Unlocking Quadratic Equations Identifying A, B, And C In Y=-3x^2+6x+17

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In the realm of mathematics, quadratic equations hold a significant place, serving as fundamental tools for modeling various real-world phenomena, from the trajectory of a projectile to the shape of a suspension bridge. Understanding the properties of these equations is crucial for unlocking their potential and applying them effectively. This article delves into the intricacies of quadratic equations, focusing on the specific example of $y = -3x^2 + 6x + 17$, to illustrate how to identify and interpret the key parameters that define their behavior.

Understanding Quadratic Equations: The General Form

Before we delve into the specifics of our example equation, let's establish a solid foundation by revisiting the general form of a quadratic equation. A quadratic equation is a polynomial equation of the second degree, meaning that the highest power of the variable is 2. The general form of a quadratic equation is expressed as:

$y = ax^2 + bx + c$

where:

• x is the variable.
• y is the value of the quadratic equation for a given x.
• a, b, and c are constants, with a ≠ 0. These constants play pivotal roles in determining the shape and position of the quadratic equation's graph, which is a parabola.

The coefficients a, b, and c each hold unique significance:

• a: This coefficient dictates the parabola's direction (whether it opens upwards or downwards) and its vertical stretch or compression. If a is positive, the parabola opens upwards, forming a U-shape. Conversely, if a is negative, the parabola opens downwards, resembling an inverted U-shape. The magnitude of a also influences the parabola's width; a larger absolute value of a results in a narrower parabola, while a smaller absolute value leads to a wider parabola.
• b: This coefficient, in conjunction with a, influences the horizontal position of the parabola's vertex (the point where the parabola changes direction). It plays a crucial role in determining the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves.
• c: This constant represents the y-intercept of the parabola, which is the point where the parabola intersects the y-axis. It provides a direct indication of the parabola's vertical position on the coordinate plane.

Identifying the Coefficients in $y = -3x^2 + 6x + 17$

Now that we have a firm grasp of the general form and the significance of the coefficients, let's apply this knowledge to our specific example equation: $y = -3x^2 + 6x + 17$. By comparing this equation to the general form, we can readily identify the values of a, b, and c:

• a = -3
• b = 6
• c = 17

These coefficients hold the key to understanding the properties of the parabola represented by this equation. Let's delve deeper into how these values shape the graph and behavior of the quadratic function.

The Significance of a = -3

The coefficient a, in this case, is -3. This negative value immediately tells us that the parabola opens downwards, creating an inverted U-shape. The magnitude of a, which is 3, indicates a relatively narrow parabola compared to parabolas with smaller absolute values of a. This means that the graph will rise and fall more steeply than a parabola with, say, a = -1.

The Role of b = 6

The coefficient b, being 6, contributes to the horizontal positioning of the parabola. While b alone doesn't directly reveal the vertex's x-coordinate, it plays a crucial role in determining the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two symmetrical halves. The x-coordinate of the vertex can be calculated using the formula:

$x = -b / (2a)$

In our case, this translates to:

$x = -6 / (2 * -3) = -6 / -6 = 1$

This calculation reveals that the axis of symmetry is the vertical line x = 1, and the vertex of the parabola lies on this line.

Understanding c = 17

The constant c, which is 17 in our equation, provides a straightforward piece of information: the y-intercept of the parabola. This means that the parabola intersects the y-axis at the point (0, 17). This point serves as a crucial reference point when sketching the graph of the parabola.

Putting It All Together: Visualizing the Parabola

By analyzing the coefficients a, b, and c, we've gathered valuable insights into the properties of the parabola represented by $y = -3x^2 + 6x + 17$. We know that:

• The parabola opens downwards (due to a = -3).
• The parabola is relatively narrow (due to the magnitude of a being 3).
• The axis of symmetry is the line x = 1 (calculated using a and b).
• The y-intercept is the point (0, 17) (determined by c = 17).

With this information, we can begin to visualize the parabola. We know it's an inverted U-shape, centered around the line x = 1, and intersects the y-axis at (0, 17). To get a more precise picture, we can determine the vertex, which is the highest point on the parabola. We already know the x-coordinate of the vertex is 1 (from the axis of symmetry). To find the y-coordinate, we substitute x = 1 into the equation:

$y = -3(1)^2 + 6(1) + 17 = -3 + 6 + 17 = 20$

Therefore, the vertex of the parabola is the point (1, 20). This is the highest point on the graph, and it confirms that the parabola opens downwards.

Additional Properties and Insights

Beyond the coefficients a, b, and c, there are other important properties of quadratic equations that we can explore. These include:

The Discriminant

The discriminant, denoted as Δ (delta), is a crucial value that reveals the nature of the roots (or solutions) of the quadratic equation. It is calculated using the formula:

$Δ = b^2 - 4ac$

The discriminant provides the following information:

• If Δ > 0: The equation has two distinct real roots, meaning the parabola intersects the x-axis at two different points.
• If Δ = 0: The equation has one real root (a repeated root), meaning the parabola touches the x-axis at exactly one point (the vertex).
• If Δ < 0: The equation has no real roots, meaning the parabola does not intersect the x-axis.

For our equation, $y = -3x^2 + 6x + 17$, the discriminant is:

$Δ = (6)^2 - 4(-3)(17) = 36 + 204 = 240$

Since Δ > 0, we know that the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points. To find these roots, we would use the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is expressed as:

$x = (-b ± √(b^2 - 4ac)) / (2a)$

Substituting the values from our equation, we get:

$x = (-6 ± √240) / (2 * -3)$

$x = (-6 ± √(16 * 15)) / -6$

$x = (-6 ± 4√15) / -6$

This gives us two roots:

$x₁ = (3 - 2√15) / 3 ≈ -1.58$

$x₂ = (3 + 2√15) / 3 ≈ 3.58$

These roots represent the x-coordinates of the points where the parabola intersects the x-axis.

Conclusion: A Comprehensive Understanding of Quadratic Properties

By meticulously analyzing the coefficients a, b, and c, calculating the discriminant, and applying the quadratic formula, we've gained a comprehensive understanding of the properties of the quadratic equation $y = -3x^2 + 6x + 17$. We've determined that the parabola opens downwards, is relatively narrow, has an axis of symmetry at x = 1, a y-intercept at (0, 17), a vertex at (1, 20), and intersects the x-axis at approximately x = -1.58 and x = 3.58.

This detailed analysis highlights the power of understanding the individual components of a quadratic equation and how they contribute to the overall behavior of the function. By mastering these concepts, you can confidently tackle a wide range of quadratic equation problems and apply them to various real-world scenarios. The knowledge gained here provides a solid foundation for further exploration of more advanced mathematical concepts and applications.

By understanding these properties, we can effectively model and solve problems involving parabolic trajectories, optimization, and various other applications. The ability to dissect and interpret quadratic equations is a valuable skill in mathematics and beyond.