Copy And Complete The Table For The Relation Y = 7 + 3x - X² For The Interval -3 < X < 8.

Introduction

In the realm of mathematics, understanding relations and functions is paramount. This article delves into the intricacies of the quadratic relation y = 7 + 3x - x², providing a comprehensive exploration of its behavior within the interval -3 < x < 8. Our journey begins with completing a table of values, a foundational step in visualizing and analyzing the function's characteristics. This exploration extends to sketching the graph of the relation, identifying key features such as the vertex and intercepts, and ultimately, solving related equations and inequalities. By meticulously examining this specific quadratic relation, we aim to solidify your understanding of quadratic functions in general and their applications in various mathematical contexts.

Completing the Table of Values for y = 7 + 3x - x²

The first step in understanding the behavior of the relation y = 7 + 3x - x² within the given interval is to construct a table of values. This involves substituting various values of x within the interval -3 < x < 8 into the equation and calculating the corresponding y values. The table provides a discrete set of points that we can then use to visualize the function's graph and identify trends. Let's meticulously calculate the y values for each given x value:

• When x = -3:
  • y = 7 + 3(-3) - (-3)² = 7 - 9 - 9 = -11
• When x = -2:
  • y = 7 + 3(-2) - (-2)² = 7 - 6 - 4 = -3
• When x = -1:
  • y = 7 + 3(-1) - (-1)² = 7 - 3 - 1 = 3
• When x = 0:
  • y = 7 + 3(0) - (0)² = 7 + 0 - 0 = 7
• When x = 1:
  • y = 7 + 3(1) - (1)² = 7 + 3 - 1 = 9
• When x = 2:
  • y = 7 + 3(2) - (2)² = 7 + 6 - 4 = 9
• When x = 3:
  • y = 7 + 3(3) - (3)² = 7 + 9 - 9 = 7
• When x = 4:
  • y = 7 + 3(4) - (4)² = 7 + 12 - 16 = 3
• When x = 5:
  • y = 7 + 3(5) - (5)² = 7 + 15 - 25 = -3
• When x = 6:
  • y = 7 + 3(6) - (6)² = 7 + 18 - 36 = -11

Now, let's present these calculated values in a completed table:

This completed table provides a clear numerical representation of the relation y = 7 + 3x - x² within the specified interval. The symmetrical nature of the y values around x = 1.5 hints at the parabolic shape of the graph and the presence of a maximum point. This initial step is crucial for a deeper understanding of the quadratic relation. Understanding this data is essential for visualizing the graph and predicting the function's behavior.

Unveiling the Graph of y = 7 + 3x - x²

Having populated the table of values, we now transition to the visual representation of the quadratic relation y = 7 + 3x - x²: its graph. Plotting the points from the completed table onto a coordinate plane allows us to observe the curve formed by the relation. This graphical representation is invaluable for understanding the function's behavior, identifying key features, and making predictions about its values.

The graph of y = 7 + 3x - x² is a parabola, a characteristic shape of quadratic functions. Parabolas are symmetrical curves, and this particular one opens downwards due to the negative coefficient of the x² term. This downward orientation indicates that the function has a maximum value.

By carefully plotting the points from the table, we can sketch the curve. The points (-3, -11), (-2, -3), (-1, 3), (0, 7), (1, 9), (2, 9), (3, 7), (4, 3), (5, -3), and (6, -11) provide a clear outline of the parabola. We can observe the symmetry around the vertical line x = 1.5, which represents the axis of symmetry.

The highest point on the parabola is the vertex, which represents the maximum value of the function. From the graph, we can estimate the vertex to be around the point (1.5, 9.25). The y-intercept, where the graph crosses the y-axis, is (0, 7), as evident from both the table and the equation. The x-intercepts, where the graph crosses the x-axis, are the solutions to the equation 7 + 3x - x² = 0. These points can be estimated from the graph or calculated algebraically.

The graph serves as a powerful tool for visualizing the relationship between x and y. It allows us to quickly estimate function values, identify the maximum point, and understand the overall trend of the quadratic relation. This visual understanding is crucial for solving equations and inequalities related to the function. Analyzing the graph provides a deeper insight into the nature of quadratic functions and their properties. This graphical representation makes it easier to predict the behavior and values of the function, especially for solving complex equations and inequalities.

Solving Equations and Inequalities Involving y = 7 + 3x - x²

The exploration of the quadratic relation y = 7 + 3x - x² extends beyond graphing and table completion to solving equations and inequalities. This involves finding the values of x that satisfy specific conditions related to the function's output, y. These solutions have practical applications in various fields, making this a crucial aspect of understanding quadratic relations.

Solving Quadratic Equations

A fundamental task is to solve the quadratic equation 7 + 3x - x² = 0. This corresponds to finding the x-intercepts of the graph, the points where the parabola intersects the x-axis. There are several methods to solve quadratic equations, including:

1. Factoring: This method involves expressing the quadratic as a product of two linear factors. However, not all quadratic equations can be easily factored.

2. Quadratic Formula: This formula provides a general solution for any quadratic equation in the form ax² + bx + c = 0:

   • x = (-b ± √(b² - 4ac)) / (2a)

   In our case, a = -1, b = 3, and c = 7. Substituting these values into the quadratic formula, we get:

   • x = (-3 ± √(3² - 4(-1)(7))) / (2(-1))
   • x = (-3 ± √(9 + 28)) / (-2)
   • x = (-3 ± √37) / (-2)

   This gives us two solutions:

   • x ≈ (-3 + 6.08) / (-2) ≈ -1.54
   • x ≈ (-3 - 6.08) / (-2) ≈ 4.54

   These values represent the x-intercepts of the graph, where the function's output is zero.

3. Completing the Square: This method involves rewriting the quadratic expression in a form that allows for easy solution. It's a useful technique for understanding the structure of quadratic equations.

Solving Quadratic Inequalities

We can also solve inequalities such as 7 + 3x - x² > 0 or 7 + 3x - x² < 0. These inequalities ask for the intervals of x where the function's output is either positive or negative. Graphically, this corresponds to the portions of the parabola that lie above or below the x-axis.

To solve 7 + 3x - x² > 0, we first find the roots of the equation 7 + 3x - x² = 0, which we already calculated to be approximately -1.54 and 4.54. Since the parabola opens downwards, the function is positive between these roots. Therefore, the solution to the inequality is -1.54 < x < 4.54.

Similarly, to solve 7 + 3x - x² < 0, we look for the intervals where the parabola lies below the x-axis. This occurs when x < -1.54 or x > 4.54.

Solving these inequalities provides a comprehensive understanding of the function's behavior across its domain. It's a crucial aspect of mastering quadratic functions. Understanding and solving quadratic inequalities helps in predicting where the function will have positive or negative values, providing a more complete picture of the function's behavior. This analytical approach is vital for real-world applications where understanding the range of possible outcomes is essential.

Conclusion

Our exploration of the quadratic relation y = 7 + 3x - x² has taken us through several key steps, from completing a table of values to sketching its graph and solving related equations and inequalities. This comprehensive analysis has not only provided insights into this specific function but also strengthened our understanding of quadratic relations in general. The ability to manipulate and interpret quadratic functions is essential in various fields, including physics, engineering, and economics. By mastering these concepts, you are well-equipped to tackle more complex mathematical problems and real-world applications. This journey through quadratic relations equips you with the skills necessary to solve complex mathematical problems and apply these concepts in various real-world contexts.