Problem 123 - Algebra and Trigonometry

Question

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned}&-x^{4}+4 x^{3}-4 x^{2}=0\\\&[-6,6,1] \text { by }[-9,2,1]\end{aligned}$$

Step-by-Step Solution

Answer

The solutions are the x-values where the plot crosses the x-axis which are validated if they equal to zero when plugged back into the original equation.

Step 1: Graph the Equation

For the equation \(-x^{4}+4 x^{3}-4 x^{2}=0\), set up your graphing utility to the given viewing rectangle [-6,6,1] by [-9,2,1]. This means your x-values will range from -6 to 6 and your y-values will range from -9 to 2. Once you have set up your graphing utility, plot the equation.

Step 2: Identify the x-intercepts

The x-intercepts of any plot or graph on a Cartesian plane are points where the function crosses the x-axis, meaning where the y-value is zero. Identify these points on your graph, which are the real solutions for the equation.

Step 3: Verify the Solutions

Now, directly substitute the identified x-values from step two into the original equation. These values should make the equation equal to zero. If they do, then they are the solutions to the equation. If not, revisualize the diagram and identify the intersect values again.

Key Concepts

Understanding x-intercepts

In any graph of a polynomial equation, the x-intercepts are crucial features to identify. These are the points on the graph where the curve crosses the x-axis, meaning the y-value is zero here.
To find the x-intercepts, simply look for the values of \(x\) where the output of the function (\(y\)) is zero. For the polynomial equation \(-x^4 + 4x^3 - 4x^2 = 0\), find the points on the plotted graph where the line meets the x-axis. This is where the x-intercepts are located.

• The x-intercepts represent the real roots of the equation.
• These points give an insight into how the graph behaves and provide the solutions to the equation.
• They help us understand the relationship between factors and the overall polynomial.

In many cases, especially for complex polynomial equations, accurately finding and identifying these intercepts is key to solving the equation.

Utilizing a graphing utility

A graphing utility is a powerful tool for visualizing and solving polynomial equations. This tool allows you to plot the function, making it easier to see where the graph intersects with the x-axis.

• To use a graphing utility, input your equation and set the viewing rectangle to encompass a range of values for \(x\) and \(y\).
• For the equation \(-x^4 + 4x^3 - 4x^2 = 0\), the given range is \([-6,6,1]\) by \([-9,2,1]\), meaning \(x\) ranges from -6 to 6, and \(y\) from -9 to 2.
• The utility helps you visually find solutions by clearly showing the x-intercepts.

Using graphing utilities can simplify the process of solving equations by providing a clear graphical representation, which can be indispensable for more complicated polynomials with multiple intercepts or complex behavior.

Working on a Cartesian plane

The Cartesian plane is a fundamental concept in graphing polynomial equations. It allows us to visually represent equations like \(-x^4 + 4x^3 - 4x^2 = 0\) and understand their solutions.

• The plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
• Each point on this plane represents a pair of values \((x, y)\), providing a visual method to solve equations.
• On this plane, the x-intercepts are easily identifiable as they appear where the curve crosses the x-axis.

The Cartesian plane facilitates the comprehension of functions by transforming abstract algebraic equations into tangible geometrical representations. This visual aid is essential for identifying patterns and solutions in polynomial graphs.

Direct substitution for verification

Direct substitution is a method used to verify solutions obtained from a graph, specifically x-intercepts. After identifying these intercepts, substituting them back into the original equation confirms their validity.

• For the equation \(-x^4 + 4x^3 - 4x^2 = 0\), when you find x-intercepts such as \(x=a\), substitute \(a\) in place of \(x\) in the equation.
• If the equation results in 0 after substitution, it means that the identified \(x\)-value is indeed correct.
• This method ensures that the solutions are not only visually apparent but also algebraically valid.

Using direct substitution as a verification tool helps confirm that the visual matches the algebraic representation, providing confidence in the accuracy of your solutions.