Problem 68 - 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. $$y=x^{4}-29 x^{2}+100$$

Step-by-Step Solution

Answer

The graph of the function \(y=x^{4}-29 x^{2}+100\) has four x-intercepts: \(x = -5, -2, 2, 5\).

Step 1: Graph the Equation

Utilize a graphing utility or software that is available to you, to graph the equation \(y = x^{4}-29x^{2}+100\). Make sure the values on the axes are set so the graph covers a reasonable range.

Step 2: Approximate the \(x\)-Intercepts

Observe the points at which the graph crosses the x-axis. These points are the \(x\)-intercepts. Save the approximate coordinates of these points.

Step 3: Solve the Equation for \(x\)-Intercepts

To find the \(x\)-intercepts mathematically, set \(y = 0\) in the equation and solve for \(x\). The equation becomes \(0 = x^{4}-29x^{2}+100\). It can be easier to solve this if it's rewritten as a quadratic equation in terms of \(x^2\). Let \(z = x^2\), then the equation becomes \(0 = z^2 - 29z + 100\). This can be factored to \((z-25)(z-4)=0\). Hence, \(z=25\) and \(z=4\). Substituting \(z\) with \(x^2\), we have \(x^2 = 25\) and \(x^2 = 4\). The solutions to these are \(x = ±5\) and \(x = ±2\).

Step 4: Compare the Results

Compare the mathematically calculated \(x\)-intercepts with those approximated from the graph. They should be very close to each other if not the same.

Key Concepts

x-intercepts

In a graph, the x-intercepts of a function represent the points where the graph crosses the x-axis. These points happen when the value of the function (y) equals zero. Understanding and finding these points is crucial, as they give us valuable information about the roots of the equation.

When looking for x-intercepts:

• Set the equation equal to zero and solve for x.
• The solutions will give the x-coordinates of the points where the graph touches or crosses the x-axis.
• For example, in our exercise with the equation \(y = x^4 - 29x^2 + 100\), we set it to zero and solve the resulting equation \(x^4 - 29x^2 + 100 = 0\) to find the x-intercepts.

These intercepts are not just mere points; they are the key to solving many polynomial equations.

quadratic equation

A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\). In our exercise, by substituting \(z = x^2\), we transformed the original polynomial equation into a quadratic form. This is a common technique that simplifies the process of solving higher-degree polynomials.

Here's the transformation process:

• Rewrite the polynomial equation in terms of another variable, such as \(z\).
• Express the polynomial as a quadratic in the new variable.
• In our specific exercise, the equation \(x^4 - 29x^2 + 100 = 0\) became \(z^2 - 29z + 100 = 0\) where \(z = x^2\).
• Once the equation is in quadratic form, you can use methods like factoring, completing the square, or the quadratic formula to find the roots.

Factoring, in this case, allows for the equation to be split into two separate linear factors, which helps in quickly finding the x-intercepts.

graphing utility

Graphing utilities, like graphing calculators or specialized software, are powerful tools used to visualize equations. They help us understand the geometric aspects of mathematical functions, making them far more tangible.

When using a graphing utility, consider the following:

• Input the equation accurately into the system.
• Adjust the viewing window to properly display the critical features of the graph.
• Look for important attributes such as intercepts, turning points, and asymptotes.
• In the given problem, a graphing utility aids in approximating the x-intercepts by showing where the graph crosses the x-axis.

These tools not only show solutions visually but also confirm the solutions derived algebraically, providing a dual-check mechanism.

polynomial function

Polynomial functions are expressions consisting of variables and coefficients, with operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of the polynomial is the highest power of the variable in the expression.

Let's delve deeper into how polynomial functions operate:

• The function \(y = x^4 - 29x^2 + 100\) in the exercise is a polynomial function of degree 4.
• These functions may have several terms, and each term has a coefficient and a variable part raised to a power.
• Graphing polynomial functions often exhibit curves with turning points, and the number of turning points is generally the degree minus one.
• Understanding the behaviour of these functions helps in solving for roots, sketching graphs, and analyzing changes in values and trends.

The manipulation of polynomial functions forms the basis for various algebraic techniques and applications.