Question

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-2 x^{2}+12 x\)

Short answer

The function has a maximum value of 18.

Step-by-step solution

Determine the form of the quadratic function

The given quadratic function is \(f(x)=-2 x^{2}+12 x\). It is in the standard form \(f(x)=ax^2 + bx + c\).Here, \(a = -2\), \(b = 12\), and \(c = 0\).

Identify the coefficient of the quadratic term

Look at the coefficient of the quadratic term, which is \(a=-2\). Since \(a\) is negative, the parabola opens downwards. This means the function has a maximum value.

Find the vertex of the quadratic function

The vertex of a quadratic function \(f(x) = ax^2 + bx + c\) can be found at \(x = -\frac{b}{2a}\). Plug \(a = -2\) and \(b = 12\) into the formula:\[ x = -\frac{12}{2(-2)} = \frac{12}{4} = 3 \]

Calculate the value of the function at the vertex

Substitute \(x = 3\) back into the function to find the maximum value:\[ f(3) = -2(3)^2 + 12(3) = -2(9) + 36 = -18 + 36 = 18 \]So, the maximum value of the function is 18.

Key concepts

vertex formula

In a quadratic function, the vertex is a crucial point where the function reaches its maximum or minimum value. The vertex formula helps us find the x-coordinate of this vertex. For any quadratic function in the standard form: \( f(x) = ax^2 + bx + c \), you can find the x-coordinate of the vertex using the formula:
This formula plays a significant role in analyzing quadratic functions. Plugging the constants from the quadratic equation into this formula directly provides us with the vertex x-coordinate. Knowing this, you can easily determine the function's value at this point to find extremums.

maximum value

In the given quadratic function \( f(x) = -2x^2 + 12x \), we first noted that the coefficient of \(x^2\) is negative (\(a = -2\)). If the leading coefficient \(a\) is negative, the parabola opens downwards, indicating a maximum value at the vertex.
After determining the parabola opens downwards, we've calculated the x-coordinate of the vertex using the vertex formula. By substituting this x-coordinate back into the original function, we determine the maximum function value.
We substituted \(x = 3\) back into the function to get \( f(3) = -2(3)^2 + 12(3) = 18 \), giving us the maximum value of \(18\).

parabola

A quadratic function graph is called a parabola. For the given function \( f(x) = -2x^2 + 12x \), the parabola opens downwards because the coefficient of \(x^2\) (which is \(a\)) is negative. This influences how we determine the vertex and the extremums of the function.
Parabolas have some unique properties:

• The direction of opening (upwards or downwards) depends on the sign of \(a\).
• The vertex is the highest or lowest point of the parabola.
• The axis of symmetry of a parabola is a vertical line that passes through the vertex.

Understanding how the parabola's direction and the vertex's position affect the quadratic function is important. It helps to find the maximum or minimum values without graphing. In summary, recognizing these properties aids in solving and graphing quadratic equations effectively and efficiently.