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. $$f(x)=\frac{1}{3}-\frac{1}{3} x^{2}$$

Short answer

The real zeros of the function are \(x = 1\) and \(x = -1\).

Step-by-step solution

Set Up The Equation

The first step is to set up the equation that we need to solve. In this case, we are looking for the zeros of the function \(f(x) = \frac{1}{3} - \frac{1}{3}x^{2}\), so we'll set the function equal to zero: \(\frac{1}{3} - \frac{1}{3}x^{2} = 0\).

Solve For \(x\)

Next, solve for \(x\). First, clear up the fractions by multiplying every term by 3: \(-x^{2} + 1 = 0\). After rearranging this equation, we get \(x^{2} = 1\). Take the square root of both sides to solve for \(x\), remembering to consider both the positive and negative roots. This gives us \(x = ±1\).

Confirming Your Result with a Graph

To confirm the solutions, graph the equation \(y = \frac{1}{3} - \frac{1}{3}x^{2}\) in a graphing utility. The graph will cross the x-axis at \(x = 1\) and \(x = -1\), which confirms the solutions found algebraically.

Key concepts

Algebraic Methods

When determining the real zeros of a quadratic function like \( f(x) = \frac{1}{3} - \frac{1}{3}x^{2} \), we primarily use algebraic methods.
These methods involve solving equations to find the values of \( x \) that make the function equal to zero.
The first step in this approach is setting up the equation \( \frac{1}{3} - \frac{1}{3}x^{2} = 0 \).

This equation implies finding where the output \( f(x) \) equals zero (real zeros).
We simplify by eliminating the fraction: multiply each term by 3, giving us \(-x^{2} + 1 = 0\).
Rearrange this to get \(x^{2} = 1\).
Taking the square root of both sides, remember that quadratic equations often have two solutions: positive and negative.
This results in \(x = ±1\).

• Write down the function and set to zero.
• Simplify by multiplying to remove fractions.
• Solve for \( x \), remembering both positive and negative roots.

Graphical Confirmation

Once you've determined potential zeros algebraically, it's helpful to use a graphing utility to confirm these solutions.
This is because a graphical view can affirm our algebraic solutions by showing the points where the graph intersects the x-axis.
For our quadratic function \( f(x) = \frac{1}{3} - \frac{1}{3}x^{2} \), when graphed, the curve forms an upside-down parabola due to the negative coefficient of \( x^{2} \).

By plotting this function, identifying the x-intercepts at \( x = 1 \) and \( x = -1 \), we confirm our previous algebraic findings.
This visual confirmation is a very useful check.
If the graph passes through the x-axis exactly where your calculations predicted, it supports your algebraic work.

• Use the graph to visualize the zeros.
• Look for points where the curve intersects the x-axis.
• Confirm these graphical results align with the algebraic solutions.

Quadratic Functions

Quadratic functions, like \( f(x) = \frac{1}{3} - \frac{1}{3}x^{2} \), are a fundamental class of polynomial functions characterized by their degree of 2.
The general form of a quadratic function is \( ax^{2} + bx + c \), where \( a \), \( b \), and \( c \) are constants.
In the given problem, \( a = -\frac{1}{3} \), \( b = 0 \), and \( c = \frac{1}{3} \).

Quadratics graph as parabolas, which can open upwards or downwards based on the sign of \( a \).
When \( a < 0 \), the parabola opens downward, as seen with this function.
This opening direction can help confirm the nature of the zeros found, ensuring they are real and potential turning points of the graph.

Finding zeros of quadratic functions is crucial, as it helps predict where the function's value equals zero—where the graph meets the x-axis.
Such analysis is instrumental in various applications, including physics and engineering, where these functions model different phenomena.

• Recognize the form of the quadratic function.
• Understand how the coefficient \( a \) affects the parabola's direction.
• Apply zeros' concepts to solve real-world problems.