Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Identify any symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Determine the number of \(x\) -intercepts of the graph. \(h(x)=x^{3}(x-3)^{2}\)

Step-by-Step Solution

Answer

The graph of function \(h(x)=x^{3}(x-3)^{2}\) has no symmetry with respect to the x-axis, y-axis, or the origin. The graph of the function has two x-intercepts at x = 0 and x = 3.

Step 1: Graph the function

Use a graphing utility to plot the function \(h(x)=x^{3}(x-3)^{2}\).

Step 2: Identify symmetry

To check whether the graph is symmetric with respect to the y-axis, replace \(x\) with \(-x\) in the equation. \(h(-x) = (-x)^{3}((-x)-3)^{2} = -x^{3}(x+3)^{2}\). This is not equal to the equation \(h(x) = x^{3}(x-3)^{2}\), therefore, the graph is not symmetric with respect to the y-axis. To check symmetry with respect to the origin, compare \(h(-x)\) with \(-h(x)\). We have \(-h(x) = -(x^{3}(x-3)^{2}) = -x^{3}(x-3)^{2}\), which is not equal to \(h(-x)\), so the graph is not symmetric with respect to the origin.

Step 3: Determine the number of x-intercepts

Find the x-intercepts by setting the equation equal to zero. \(0 = x^{3}(x-3)^{2}\). The solutions to this equation are the x-intercepts. We get x = 0 and x = 3 as the solutions, so there are two x-intercepts.

Key Concepts

Symmetry in Graphs

When studying graphs, symmetry helps us understand the shape and behavior of a function. Symmetry refers to the property where one part of the graph mirrors another part.

• Y-axis symmetry: This occurs when a graph mirrors itself over the y-axis. To determine this, replace every instance of \(x\) in the function with \(-x\) and compare the resulting equation. If \(h(-x) = h(x)\), the graph is symmetric about the y-axis.
• Origin symmetry: A graph has origin symmetry if rotating it 180 degrees around the origin doesn't change the graph. This means that for origin symmetry, \(h(-x) = -h(x)\).

In the case of the function \(h(x) = x^{3}(x-3)^{2}\), neither y-axis nor origin symmetry is present. Calculating both \(h(-x)\) and \(-h(x)\) yields different results from \(h(x)\), confirming the lack of symmetry.

Finding X-Intercepts

X-intercepts are critical points where a graph crosses or touches the x-axis. They represent real solutions to the equation when the output of the function, \(h(x)\), is zero.

• To find the x-intercepts, set the function equal to zero: \(x^3(x-3)^2 = 0\).
• Factor the equation if necessary and solve each part: in this case, \(x = 0\) and \(x - 3 = 0\).
• From this setup, the solutions are \(x = 0\) and \(x = 3\).

These solutions suggest that the graph of the function has x-intercepts at the points \((0,0)\) and \((3,0)\). Each intercept corresponds to a significant change in the graph's direction or curvature as it crosses the x-axis.

Exploring Polynomial Functions

Polynomial functions consist of variables raised to whole number exponents, combined using addition, subtraction, and multiplication. They are fundamental in algebra and calculus due to their straightforward properties and the ability to model numerous practical situations.

• Degree: The highest power of the variable \(x\) in a polynomial gives its degree, affecting its graph's shape. For \(h(x) = x^3(x-3)^2\), the highest power of \(x\) is 5, making it a fifth-degree polynomial.
• Roots: Roots or zeros of a polynomial like this one are the solutions to \(h(x) = 0\). They indicate where the graph intersects the x-axis.
• End behavior: This describes how the graph behaves as \(x\) approaches infinity or negative infinity, influenced by the leading term, \(x^5\), in the polynomial.

Understanding these features supports a comprehensive grasp of how polynomial functions behave graphically and analytically. As they are inherently continuous and smooth curves, polynomials provide a clear picture of mathematical relationships.