Problem 47 - Algebra 2

Question

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=-7-3 x^{2}+12 x $$

Step-by-Step Solution

Answer

The function has a maximum value of 5. Domain: \((-\infty, \infty)\), Range: \((-\infty, 5]\).

Step 1: Identify the Type of Function

The given function is a quadratic function in the form \( f(x) = ax^2 + bx + c \). Here, \( a = -3 \), \( b = 12 \), and \( c = -7 \).

Step 2: Determine Maximum or Minimum

For quadratic functions of the form \( ax^2 + bx + c \), if \( a > 0 \) the parabola opens upwards, indicating a minimum. If \( a < 0 \), the parabola opens downwards, indicating a maximum. Since \( a = -3 \), the function has a maximum value.

Step 3: Find the Vertex

The vertex formula for the x-value is \( x = -\frac{b}{2a} \). Substitute \( b = 12 \) and \( a = -3 \) to find \( x = -\frac{12}{2(-3)} = 2 \).

Step 4: Evaluate the Function at the Vertex

Substitute \( x = 2 \) into the function to find the maximum value: \( f(2) = -7 - 3(2)^2 + 12(2) = -7 - 12 + 24 = 5 \). Thus, the maximum value of the function is 5.

Step 5: State the Domain

The domain of any polynomial function, including quadratic functions, is all real numbers: \(( -\infty, \infty )\).

Step 6: State the Range

Since the function has a maximum value of 5 and opens downward, the range is \(( -\infty, 5 ]\).

Key Concepts

Maximum or Minimum Value in Quadratic Functions

In quadratic functions, determining whether a function has a maximum or a minimum value is based on the coefficient of the squared term, denoted as \( a \) in the standard quadratic form \( ax^2 + bx + c \).
If \( a > 0 \), the parabola opens upwards, creating a U-shape with a lowest point at the vertex.

• This implies a **minimum value** is present at the vertex.

If \( a < 0 \), the parabola opens downwards, forming an inverted U-shape.

• This indicates the existence of a **maximum value** at the vertex.

In the provided exercise, the quadratic function \( f(x) = -7 - 3x^2 + 12x \) has \( a = -3 \), confirming that the parabola opens downward and possesses a maximum value at its vertex. This vertex is the peak point of the parabola's curve where the highest value of the function occurs.

Domain of a Function

The domain of a function refers to all the possible input values (x-values) for which the function is defined. Quadratic functions, which are forms of polynomial functions, inherently have domains that encompass all real numbers.

This means that there are no restrictions on the values that \( x \) can take, as the function's expression is valid for every real number. Therefore, for any quadratic function, including \( f(x) = -7 - 3x^2 + 12x \) given in the exercise, the domain is expressed as:

• \((-\infty, \infty)\)

Understanding this ensures you can plug any real number into the function without encountering undefined operations.

Range of a Function

The range of a function refers to all the possible output values (y-values) that the function can produce.
In quadratic functions, much like determining the maximum or minimum value, the range depends on the direction the parabola opens.

• If a quadratic function opens upward \((a > 0)\), the smallest y-value is at the vertex, leading to a range from the vertex's y-value to infinity \(( [y_{min}, \infty) )\).
• Conversely, if the parabola opens downward \((a < 0)\), the highest y-value is at the vertex, giving a range from negative infinity up to the vertex’s y-value \(( (-\infty, y_{max}] )\).

For \( f(x) = -7 - 3x^2 + 12x \), the maximum y-value at the vertex is 5, leading to its range being precisely:

• \( (-\infty, 5] \)

Understanding the range allows one to recognize all potential y-values that the function can achieve.

Vertex of a Parabola

The vertex of a parabola describes the "turning point" of a quadratic function and is crucial for determining both the maximum or minimum value and the range of the function. For any quadratic function in the form \( ax^2 + bx + c \), the x-coordinate of the vertex is derived using the formula:

\[ x = -\frac{b}{2a} \]
This formula allows us to calculate precisely where along the x-axis the maximum or minimum point lies.

In the exercise's quadratic function \( f(x) = -7 - 3x^2 + 12x \), with \( a = -3 \) and \( b = 12 \), substituting these values yields:

• \( x = -\frac{12}{2(-3)} = 2 \)

Once the x-value of the vertex is determined, substituting it back into the quadratic equation reveals the y-coordinate, fully identifying the vertex. For \( f(x) \), the y-coordinate is:

• \( f(2) = 5 \)

This vertex (\( x=2, y=5 \)) represents the peak of the parabola defined by the exercise, giving key insights into its structure and behavior.