Problem 58 - Understanding Elementary Algebra with Geometry

Question

Plot a few points that satisfy the equation \(y=x^{3} .\) Do you think the graph of this equation is a straight line? Explain.

Step-by-Step Solution

Answer

The graph of \( y = x^{3} \) is a curve, not a straight line.

Step 1 - Understanding the Equation

The equation given is a cubic function: \( y = x^{3} \). A cubic function represents a polynomial of degree 3.

Step 2 - Choose Values for x

Select a few values for x to calculate corresponding values of y. For example, choose x = -2, -1, 0, 1, 2.

Step 3 - Calculate Corresponding Values of y

Substitute the chosen values of x into the equation to find y. \[ \text{For } x = -2, y = (-2)^{3} = -8 \] \[ \text{For } x = -1, y = (-1)^{3} = -1 \] \[ \text{For } x = 0, y = 0^{3} = 0 \] \[ \text{For } x = 1, y = 1^{3} = 1 \] \[ \text{For } x = 2, y = 2^{3} = 8 \]

Step 4 - Plot the Points

Plot the points found in Step 3 on a coordinate plane. The points are: (-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8).

Step 5 - Analyze the Plot

Examine the plotted points. Notice that they do not form a straight line. The points curve upwards and increase in steepness on both ends.

Step 6 - Conclusion

Since the plotted points do not lie on a straight line, the graph of the equation \( y = x^{3} \) is not a straight line but a curve. This is typical of cubic functions.

Key Concepts

polynomial equations

Polynomial equations are expressions consisting of variables and coefficients, involving operations like addition, subtraction, multiplication, and non-negative integer exponents. For example, the cubic function given in the exercise, \(y = x^{3}\), is a third-degree polynomial because the highest degree of exponent is 3.

A polynomial can have one or more terms. Each term includes a variable raised to a power and a coefficient. These equations can be used to describe a wide range of real-world situations and are the basis for more advanced mathematical concepts.

The general form of a polynomial equation is:
\[ a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x + a_{0} = 0 \]

Where:
- \(a_{n}\) are the coefficients
- \(x\) is the variable
- \(n\) is a non-negative integer representing the degree of the polynomial

The solution provided demonstrates how to handle a cubic polynomial by choosing values for \(x \), substituting them into the equation, and finding corresponding values for \(y\). This method can be applied to other polynomial equations, too, regardless of their degree.

graphing functions

Graphing functions is a critical aspect of understanding how a function behaves. To graph a function, you plot points that satisfy the equation and then connect them to visualize the relationship between the variables. This helps you understand the overall shape and characteristics of the function.

For example, in the exercise, we used the cubic function \(y = x^{3}\). After selecting some values of \(x\) and calculating the corresponding \(y\) values, we plotted the points \((-2, -8), (-1, -1), (0, 0), (1, 1), (2, 8)\) on the coordinate plane.

Once the points were plotted:
- The curve increases steeply, highlighting the rapid growth of \(y\) as \(x\) moves away from zero.
- The graph forms a curve rather than a straight line, typical of cubic functions.

Understanding the shape of a graph is essential when analyzing functions. Different types of functions, such as linear, quadratic, and cubic, have distinct shapes:

• Linear functions produce straight lines.
• Quadratic functions form parabolas.
• Cubic functions create S-shaped curves.

By graphing a function, you can better understand its properties, such as where it increases or decreases, and identify any symmetries or asymptotes.

coordinate plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It is used to graph and analyze relationships between two variables.

Each point on the coordinate plane is represented by an ordered pair \((x, y)\), where \(x\) is the coordinate on the horizontal axis and \(y\) is the coordinate on the vertical axis.

In the exercise, we plotted points on the coordinate plane:

• On the horizontal axis (x-axis), we placed the values of \(-2, -1, 0, 1, 2\).
• On the vertical axis (y-axis), we placed the values of \(-8, -1, 0, 1, 8\).

The intersection of these values marks the points where each ordered pair lies. By plotting each point on the coordinate plane, we can visualize the relationship described by our function \(y = x^{3}\).

The coordinate plane is essential for graphing because it allows us to see how one variable depends on another and identify patterns or trends in data. It is a fundamental tool in mathematics and many applications in science, engineering, and economics.