Graphing Quadratic Functions Using Calculators Finding Solutions

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In this article, we will explore how to use a graphing calculator to graph a quadratic function and find its solutions. Quadratic functions are polynomial functions of the second degree, and their graphs are parabolas. The solutions of a quadratic equation are the x-intercepts of the parabola, which are the points where the graph intersects the x-axis. We will use the quadratic function y = x² - 5x - 36 as an example to illustrate the process. We will also determine which of the given values (-9, -4, 4, 6) is a solution of the quadratic equation 0 = x² - 5x - 36.

Understanding Quadratic Functions

Before diving into graphing calculators, let's briefly review quadratic functions. A quadratic function is generally expressed in the form: f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is the point where the parabola changes direction, and its x-coordinate is given by –b / 2a. The solutions or roots of a quadratic equation ax² + bx + c = 0 are the x-values where the parabola intersects the x-axis. These points are also known as the x-intercepts or zeros of the function.

Key Characteristics of Quadratic Functions

  • Parabola: The U-shaped curve that represents the function.
  • Vertex: The highest or lowest point on the parabola, representing the maximum or minimum value of the function.
  • Axis of Symmetry: A vertical line that passes through the vertex, dividing the parabola into two symmetrical halves.
  • X-Intercepts (Roots or Zeros): The points where the parabola intersects the x-axis, representing the solutions of the quadratic equation.
  • Y-Intercept: The point where the parabola intersects the y-axis, found by setting x = 0 in the function.

Understanding these characteristics is crucial for interpreting the graph of a quadratic function and finding its solutions. In the following sections, we will use the graphing calculator to visualize these features and determine the solutions of the given equation.

Using a Graphing Calculator to Graph the Quadratic Function

Graphing calculators are powerful tools for visualizing functions and finding their solutions. To graph the quadratic function y = x² - 5x - 36, follow these steps:

  1. Turn on the Calculator: Press the "ON" button to turn on your graphing calculator.
  2. Access the Graphing Menu: Press the "Y=" button to access the function editor where you can enter the equation.
  3. Enter the Equation: Input the quadratic function y = x² - 5x - 36 into the Y1 slot. Use the "X, T, θ, n" button for the variable x, the "^" button for exponentiation (to square x), the "-" button for subtraction, and the appropriate numerical keys for the coefficients and constants.
  4. Adjust the Window Settings: Press the "WINDOW" button to adjust the viewing window. Set appropriate values for Xmin, Xmax, Ymin, and Ymax to ensure that the important features of the graph (such as the vertex and x-intercepts) are visible. A standard window setting of -10 to 10 for both x and y may not always be ideal, so you might need to experiment with different values.
  5. Graph the Function: Press the "GRAPH" button to display the graph of the quadratic function.

Interpreting the Graph

Once the graph is displayed, observe the parabola. Note the following:

  • Direction of Opening: Since the coefficient of x² is positive (1), the parabola opens upwards.
  • Vertex: Identify the vertex, which is the lowest point on the graph. This represents the minimum value of the function.
  • X-Intercepts: These are the points where the parabola crosses the x-axis. These points represent the solutions of the equation 0 = x² - 5x - 36.
  • Y-Intercept: This is the point where the parabola crosses the y-axis. It can be found by setting x = 0 in the equation.

By carefully observing the graph, you can estimate the solutions of the quadratic equation. To find more precise solutions, you can use the calculator's built-in functions, which we will discuss in the next section.

Finding Solutions Using the Graphing Calculator

To find the solutions of the equation 0 = x² - 5x - 36 more precisely, we can use the graphing calculator's "zero" or "root" finding function. This function helps identify the x-intercepts of the graph, which are the solutions we are looking for. Here’s how to use this feature:

  1. Access the Calculate Menu: After graphing the function, press the "2nd" button followed by the "TRACE" button (which usually has "CALC" above it) to access the calculate menu.
  2. Select "zero" or "root": Choose the "zero" option (usually option 2) to find the x-intercepts.
  3. Set the Boundaries: The calculator will prompt you to set a left bound and a right bound. Use the arrow keys to move the cursor to a point on the graph that is to the left of the x-intercept you want to find, and press "ENTER". Then, move the cursor to a point on the graph that is to the right of the x-intercept and press "ENTER" again.
  4. Provide a Guess: The calculator will then ask for a guess. Move the cursor close to the x-intercept you are trying to find and press "ENTER".
  5. Read the Solution: The calculator will display the x-coordinate of the x-intercept, which is the solution to the equation.

Applying the Steps to the Given Equation

Repeat this process for each x-intercept of the graph of y = x² - 5x - 36. You will find that the parabola intersects the x-axis at two points. These points correspond to the solutions of the equation 0 = x² - 5x - 36.

By using the zero-finding function, you can accurately determine the solutions. This method is particularly useful for quadratic equations that are difficult or impossible to solve algebraically. In the next section, we will compare the solutions found using the graphing calculator with the given values to determine which one is a correct solution.

Identifying the Correct Solution from the Given Values

After using the graphing calculator to find the solutions of the equation 0 = x² - 5x - 36, we need to compare our results with the given values: -9, -4, 4, and 6. The solutions we obtained from the graphing calculator should match one or more of these values if they are correct.

Comparing with Graphing Calculator Results

Let’s assume that when using the graphing calculator, we found the x-intercepts (solutions) to be approximately -4 and 9. Now, we compare these values with the provided options.

  • -9: This value is not a solution we found using the graphing calculator.
  • -4: This value matches one of the x-intercepts we found, so it is a potential solution.
  • 4: This value does not match any of our solutions.
  • 6: This value is also not a solution we found using the graphing calculator.

Verifying the Solution

To confirm that -4 is indeed a solution, we can substitute it back into the original equation:

0 = x² - 5x - 36 0 = (-4)² - 5(-4) - 36 0 = 16 + 20 - 36 0 = 36 - 36 0 = 0

Since the equation holds true, -4 is indeed a solution of the quadratic equation 0 = x² - 5x - 36.

Additional Verification

Similarly, if we substitute 9 into the equation:

0 = (9)² - 5(9) - 36 0 = 81 - 45 - 36 0 = 81 - 81 0 = 0

Thus, 9 is also a solution. However, since 9 is not among the given options, -4 remains the correct answer from the provided choices.

In this process, we've demonstrated how to use the graphing calculator to visually and numerically solve quadratic equations and verify the solutions against a set of given values. This method is invaluable for solving equations that may be challenging to solve algebraically or for quickly checking solutions.

Conclusion

In this article, we have demonstrated how to use a graphing calculator to graph the quadratic function y = x² - 5x - 36 and find the solutions of the equation 0 = x² - 5x - 36. Graphing calculators are essential tools for visualizing and solving quadratic equations. We learned how to input the function, adjust the window settings, and interpret the graph. We also used the calculator's zero-finding function to determine the x-intercepts, which are the solutions of the equation. By comparing the solutions obtained from the graphing calculator with the given values, we identified that -4 is a solution of the equation. This process highlights the power of graphing calculators in solving mathematical problems and provides a clear method for finding solutions to quadratic equations. Understanding quadratic functions and their graphical representations is crucial in various fields, including mathematics, physics, engineering, and computer science. The ability to use graphing calculators effectively enhances one's problem-solving skills and provides a visual approach to understanding complex concepts. The combination of algebraic understanding and graphical analysis allows for a more comprehensive grasp of quadratic equations and their applications. This method not only aids in finding solutions but also in understanding the behavior of quadratic functions, including their vertex, axis of symmetry, and direction of opening. By mastering these techniques, students and professionals can confidently tackle a wide range of mathematical problems involving quadratic functions.