axis (represents output or dependent variable).

The point where these axes intersect is called the origin, denoted as (0,0). Every point on the plane has an address, or set of coordinates, in the form of \(x, y\). By utilizing the coordinate plane, one can:

• Identify locations for various points, such as \((2, 8)\) or \((0, 0)\).
• Clearly show relationships using graphs.

The coordinate plane is vital for visualizing how one variable affects another, and for solving practical problems through graphical analysis.