Problem 98 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find three points that lie on the graph of the equation. (There are many correct answers.) $$y=\frac{1}{5} x^{3}-4 x^{2}+1$$

Step-by-Step Solution

Answer

The three points that lie on the graph of the given equation are (-1,-3), (0,1), and (1, -2.2).

Step 1: Choose x values

We will choose three x values, any numbers could work but for simplicity we will take -1, 0 and 1.

Step 2: Calculate the y value for x = -1

To calculate the y value for x = -1, we substitute x = -1 into the equation, yielding \(y=\frac{1}{5}*(-1)^{3}-4*(-1)^{2}+1 = -1/5 - 4 + 1 = -15/5 = -3\). So, the first point is (-1,-3).

Step 3: Calculate the y value for x = 0

When x = 0, only the constant remains after substitution, so y = 1. This gives us the second point (0,1).

Step 4: Calculate the y value for x = 1

When x = 1, after substitution, we get \(y=\frac{1}{5} * 1^3 - 4 * 1^2 + 1 = 1/5 - 4 + 1 = -11/5 = -2.2\). Therefore, the third point is (1, -2.2).

Key Concepts

Graphing Polynomials

Graphing polynomial functions is essential to visualize their behavior and characteristics. This includes understanding their shape, intercepts, and points of interest. Let's break down how to graph a basic polynomial, such as the one presented in the exercise: \( y = \frac{1}{5}x^3 - 4x^2 + 1 \).

To start graphing a polynomial, we must identify key features like:

• Intercepts: Where the graph crosses the axes. The y-intercept is found by evaluating the polynomial at \( x = 0 \). The x-intercepts are where \( y = 0 \), but solving this can be complex, especially for higher-degree polynomials.
• Turning Points: Points of local maxima or minima, where the graph changes direction. These are often found by using calculus, differentiating the polynomial to find critical points.
• End Behavior: How the graph behaves as \( x \) approaches positive or negative infinity. For cubic functions, the end behavior is determined by the leading term \( x^3 \), indicating the graph will go to positive infinity in one direction and negative infinity in the other.

To effectively graph, plot a few points by calculating \( y \) values for chosen \( x \) values, as seen in the exercise, to create a rough sketch and visualize the full curve.

Finding Points on Graphs

Finding points on the graph of a polynomial equation involves evaluating the function at specific x-values. This simple process helps confirm the overall shape and behavior of the graph.

Consider the example polynomial from the exercise: \( y = \frac{1}{5}x^3 - 4x^2 + 1 \). To find some points:

• First, choose x-values, which can be any numbers. In the exercise, the values chosen are -1, 0, and 1 for simplicity.
• Next, substitute each x-value into the polynomial equation to find the corresponding y-value.
• For \( x = -1 \), substituting gives \( y = -3 \), so the point is (-1, -3).
• For \( x = 0 \), the y-value is \( 1 \), providing the point (0, 1).
• Lastly, for \( x = 1 \), the y-value calculated is \( -2.2 \), resulting in the point (1, -2.2).

These points can be plotted to enhance understanding of the curve and ensure accuracy when graphing.

Cubic Equations

Cubic equations are mathematical expressions that involve a variable raised to the third power. They take the general form \( ax^3 + bx^2 + cx + d = 0 \), where \( a, b, c, \) and \( d \) are constants, with \( a eq 0 \). Understanding the nature of cubic equations is vital in various areas of algebra and calculus.

Some characteristics of cubic equations include:

• Roots: A cubic equation can have up to three real roots. These are the solutions to the equation where the graph intersects the x-axis if real.
• Curve Shape: The graph of a cubic function typically has an 'S' shape, with varying degrees of steepness and curvature influenced by the coefficients. The leading coefficient \( a \) dictates the end behavior, making the graph rise or fall across the x-axis.
• Symmetry and Turns: Unlike quadratic equations, cubic equations can cross the x-axis up to three times, creating potential maxima and minima.

In the context of the exercise, the given cubic equation \( y = \frac{1}{5}x^3 - 4x^2 + 1 \) shows how altering coefficients changes the shape and position of the graph, influencing critical points and intercepts.