Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x^{4}-10 x^{2}+9$$

Step-by-Step Solution

Answer

The \(x\)-intercepts of the graph, corresponding to the roots of the equation, are \(x = -3, -1, 1\) and \(3\).

Step 1: Graphing the equation

Start by using a graphing utility to plot the equation \(y = x^{4} - 10x^{2} + 9\). The resulting graph is a polynomial curve with peaks and valleys.

Step 2: Approximating the \(x\)-intercepts

The \(x\)-intercepts of the graph are the points where the graph crosses the x-axis, in other words, where \(y=0\). By observing the graph, it can be seen that \(x\)-intercepts are approximately at \(x = -3, -1, 1\) and \(3\).

Step 3: Solving the equation with \(y=0\)

Setting \(y = 0\) leads to the equation \(0 = x^{4} - 10x^{2} + 9\). Solve this by factoring, firstly by setting \(z = x^{2}\) leading to \(0 = z^{2} - 10z + 9\) which can be factored to \(0 = (z-1)(z-9)\). Solve for \(z\) to get \(z=1, 9\). Re-substitute \(z = x^{2}\) into these solutions to get:\(x^{2} = 1\) (solutions: \(x = 1, -1\)), and \(x^{2} = 9\) (solutions: \(x = 3, -3\)).

Step 4: Compare results

The solutions obtained from solving the equation match the approximated \(x\)-intercepts from the graph, validating that the \(x\)-intercepts of a graph are the exact roots of the function when \(y=0\).

Key Concepts

X-Intercepts

Understanding x-intercepts is crucial when graphing polynomial functions. These intercepts represent the points where the graph crosses the x-axis, indicating that the value of the function (or y) is zero at these points. In the graph of the equation y = x^4 - 10x^2 + 9, the x-intercepts are found by closely examining where the curve meets the x-axis.

Using a graphing utility, we can visually approximate these intercepts. However, for more precision, algebraic methods such as factoring the polynomial or applying numerical algorithms can provide the exact x-intercepts. These intercepts are crucial because they offer insight into the roots of the polynomial, showing where the function changes from positive to negative or vice versa.

Factoring Polynomials

When we have an equation, such as y = x^4 - 10x^2 + 9, factoring polynomials is a powerful algebraic tool to determine the roots of the function. Factoring converts the polynomial into a product of simpler polynomials, making it easier to solve for x. In our example, we let z = x^2 to reduce the quartic polynomial to a quadratic, which seems more manageable. We obtain z^2 - 10z + 9, which further factors into (z - 1)(z - 9).

After finding the roots for z, we revert to the original variable by taking the square root of each solution. Factoring polynomials simplifies finding the roots and contributes to a deeper understanding of the polynomial's behavior.

Graphing Utility

A graphing utility is an invaluable tool for visualizing polynomial functions, such as the given example y = x^4 - 10x^2 + 9. It offers a graphical representation which helps us identify features like the x-intercepts, turning points, and end behavior. Initially, the graphing utility presents us with a curve that shows the general behavior of the polynomial.

While a graphing utility provides a visual approximation, it's also essential to confirm any visual findings algebraically. This dual approach of using both graphing and algebra ensures the accuracy of our observations about the polynomial function.

Setting Equations to Zero

The process of setting equations to zero is central when finding the roots of a polynomial equation. If we have a function y = f(x), finding where the function crosses the x-axis (the x-intercepts) involves setting y to zero and solving for x. Through this process, we're able to determine the points at which the graph of the polynomial will touch or cross the x-axis.

Setting the equation to zero and solving it, either by factoring as we did with z = x^2, or other methods, gives us the exact values of the x-intercepts. This step is crucial for understanding the roots of the polynomial and its graph.