Problem 49 - Algebra 1: Concepts and Skills

Question

Use a graphing calculator to approximate the solutions of the equation. $$-\frac{1}{2} x^{2}+2 x+16=0$$

Step-by-Step Solution

Answer

The solutions are the x-values where the graph of the equation \(-\frac{1}{2} x^2 + 2x + 16\) intersects with the x-axis.

Step 1: Identify the equation

Identify the quadratic equation which is \(-\frac{1}{2}x^2 + 2x +16 = 0\).

Step 2: Use graphing calculator

Enter the equation \(-\frac{1}{2}x^2 + 2x +16 = 0\) into the graphing calculator. Select the 'GRAPH' function to view the graph of the equation. You would see a parabolic graph.

Step 3: Use 'ZERO' feature

By using the 'ZERO' feature on your graphing calculator, identify the x-values where the graph intersects the x-axis. These points are called the zeros or the roots of the equation.

Step 4: Interpret the result

The x-values where the graph intersects the x-axis are the solutions to the equation.

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool in solving quadratic equations, especially when you want to visually interpret the equation's behavior. It displays functions graphically, making it easier to locate the points where the function meets specific criteria, such as zeros or intersection points.

To use a graphing calculator effectively, follow these steps:

• First, input the quadratic equation into the calculator. For example, enter \(-\frac{1}{2}x^2 + 2x + 16 = 0\).
• Then, use the 'GRAPH' function to plot the equation. This will display a parabola on the graph screen.
• Explore the graph using navigation keys to better understand the curve's shape and behavior.
• Use built-in features like 'TRACE' to gather more information about points along the curve.

Using a graphing calculator can greatly enhance your understanding of the mathematical relationship within quadratic functions.

Quadratic Function

A quadratic function is a type of polynomial function characterized by a specific form, \(ax^2 + bx + c\), where \(a eq 0\). These functions graph as parabolas and are often used to model real-world scenarios like projectile motion.

For any quadratic function, the coefficient \(a\) determines the parabola’s direction:

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, the parabola opens downwards, as in the original equation \(-\frac{1}{2}x^2 + 2x + 16 = 0\).

The parameters \(b\) and \(c\) influence the parabola's position and shape but not the opening direction.

Understanding quadratic functions involves analyzing their parts to determine motion and direction, which aids in solving equations graphically.

Roots of the Equation

The roots of a quadratic equation are the solutions to the equation when its value equals zero. These solutions are the x-values where the parabola intersects the x-axis on a graph. In simpler terms, they are where the function equals zero.

To find these roots using a graphing calculator, use the 'ZERO' feature:

• View the parabola on the graph.
• Utilize the 'ZERO' function to pinpoint where the graph meets the x-axis.
• These intersection points are your x-values, or roots.

For the equation \(-\frac{1}{2}x^2 + 2x + 16 = 0\), the roots are found where the graph hits the x-axis, providing solutions to the equation. Understanding these roots gives insight into the equation's behavior and real-world applications.