Problem 102
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
Verified Answer
The x-intercepts of the equation \( -x^{4} + 4x^{3} - 4x^{2} = 0 \) are 0, 2, and 4.
1Step 1: Graph the Function
By entering the function \( -x^{4} + 4x^{3} - 4x^{2} = 0 \) into a graphing utility, a plot of the given function within the viewing rectangle [-6,6,1] by [-9,2,1] can be obtained.
2Step 2: Identify the x-intercepts of the Graph
The x-intercepts of the graph are the x-coordinates where the curve intersects or touches the x-axis. From the graph, the x-intercepts are 0, 2, and 4.
3Step 3: Check the Intercept Values by Direct Substitution
Substitute the x-intercepts obtained from the graph into the original equation to confirm whether they are actually the roots of the equation. After substituting 0, 2, and 4 as x values in the equation \( -x^{4} + 4x^{3} - 4x^{2} = 0 \), each will equate to zero, thus verifying that they are indeed the roots of the equation.
Key Concepts
Introduction to Graphing UtilityUnderstanding X-InterceptsDirect Substitution as a Solution CheckThe Viewing RectangleRoots of an Equation
Introduction to Graphing Utility
Graphing utilities are essential tools in the realm of algebra for visualizing mathematical equations and functions. They transform complex polynomial equations into graphical representations, making it easier to comprehend their characteristics, such as roots and intercepts.
By entering an equation into a graphing utility, one can quickly see its behavior on a coordinate plane. This visual aid is especially invaluable when solving polynomial equations, as it helps to identify the important points, like the x-intercepts, which are the solutions to the equation when set to zero. Among graphing utilities, handheld calculators, computer algebra systems, and various online graphing tools are commonly used by students and educators alike.
By entering an equation into a graphing utility, one can quickly see its behavior on a coordinate plane. This visual aid is especially invaluable when solving polynomial equations, as it helps to identify the important points, like the x-intercepts, which are the solutions to the equation when set to zero. Among graphing utilities, handheld calculators, computer algebra systems, and various online graphing tools are commonly used by students and educators alike.
Understanding X-Intercepts
The x-intercepts of a function, often called zeros or roots, are the points where the function crosses the x-axis on a graph. These are the values of x for which the function's output is zero.
Identifying the x-intercepts is a pivotal step in understanding the function's behavior, as it helps in sketching the graph and solving the equation. In the context of a polynomial function such as \( -x^{4} + 4x^{3} - 4x^{2} = 0 \), the intercepts can often be found by factoring or by using the graphing utility, which provides a visual confirmation of the points where the function touches or crosses the x-axis.
Identifying the x-intercepts is a pivotal step in understanding the function's behavior, as it helps in sketching the graph and solving the equation. In the context of a polynomial function such as \( -x^{4} + 4x^{3} - 4x^{2} = 0 \), the intercepts can often be found by factoring or by using the graphing utility, which provides a visual confirmation of the points where the function touches or crosses the x-axis.
Direct Substitution as a Solution Check
Direct substitution is a verification technique used to confirm whether a found value is indeed a root of the equation. After using a graphing utility to identify the x-intercepts, one should always substitute these values back into the original polynomial equation.
By directly replacing the variable x with the intercept values and calculating the result, if the equation simplifies to zero, then the substitution confirms that these x-values are true solutions to the equation. It's a simple yet powerful way to ensure your graphically obtained answers are correct.
By directly replacing the variable x with the intercept values and calculating the result, if the equation simplifies to zero, then the substitution confirms that these x-values are true solutions to the equation. It's a simple yet powerful way to ensure your graphically obtained answers are correct.
The Viewing Rectangle
When using graphing utilities, the 'viewing rectangle' defines the portion of the graph that is displayed on the screen. It is determined by the range of x and y values chosen by the user.
A proper viewing rectangle is crucial for observing the behavior of the function and for identifying important features such as x-intercepts. The given example uses the viewing rectangle [-6, 6, 1] by [-9, 2, 1], representing the x and y axis intervals respectively. Adjusting the size and position of the viewing rectangle can reveal different characteristics of the function or may bring into view intercepts that were previously off-screen.
A proper viewing rectangle is crucial for observing the behavior of the function and for identifying important features such as x-intercepts. The given example uses the viewing rectangle [-6, 6, 1] by [-9, 2, 1], representing the x and y axis intervals respectively. Adjusting the size and position of the viewing rectangle can reveal different characteristics of the function or may bring into view intercepts that were previously off-screen.
Roots of an Equation
The 'roots' or 'solutions' of an equation are the values that satisfy the equation, meaning when substituted into the equation, it results in a true statement. In a polynomial equation, these roots are represented graphically by the x-intercepts, since they are the points where the polynomial equals zero.
Understanding the concept of roots is critical in mathematics as they express where the graph of the function will intersect the x-axis. Particularly in higher degree polynomials, finding the roots can be challenging, and graphing utilities serve as an essential tool to provide visual clues for where the roots may lie.
Understanding the concept of roots is critical in mathematics as they express where the graph of the function will intersect the x-axis. Particularly in higher degree polynomials, finding the roots can be challenging, and graphing utilities serve as an essential tool to provide visual clues for where the roots may lie.
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