Analyzing G(x) = X^2 - 10x + 24: Min/Max Value & Location

Hey guys! Let's dive into analyzing the quadratic function g(x) = x² - 10x + 24. We're going to figure out if this function has a minimum or maximum value, where that value occurs, and what the actual minimum or maximum value is. Buckle up, because we're about to tackle some math!

(a) Does the Function Have a Minimum or Maximum Value?

To determine whether the function has a minimum or maximum value, we need to look at the leading coefficient of the quadratic. Remember, a quadratic function is generally expressed in the form g(x) = ax² + bx + c. In our case, a is the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term.

In our function, g(x) = x² - 10x + 24, the coefficient a is 1 (since there's an implied 1 in front of x²). The sign of a tells us whether the parabola opens upwards or downwards. If a is positive, the parabola opens upwards, meaning it has a minimum value. If a is negative, the parabola opens downwards, meaning it has a maximum value. Think of it like a smiley face (positive a, minimum) or a frowny face (negative a, maximum).

Since a = 1, which is positive, the parabola opens upwards. This means our function g(x) has a minimum value. This is because as x moves further away from the vertex in either direction, the value of g(x) will increase, creating a U-shape. The bottom of the U is the minimum point. To solidify this understanding, visualize a parabola opening upwards; it clearly has a lowest point but no highest point. This lowest point represents the minimum value of the function. Recognizing the relationship between the leading coefficient and the parabola's orientation is crucial for quickly determining whether a quadratic function has a minimum or maximum. It's a fundamental concept in understanding quadratic behavior and forms the basis for many subsequent calculations and applications. So, the answer to part (a) is definitively a minimum.

(b) Where Does the Minimum or Maximum Value Occur?

Now that we know we have a minimum value, let's find out where it occurs. The minimum (or maximum) value of a quadratic function occurs at its vertex. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Remember, a and b are the coefficients from our quadratic equation g(x) = ax² + bx + c.

In our function, g(x) = x² - 10x + 24, we have a = 1 and b = -10. Plugging these values into the formula, we get:

x = -(-10) / (2 * 1) = 10 / 2 = 5

So, the minimum value occurs at x = 5. This means that the lowest point on the parabola is directly above or below the x-axis at the point where x equals 5. This is the axis of symmetry for the parabola – a vertical line that runs through the vertex, dividing the parabola into two symmetrical halves. Finding the x-coordinate of the vertex is a critical step in understanding the behavior of the quadratic function. It not only tells us where the extreme value (minimum or maximum) occurs but also provides valuable information about the parabola's symmetry and position on the coordinate plane. This value will be used in the next step to calculate the actual minimum value of the function. To truly grasp this, picture the parabola – the vertex is the turning point, and its x-coordinate is precisely what we've calculated here.

(c) What is the Function's Minimum or Maximum Value?

Finally, let's determine what the actual minimum value is. To do this, we simply plug the x-coordinate of the vertex (which we found in part (b) to be x = 5) back into the original function, g(x) = x² - 10x + 24.

So, we need to calculate g(5):

g(5) = (5)² - 10(5) + 24 = 25 - 50 + 24 = -1

Therefore, the minimum value of the function is -1. This means the lowest point on the parabola representing g(x) is at the coordinate (5, -1). The y-coordinate of the vertex is the minimum value of the function because the parabola opens upwards. This point is the absolute lowest the function will ever go, and all other points on the parabola will have a y-value greater than -1. This calculation completes our analysis of the quadratic function, giving us a full understanding of its extreme value. To put it all together, the vertex (5, -1) is the key to understanding this function's behavior. The x-coordinate gives the location of the minimum, and the y-coordinate gives the minimum value itself. This final piece of information is critical for applications of quadratic functions in various fields, such as physics, engineering, and economics. Understanding how to calculate the minimum or maximum value is essential for optimization problems, where you want to find the best possible outcome.

Summary

In summary, for the function g(x) = x² - 10x + 24:

• (a) The function has a minimum value.
• (b) The minimum value occurs at x = 5.
• (c) The function's minimum value is -1.

We successfully determined the minimum value and its location by analyzing the quadratic function's coefficients and using the vertex formula. Keep practicing, and you'll be a quadratic function pro in no time!