Problem 59 - Algebra for College Students

Question

Find the maximum or minimum value of \(y\) for each function. $$y=-2 x^{2}-4 x$$

Step-by-Step Solution

Answer

The maximum value of the function is -6.

Step 1 - Identify the type of function

The given function is a quadratic function in the form of \(y = ax^2 + bx + c\). Here, \(a = -2\), \(b = -4\) and \(c = 0\).

Step 2 - Determine the direction of the parabola

Since \(a = -2\), which is less than 0, the parabola opens downwards. Therefore, the function has a maximum value.

Step 3 - Find the vertex of the parabola

For a quadratic function in the form \(y = ax^2 + bx + c\), the x-coordinate of the vertex (where the maximum or minimum occurs) is found using \(x = -\frac{b}{2a}\). Substituting the given values, \(x = -\frac{-4}{2(-2)} = 1\).

Step 4 - Calculate the y-coordinate of the vertex

Substitute \(x = 1\) back into the original equation to find \(y\). \(y = -2(1)^2 - 4(1) = -2 - 4 = -6\). So, the vertex is \((1, -6)\).

Step 5 - State the maximum value

The maximum value of the function is the y-coordinate of the vertex. Therefore, the maximum value is \(-6\).

Key Concepts

Maximum Value of a Quadratic Function

Understanding the maximum or minimum value of a quadratic function is crucial in analyzing the behavior of parabolas. For a quadratic function in the standard form \(y = ax^2 + bx + c\):

- If \(a < 0\), the parabola opens downwards, and the function has a maximum value.
- If \(a > 0\), the parabola opens upwards, and the function has a minimum value.

The maximum or minimum value occurs at the vertex of the parabola. In the original exercise, since \(a = -2\) which is less than 0, the parabola opens downwards. Therefore, the function \(y = -2x^2 - 4x\) has a maximum value.

Parabola Direction

The direction of a parabola depends on the coefficient \(a\) in the quadratic function \(y = ax^2 + bx + c\). This coefficient determines whether the parabola opens upwards or downwards:

- When \(a > 0\), the parabola opens upwards like a U-shape.
- When \(a < 0\), the parabola opens downwards like an upside-down U-shape.

In the provided problem, \(a = -2\). Since \(a < 0\), the parabola opens downwards. This downwards opening indicates that the quadratic function will have a maximum value instead of a minimum value.

Vertex Form of a Quadratic Function

The vertex of a parabola represents either the highest or lowest point of the graph, depending on the direction the parabola opens. To find the vertex of a quadratic function in the form \(y = ax^2 + bx + c\), use the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\). Once you have the x-coordinate, substitute it back into the original equation to find the corresponding y-coordinate.

For the function \(y = -2x^2 - 4x\):
- Calculate the x-coordinate of the vertex: \(x = -\frac{-4}{2(-2)} = 1\).
- Substitute \(x = 1\) back into the function to get \(y = -2(1)^2 - 4(1) = -6\).

Thus, the vertex is at (1, -6), and in this case, it represents the maximum value of the quadratic function.