Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=3 x^{2}-12 x+3\)

Step-by-Step Solution

Answer

The zeros of the function \(f(x) = 3x^2 - 12x + 3\), calculated algebraically, are \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\). The zeros approximated from the graph match the calculated zeros.

Step 1: Solve for zeros algebraically

To calculate the zeros of the function algebraically, you need to set the function equal to zero and solve for \(x\). So you get the equation \(3x^2 - 12x + 3 = 0\). Factoring out a 3, the equation simplifies to \(x^2 - 4x + 1 = 0\). This can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\). Here, \(a=1, b=-4, c=1\). Substituting these values we get two zeros, \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\).

Step 2: Use a graphing utility to graph the function

Using a graphing utility, the function \(f(x) = 3x^2 - 12x + 3\) can be plotted. The zeros of the function are the x-coordinates where the function crosses the x-axis.

Step 3: Approximate zeros from graph

After plotting the graph, the zeros of the function can be observed visually. The function crosses the x-axis at two points that approximates to the zeros calculated earlier.

Step 4: Compare the zeros

Comparing the zeros calculated algebraically with those approximated from the graph to check if they are about the same, we can confirm that the results from the algebraic and graphical approaches agree.

Key Concepts

Algebraic Methods

Solving quadratic equations using algebraic methods often involves various approaches. For the given equation \(3x^2 - 12x + 3 = 0\), we focus on the method of factoring and the quadratic formula.
To find the zeros algebraically, first set the equation to zero:

• Factor out the greatest common factor, 3, from each term to simplify: \(x^2 - 4x + 1 = 0\). This makes solving more manageable.
• Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), to solve the simplified equation. Substitute \(a = 1, b = -4, c = 1\).
• Calculate the discriminant \(b^2 - 4ac\). Here, \((-4)^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12\).
• Substitute into the quadratic formula to find the zeros: \(x = 2 \pm \sqrt{3}\).

These roots tell us where the function intersects the x-axis, providing the solutions \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\).

Graphing Utilities

Graphing utilities, such as graphing calculators or software, are powerful tools for visualizing functions. For function \(f(x) = 3x^2 - 12x + 3\), these tools help to graph the curve and visualize its behavior.

• Begin by entering the function into the graphing utility.
• Plot the graph to locate where the curve intersects the x-axis. These intersections suggest the function's zeros.

Using a graphing utility provides a visual confirmation of the results obtained algebraically. It allows us to see the shape of the graph around the zeros, which appear as points where the line crosses the x-axis.

Zero of a Function

The zero of a function is truly important in mathematics, as it represents the \(x\)-values where the function outputs zero. This means that the graph of the function will touch or cross the x-axis at these points.

• Zero points illustrate solutions to the equation \(f(x) = 0\).
• For the function \(f(x) = 3x^2 - 12x + 3\), we calculate these algebraically, finding \(x = 2 + \sqrt{3}\) and \(x = 2 - \sqrt{3}\).
• Graphically, these zeros are observed at the points where the graph crosses the x-axis.

Understanding the zeros helps us analyze the function’s behavior, helping predict and explain areas where the function has either positive or negative outputs.