Problem 34 - Algebra and Trigonometry

Question

\(29-38=\) Find the maximum or minimum value of the function. $$ g(x)=100 x^{2}-1500 x $$

Step-by-Step Solution

Answer

The minimum value of the function is -5625, occurring at \(x = 7.5\).

Step 1: Identify Characteristics of the Function

The function given is a quadratic function in the form of \(f(x) = ax^2 + bx + c\). Here, \(a = 100\), \(b = -1500\), and \(c = 0\). Because \(a = 100 > 0\), the parabola opens upwards, which means the function will reach a minimum value at its vertex.

Step 2: Find the Vertex Using the Vertex Formula

To find the vertex of a quadratic function \(ax^2 + bx + c\), use the vertex formula \(x = -\frac{b}{2a}\). Substitute \(b = -1500\) and \(a = 100\) into the formula:\[x = -\frac{-1500}{2 \times 100} = \frac{1500}{200} = 7.5\]

Step 3: Calculate the Minimum Value of the Function

Now that we know the \(x\)-coordinate of the vertex is 7.5, we substitute \(x = 7.5\) back into the function to find the minimum value of the function:\[g(7.5) = 100(7.5)^2 - 1500(7.5)\]Calculate further:\[g(7.5) = 100 imes 56.25 - 11250\]\[= 5625 - 11250 = -5625\]

Step 4: Conclusion

The minimum value of the function \(g(x) = 100x^2 - 1500x\) is -5625 and it occurs at \(x = 7.5\). This concludes that the vertex is the point (7.5, -5625).

Key Concepts

Vertex of a Parabola

In quadratic functions, identifying the vertex is crucial as it represents either the maximum or minimum point of the parabola. A quadratic function is generally expressed as \(f(x) = ax^2 + bx + c\). The "vertex" refers to the topmost or bottommost point of the parabola.

• If the coefficient \(a\) is greater than zero (\(a > 0\)), the parabola opens upwards and the vertex is the minimum point.
• If \(a\) is less than zero (\(a < 0\)), the parabola opens downwards and the vertex is the maximum point.

To find the vertex, we use the vertex formula \(x = -\frac{b}{2a}\). It provides the \(x\)-coordinate of the vertex. Once you have this value, plug it back into the original quadratic equation to find the corresponding \(y\)-coordinate of the vertex. Together, these coordinates form the vertex of the parabola. For example, for the quadratic function \(g(x) = 100x^2 - 1500x\), the vertex is found at the coordinates \((7.5, -5625)\). This is the point where the parabola has its minimum value.

Minimum Value

The minimum value of a quadratic function occurs at the vertex when the parabola opens upwards, which is determined by the sign of the coefficient \(a\). Since the function \(g(x) = 100x^2 - 1500x\) has \(a = 100\), it's clear that the parabola opens upwards. Therefore, it has a minimum value.To locate this minimum, utilize the \(x\)-coordinate found using the vertex formula \(x = -\frac{b}{2a}\). Then substitute this \(x\)-value into the function to find the minimum value. In our given function:1. Calculate the \(x\)-coordinate of the vertex: \(x = \frac{1500}{200} = 7.5\).2. Substitute \(x = 7.5\) back into the function to find the minimum value: \[g(7.5) = 100(7.5)^2 - 1500 \times 7.5 = -5625\]Thus, the function achieves its minimum value at \(-5625\), and it occurs at the \(x\)-coordinate of \(7.5\). This process highlights how the vertex determines whether a point is the lowest (or the highest) on the graph of a quadratic function.

Quadratic Formula

The quadratic formula is an essential tool used for finding the roots or solutions of a quadratic equation. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is derived from the standard form of a quadratic function \(ax^2 + bx + c = 0\) and helps determine where the function touches the \(x\)-axis.

• If the discriminant \(b^2 - 4ac\) is positive, the quadratic equation has two distinct real solutions.
• If the discriminant is zero, the equation has exactly one real solution or a repeated root. This occurs at the vertex.
• If the discriminant is negative, there are no real solutions; the equation has complex or imaginary roots.

Even in cases where you are interested in finding either a maximum or minimum value, knowing the roots can inform you of key attributes about the parabola’s shape and position on the graph. Nevertheless, to focus solely on the maximum or minimum points, it is sufficient to use the vertex formula instead of the quadratic formula.