Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=\frac{1}{3} x^{2}+\frac{1}{3} x-\frac{2}{3}$$

Step-by-Step Solution

Answer

The real zeros of the function \(f(x) = \frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3}\) are \(x = -1\) and \(x = 2\).

Step 1: Set the Function Equal to Zero

To find the real zeros of the function, one needs to set \(f(x)\) equal to zero and solve for \(x\), that is, \(\frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3} = 0\). This represents a quadratic equation in standard form, where \(a = \frac{1}{3}\), \(b = \frac{1}{3}\), and \(c = -\frac{2}{3}\).

Step 2: Apply Quadratic Formula

Since the equation is quadratic, it can be solved using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting \(a\), \(b\), and \(c\) yields \(x = \frac{-\frac{1}{3} \pm \sqrt{(\frac{1}{3})^2 - 4(\frac{1}{3})(-\frac{2}{3})}}{2*\frac{1}{3}}\). After simplification, the solution becomes \(x = -1, 2\).

Step 3: Graph the Function

Using a graphing utility, plot the function \(f(x) = \frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3}\). Notice that the function crosses the x-axis at \(x = -1\) and \(x = 2\), confirming the algebraic solutions.

Key Concepts

Real Zeros

Real zeros play an important role in understanding the behavior of quadratic functions. To start with, a real zero is a point where the graph of the function hits the x-axis. When solving the equation \(f(x) = 0\), the solutions are the real zeros of the function. For the function \(f(x) = \frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3}\), our task is to find these values of \(x\) where the function equals zero.
The process involves setting the function equal to zero and solving for \(x\). In our example, we simplify \(\frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3} = 0\) to find these zeros. Real zeros are important because they show us where the function changes from positive to negative or vice versa. This change is observed when the graph crosses the x-axis. In simpler terms, real zeros tell us about the function's roots on a graph.
Identifying real zeros not only helps in solving equations but also in understanding the overall shape of the graph. By spotting these points, you can better grasp how the graph behaves and connects to other concepts in mathematics.

Quadratic Formula

The quadratic formula is a reliable method to find the zeros of any quadratic equation. Quadratic equations are those of the form \(ax^2 + bx + c = 0\). The quadratic formula offers a precise manner to solve them:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula automatically applies to our function \(f(x) = \frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3}\). In this instance, \(a\), \(b\), and \(c\) have values of \(\frac{1}{3}\), \(\frac{1}{3}\), and \(-\frac{2}{3}\) respectively.
Plugging these into the formula, we solve for \(x\):

• \(x = \frac{-\frac{1}{3} \pm \sqrt{(\frac{1}{3})^2 - 4(\frac{1}{3})(-\frac{2}{3})}}{2 \cdot \frac{1}{3}}\)
• After simplifying, we find \(x = -1\) or \(x = 2\)

This dual result matches our real zeros from the previous calculation. The quadratic formula not only gives precise solutions but also reveals whether the roots are real or complex. In our situation, a positive discriminant \(b^2 - 4ac\) shows that the zeros are real.

Graphing Utility

Graphing utilities are fantastic tools to visualize mathematical functions. They allow us to easily see where functions cross axes and understand their behavior. For our quadratic function \(f(x) = \frac{1}{3}x^2 + \frac{1}{3}x - \frac{2}{3}\), using such a utility lets you confirm any algebraic solutions visually.
Imagine plotting the function on a graph. The intersections with the x-axis are your real zeros, which we've previously calculated as \(x = -1\) and \(x = 2\). By using a graphing utility, you can see these points and their accuracy in a real-world context. Additionally, observing the shape of the graph—a parabola—gives you insights into the function's properties, like direction and vertex.
These tools are beyond textbooks and align with computational skills in math. They encapsulate precise data and present it visually, turning abstract concepts into tangible insights. In a learning context, graphing utilities bridge the gap between solving equations and understanding their implications thoroughly. This visual confirmation is not just helpful but crucial for mastering quadratic functions.