Problem 24 - Algebra and Trigonometry Real Mathematics, Real People

Question

The zero(s) of the function are given. Verify the zero(s) both algebraically and graphically. \(Function\) \(f(x)=x^{3}-9 x^{2}+18 x\) \(Z e r o(s)\) \(x=0,3,6\)

Step-by-Step Solution

Answer

The given x-values \(x=0,3,6\) are indeed the zeros of the function \(f(x) = x^{3}-9x^{2}+18x\), as verified algebraically by substituting the values into the function, and graphically by observing the function intersection points on the x-axis.

Step 1: Algebraic Verification

To verify the zeros algebraically, substitute each given x-value into the function and ensure it evaluates to 0. For x = 0: \(f(0) = (0)^{3} - 9(0)^{2} + 18(0) = 0\). For x = 3: \(f(3) = (3)^{3} - 9(3)^{2} + 18(3) = 0\). Lastly, for x = 6: \(f(6) = (6)^{3} - 9(6)^{2} + 18(6) = 0\). All values lead to the function returning 0, which verifies algebraically that given x-values are zeros of the function.

Step 2: Graphic Verification

To verify the zeros graphically, plot the function \(f(x) = x^{3}-9x^{2}+18x\) and mark the x-values: 0, 3, and 6. Observe the points at which the function intersects the x-axis. If these points correspond to the given x-values, then you can confirm that these are indeed the zeros of the function. On plotting the function, the function intersects the x-axis at x=0, x=3 and x=6, thereby verifying the zeros graphically.

Step 3: Conclusion

After analyzing both algebraically and graphically, it can be concluded that \(x=0,3,6\) are indeed zeros of the function \(f(x) = x^{3}-9x^{2}+18x\)

Key Concepts

Graphical Verification

Graphical verification involves plotting a function to see where its graph meets the x-axis. This is important because any point where the graph of a function intersects the x-axis represents a zero of the function.

To verify zeros graphically, follow these steps:

• First, use a graphing tool or graph paper to draw the function. The given function is cubic: \(f(x)=x^{3}-9x^{2}+18x\).
• Next, check the graph for points where it crosses the x-axis. These intersections confirm the zeros of the function.
• In this case, plotting \(f(x)\) shows intersections at \(x=0\), \(x=3\), and \(x=6\). Thus, these values are confirmed as zeros graphically.

Graphical verification is a visually intuitive way to support algebraic findings. It helps in understanding root behavior and function changes across the x-axis.

Polynomial Zeros

Polynomial zeros, often called roots, are the values of \(x\) for which the polynomial evaluates to zero. For the function \(f(x)=x^{3}-9x^{2}+18x\), the zeros are \(x=0\), \(x=3\), and \(x=6\).

Determining polynomial zeros involves solving the equation \(f(x) = 0\). In our function:

• Factor the expression: \(x(x-3)(x-6) = 0\).
• Set each factor equal to zero: \(x=0\), \(x-3 = 0\), and \(x-6 = 0\).
• Solve these simple equations to find the zeros.

Algebraically finding zeros provides exact values critical for various applications, such as calculus or solving real-world problems.

Function Intersections

Function intersections describe the points where a function meets specific axes or other functions. In the context of verifying zeros, we're looking at the function's intersection with the x-axis.

To find where a polynomial function like \(f(x)\) meets the x-axis:

• We solve for \(f(x)=0\) to find these intersection points. This is equivalent to finding the zeros.
• The function \(f(x)=x^{3}-9x^{2}+18x\) intersects the x-axis at points \(x=0\), \(x=3\), and \(x=6\).
• These intersections reveal the input values (or x-values) where the output of the function is zero.

Function intersections are significant because they indicate where the graph of the function changes direction or crosses below and above the x-axis. This helps in predicting function behavior and analyzing graphical data.