Problem 60 - Brief Applied Calculus

Question

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (Round answers to two decimal places.) $$ 5 x^{2}+14 x+20=0 $$

Step-by-Step Solution

Answer

The solutions are approximately \( x = -2 \) and \( x = -2 \).

Step 1: Set Up the Equation in the Calculator

Input the equation \( y = 5x^2 + 14x + 20 \) into the graphing calculator. Ensure that the equation is correctly entered for accurate results.

Step 2: Adjust the Window Settings

Set the graphing calculator window to \([-10, 10]\) for both the x-axis and y-axis. This will provide a good view of where the equation intersects the x-axis.

Step 3: Graph the Equation

Once the equation is input and the window is set, press the 'GRAPH' button. The parabola should appear on the screen with a downward opening due to the positive leading coefficient.

Step 4: Find the X-Intercepts

Use the functions like 'ZERO' or 'TRACE' on the calculator to find where the graph crosses the x-axis. These points are the solutions to the equation because the y-value is zero at these points.

Step 5: Use the Calculator's Zero Function

Press the 'ZERO' function key or equivalent on your calculator. Move the cursor near where the parabola appears to touch the x-axis and follow the on-screen instructions to find the precise x-values where the equation equals zero.

Step 6: Record the Solutions

The calculator will give you the x-intercepts. These are the solutions to the equation \( 5x^2 + 14x + 20 = 0 \). Round these solutions to two decimal places for the final answer.

Key Concepts

Window Settings in Graphing Calculator

When using a graphing calculator to solve equations, setting the correct window size is crucial for proper visualization. The "window" defines the portion of the graph and the range of values displayed on the screen. For most equations, especially quadratics, a typical starting window is

• X-axis: -10 to 10
• Y-axis: -10 to 10

This window allows you to capture the general shape and key features of the graph, like the vertex and x-intercepts.

Adjusting the window: - **If the graph appears too flat or steep**, changes might be needed. Use smaller intervals like [-5, 5] or larger ones like [-20, 20] depending on the function's nature. - **Experiment with different settings**, especially when the x-intercepts or vertex lie outside the visible area. By playing with the window values, you can ensure all critical points are visible for analysis.

Finding X-Intercepts

The x-intercepts of a graph are the points where the graph touches or crosses the x-axis. These intercepts are important because they represent the solutions to the quadratic equation when the function equals zero. To find x-intercepts using a graphing calculator:

• **Graph the Equation:** Input the quadratic equation into the calculator and press 'GRAPH'. Ensure the parabola is visible within the set window.
• **Use the 'ZERO' or 'ROOT' Function:** Navigate to these calculator functions, which will help you pinpoint where the graph intersects the x-axis.
• **Select or Trace the Points:** Move the calculator’s cursor close to where the curve appears to touch the x-axis. The screen will guide you with prompts to find and confirm the intercepts.

By accurately finding and rounding these intercepts to two decimal places, we get the solutions to the quadratic equation. This process is often assisted by the calculator's trace function, which can provide an approximate location, making it easier and faster to zero in on the exact intercepts with the calculator's built-in tools.

Parabola Graphing

Graphing a parabola is like drawing a bridge over the x-axis—it can peak or dip based on the equation. A parabolic graph is the geometric representation of a quadratic function, typically written as \(y = ax^2 + bx + c\). Key features of a parabola:- **Direction:** The parabola opens upward if \(a\) is positive and downward if \(a\) is negative.- **Vertex:** The highest or lowest point of the parabola.- **Axis of Symmetry:** A vertical line that divides the parabola into two symmetric halves. It passes through the vertex.Graphing tips:- Begin by creating a simple table of values for \(x\) and calculate corresponding \(y\) values.- Plot these points on the graph to sketch the curve.- Use a graphing calculator for accuracy, especially for complex coefficients.Graphing parabolas with graphing calculators allows you to visualize solutions effortlessly. This visual aid supports deeper understanding and confirmation of algebraic solutions by showing where the graph intersects the x-axis.

Quadratic Equations

Quadratic equations are polynomial equations of the form: \(ax^2 + bx + c = 0\). They play a significant role in algebra and countless real-world applications.Understanding quadratic equations involves recognizing several characteristics:

• Degree: These are second-degree equations because of the \(x^2\) term.
• Coefficients: The numbers \(a\), \(b\), and \(c\) define the parabola's shape and position. Changing these values affects the graph's steepness, width, and direction.
• Discriminant: Calculated as \(b^2 - 4ac\). It determines the number and type of solutions: - If positive, there are two real solutions. - If zero, there is one real solution (a touching point on the x-axis). - If negative, the solutions are complex or imaginary, meaning the graph doesn't touch the x-axis.

Solving quadratic equations often involves graphing to visually identify the x-intercepts. Graphing calculators assist in this by providing a quick and intuitive way to analyze and solve these equations. The graph gives visual confirmation of the algebraic solutions you derive from any method, like factoring, completing the square, or using the formula.