Analyzing G(x) = -x^2 + 14x + 39: Discriminant & Roots

Let's dive into analyzing the quadratic function g(x) = -x² + 14x + 39. We'll explore its key features, including the discriminant, the number of real roots, and its behavior at the vertex. Understanding these aspects helps us grasp the overall nature of this parabolic function. So, grab your thinking caps, and let’s get started!

Determining the Discriminant

To begin our analysis, let's focus on the discriminant. The discriminant is a crucial part of the quadratic formula that tells us about the nature of the roots of a quadratic equation. For a quadratic equation in the standard form of ax² + bx + c = 0, the discriminant (often denoted as Δ) is given by the formula: Δ = b² - 4ac. In our case, the function is g(x) = -x² + 14x + 39, which means a = -1, b = 14, and c = 39. Now, let's plug these values into the discriminant formula:

Δ = (14)² - 4(-1)(39) Δ = 196 + 156 Δ = 352

Thus, the discriminant of the function g(x) is 352. The discriminant, in simple terms, is like a detective for quadratic equations. It helps us uncover the secrets of the roots – whether they are real, imaginary, distinct, or repeated. By calculating the discriminant, we gain a valuable insight into the behavior and characteristics of the quadratic function, which is essential for further analysis and problem-solving. Now that we've successfully calculated the discriminant, we're one step closer to fully understanding our quadratic function. This value, 352, isn't just a number; it’s a key that unlocks deeper insights into the nature of the function's roots and its graph.

Finding the Number of Real Roots

Now that we've calculated the discriminant, let's use this information to determine the number of real roots. The discriminant provides valuable information about the nature and number of roots of a quadratic equation. The rules are as follows:

• If Δ > 0, the quadratic equation has two distinct real roots.
• If Δ = 0, the quadratic equation has one real root (a repeated root).
• If Δ < 0, the quadratic equation has no real roots (two complex roots).

In our case, we found that the discriminant Δ = 352. Since 352 is greater than 0, we can conclude that the function g(x) = -x² + 14x + 39 has two distinct real roots. These roots are the points where the parabola intersects the x-axis. Real roots are like the tangible solutions in the world of mathematics. They represent the actual points where the graph of the quadratic function crosses or touches the x-axis. In practical terms, these roots can signify real-world values or solutions to a problem modeled by the quadratic equation. For example, in physics, they might represent the time at which a projectile hits the ground, or in economics, they could indicate break-even points in a business model. The fact that our discriminant is positive and indicates two real roots suggests that our function has two such meaningful intersection points, each carrying its own significance in the context of the problem at hand.

Analyzing the Vertex of g(x) when x = 7

Finally, let's analyze the behavior of g(x) when x = 7. To do this, we first need to understand that the vertex of a parabola (which is the graph of a quadratic function) represents either the maximum or minimum point of the function. For a quadratic function in the form g(x) = ax² + bx + c, the x-coordinate of the vertex can be found using the formula x = -b / 2a. In our case, a = -1 and b = 14, so the x-coordinate of the vertex is:

x = -14 / (2 * -1) = 7

This confirms that x = 7 is indeed the x-coordinate of the vertex. To determine whether the function has a maximum or minimum at this point, we look at the coefficient of the x² term (which is a). Since a = -1 is negative, the parabola opens downwards, meaning the vertex represents a maximum point. Now, let's find the y-coordinate of the vertex by plugging x = 7 into the function:

g(7) = -(7)² + 14(7) + 39 g(7) = -49 + 98 + 39 g(7) = 88

So, the vertex of the function is at the point (7, 88), and since the parabola opens downwards, g(x) has a maximum value of 88 when x = 7. The vertex of a parabola is more than just a point; it’s the function’s turning point. It signifies the spot where the function changes direction, either ascending to a peak (maximum) or descending to a trough (minimum). For our function, g(x) = -x² + 14x + 39, the vertex at (7, 88) tells us that the function reaches its highest value at x = 7, and that maximum value is 88. This is a crucial piece of information because it allows us to understand the function’s range and overall behavior. The fact that this function has a maximum suggests that it could model scenarios where there’s an optimal point or a limit, such as the maximum height a ball reaches when thrown in the air or the peak profit a company can achieve.

In summary, we've analyzed the quadratic function g(x) = -x² + 14x + 39 and found that:

• The discriminant is 352.
• The function has two distinct real roots.
• The function has a maximum value of 88 when x = 7.

Understanding these characteristics provides a comprehensive view of the function's behavior and properties. For further learning about quadratic functions and their properties, you might find valuable resources on websites like Khan Academy.