Question

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-2 x^{2}+8 x+3\)

Short answer

The function has a maximum value of 11.

Step-by-step solution

Identify the Coefficient of the Quadratic Term

In the given function, identify the coefficient of the quadratic term, which is the term with the highest power of x. The given function is: \(f(x)=-2x^{2}+8x+3\)The coefficient of \(x^2\) is \( -2 \).

Determine the Direction of the Parabola

The sign of the quadratic term's coefficient determines the direction of the parabola. If the coefficient is positive, the parabola opens upwards and has a minimum value. If the coefficient is negative, the parabola opens downwards and has a maximum value. Since the coefficient is \( -2 \), which is negative, the parabola opens downwards and therefore, the function has a maximum value.

Find the Vertex using the Formula for the x-coordinate

The x-coordinate of the vertex of a parabola in the form \( ax^2 + bx + c \) can be found using the formula: \(x = -\frac{b}{2a}\). For the given function \(f(x)=-2x^{2}+8x+3\), \(a = -2\) and \(b = 8\). Substitute these values into the formula to get:\(x = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2\).

Substitute the x-coordinate back into the Function

To find the maximum value, substitute the x-coordinate of the vertex back into the original function: \(f(x)=-2x^{2}+8x+3\). Substitute \( x = 2 \) to get:\(f(2)=-2(2)^{2}+8(2)+3\).Simplify the expression:\(f(2)=-2(4)+16+3=-8+16+3=11\).

Key concepts

vertex of a parabola

Understanding the vertex of a parabola is crucial when working with quadratic functions. The vertex represents the highest or lowest point on the parabola. For a quadratic function in the form \( ax^2 + bx + c \), the x-coordinate of the vertex can be determined using the formula: \(x = -\frac{b}{2a}\).
This formula comes from completing the square or from calculus-based optimization.
In our example, \(f(x) = -2x^2 + 8x + 3\), the coefficients are \(a = -2\) and \(b = 8\). Plugging these into the formula, we get: \(x = -\frac{8}{2(-2)} = 2\).

Once we have the x-coordinate, we substitute it back into the original function to find the y-coordinate, which gives the maximum or minimum value.

maximum and minimum values

The maximum or minimum value of a quadratic function depends on the direction the parabola opens. This direction is determined by the sign of the coefficient \(a\) in the quadratic term \(ax^2\).

• If \(a > 0\), the parabola opens upward, and the function has a minimum value at the vertex.
• If \(a < 0\), the parabola opens downward, and the function has a maximum value at the vertex.

In our function \(f(x) = -2x^2 + 8x + 3\), since \(a = -2\) (which is negative), the parabola opens downward. This means our function has a maximum value.
We already calculated the x-coordinate of the vertex as \(x = 2\). Substituting \(x = 2\) back into the function gives us the y-coordinate: \[f(2) = -2(2)^2 + 8(2) + 3 = -8 + 16 + 3 = 11\].
Therefore, the maximum value of the function is 11.

coefficients

Coefficients in a quadratic equation heavily influence the shape and properties of the parabola. Let's break down their roles:

• Coefficient \(a\): Determines the direction of the parabola. If \(a\) is positive, the parabola opens upward. If \(a\) is negative, it opens downward. It also affects the width of the parabola. Larger absolute values of \(a\) make the parabola narrower, while smaller absolute values make it wider.
• Coefficient \(b\): Influences the position of the vertex and the axis of symmetry of the parabola.
• Coefficient \(c\): Represents the y-intercept of the parabola, which is the point where the parabola crosses the y-axis.

In our function \(f(x) = -2x^2 + 8x + 3\), \(a = -2\), \(b = 8\), and \(c = 3\). Understanding these coefficients allows us to predict the behavior of the function before graphing it.

For example, the negative \(a\) value informs us that the function has a maximum value, while \(b\) helps locate the vertex at \(x = 2\). The y-intercept is at \(y = 3\).