Problem 38 - Algebra and Trigonometry

Question

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 at \( x = 7.5 \).

Step 1: Identify the Type of Function

The function given is of the form \( g(x) = ax^2 + bx + c \), with \( a = 100 \), \( b = -1500 \), and \( c = 0 \). This is a quadratic function, and since \( a = 100 > 0 \), it opens upwards and has a minimum value.

Step 2: Find the Vertex

The vertex of the parabola given by \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Substitute \( b = -1500 \) and \( a = 100 \) into this formula to find the x-coordinate of the vertex: \[x = -\frac{-1500}{2 \times 100} = \frac{1500}{200} = 7.5\]

Step 3: Calculate the Vertex y-coordinate

Substitute \( x = 7.5 \) back into \( g(x) = 100x^2 - 1500x \) to find the y-coordinate of the vertex, which represents the minimum value:\[g(7.5) = 100(7.5)^2 - 1500(7.5)\]Calculate each term:\( 100 \times 7.5^2 = 100 \times 56.25 = 5625 \)\( 1500 \times 7.5 = 11250 \)So,\[g(7.5) = 5625 - 11250 = -5625\]

Step 4: State the Minimum Value

The minimum value of the function \( g(x) = 100x^2 - 1500x \) is found at the vertex. Since the function opens upwards, the minimum value is found at \( x = 7.5 \), and the minimum value is \( -5625 \).

Key Concepts

Vertex of a Parabola

In a quadratic function, the graph forms a U-shaped curve called a parabola. The point at which the parabola changes direction is known as the **vertex**. For quadratic functions of the form \(ax^2 + bx + c\), the vertex
is especially important, as it reveals the location of the maximum or minimum value of the function. To find the x-coordinate of the vertex, you can use the formula:

• \(x = -\frac{b}{2a}\)

Here, \(b\) and \(a\) are coefficients from the quadratic equation.
Once you have the x-coordinate, you can find the y-coordinate by substituting this value back into the original function. The complete pair \((x, g(x))\) then gives you the vertex.

Minimum Value

When dealing with quadratic functions, understanding the value at the vertex helps in finding either a maximum or a minimum point. If the parabola opens upwards
(meaning \(a > 0\)), it has a **minimum** value at its vertex.

• The vertex represents the smallest point on the parabola.
• This is the y-coordinate of the vertex.

For
our example function, \(g(x) = 100x^2 - 1500x\), the vertex is located at \(x = 7.5\), and the minimum value is \(-5625\). This point is where the function achieves its lowest value
on the graph and is determined by plugging the x-coordinate of the vertex back into the equation.

Quadratic Formula

The **Quadratic Formula** is not used directly in finding the vertex of a parabola, but it is an essential tool in solving equations of the form \(ax^2 + bx + c = 0\). This formula allows us to find the x-values where the function intersects the x-axis.

• The formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

These x-values are the roots of the quadratic equation.
While the vertex provides the minimum or maximum value of the function, the roots are the x-values where the function equals zero. Understanding both can give a comprehensive view of the nature of the quadratic graph
and its relationship to linear equations.

Parabolic Graphs

Quadratic functions create parabolic graphs. These are symmetrical, U-shaped curves that can open upwards or downwards. The direction of opening is determined by the coefficient \(a\):

• If \(a > 0\), the graph opens upwards, showcasing a minimum value at the vertex.
• If \(a < 0\), the graph opens downwards, presenting a maximum value at the vertex.

The vertex serves as a critical point, influencing the parabola shape and helping identify its maximum or minimum.

• In our example, the graph opens upwards since \(a = 100 > 0\), meaning the vertex is a minimum.

Understanding how a parabola behaves aids in visualizing solutions to quadratic equations and informs the potential range of values the function can take.