Problem 83 - Precalculus

Question

Determine the domain and range of the quadratic function. \(k(x)=3 x^{2}-6 x-9\)

Step-by-Step Solution

Answer

The domain is all real numbers and the range is \( [-12, \infty) \).

Step 1: Identify the Type of Function

The function given is a quadratic function, which generally has the form \( f(x) = ax^2 + bx + c \). Here, \( a = 3 \), \( b = -6 \), and \( c = -9 \). This tells us we are working with a parabola.

Step 2: Determine the Domain

The domain of a quadratic function is all real numbers, \( \mathbb{R} \), because you can plug any real number into the equation and obtain a real output.

Step 3: Determine the Vertex of the Parabola

To find the range, first calculate the vertex of the parabola because the vertex form of a quadratic will guide us to the minimum or maximum value. The x-coordinate of the vertex can be calculated using \( x = -\frac{b}{2a} \). Here \( x = -\frac{-6}{2\times3} = 1 \).

Step 4: Calculate the y-coordinate of the Vertex

Substitute \( x = 1 \) back into the function to find the y-coordinate of the vertex. \( k(1) = 3(1)^2 - 6(1) - 9 = 3 - 6 - 9 = -12 \). The vertex of the parabola is \( (1, -12) \).

Step 5: Analyze the Parabola's Orientation

Since the coefficient \( a = 3 \) is positive, the parabola opens upwards. This means the vertex represents the minimum point of the function.

Step 6: Determine the Range

With the parabola opening upwards and the vertex being the minimum point, the range of the function is all real numbers greater than or equal to -12. Thus, the range is \( [-12, \infty) \).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial that has the form:

• \( f(x) = ax^2 + bx + c \)

Here, \( a \), \( b \), and \( c \) are constants. It represents a parabola when graphed on a coordinate plane.
The leading coefficient \( a \), determines whether the parabola opens upward or downward:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Quadratic functions are essential in various applications such as physics, engineering, and economics, where relationships can form parabolic shapes.
In the given function, \( k(x) = 3x^{2} - 6x - 9 \), we identify a quadratic function with the coefficients: \( a = 3 \), \( b = -6 \), and \( c = -9 \).
This indicates that our parabola will open upwards, given that \( a \) is positive.

Vertex of a Parabola

The vertex of a parabola is a key point that represents the maximum or minimum of the function. This point helps determine the parabola’s range.
The vertex is found using a formula for the x-coordinate:

• \( x = -\frac{b}{2a} \)

For our function \( k(x) = 3x^{2} - 6x - 9 \), plugging in the values gives:

• \( x = -\frac{-6}{2 \times 3} = 1 \)

Once we have the x-value, the y-coordinate is found by substituting back into the function:

• \( k(1) = 3(1)^2 - 6(1) - 9 = -12 \)

Therefore, the vertex of the parabola is \((1, -12)\).
The parabola opens upwards, so this vertex represents the lowest point in the function's range.

Function Domain

The domain of a function is the set of all possible input values (x-values) that the function can accept. For quadratic functions,

• the domain is typically all real numbers

This means you can substitute any real number for \( x \) in \( k(x) = 3x^2 - 6x - 9 \), and the function will yield a valid output.
In other words, the domain of our quadratic function is:

• \( (-\infty, \infty) \)

This property holds because quadratic functions have no restrictions, unlike functions that could involve divisions by zero or square roots of negative numbers.

Function Range

The range of a function describes the set of possible output values (y-values). For a quadratic function like \( k(x) = 3x^{2} - 6x - 9 \), the range depends on the direction in which the parabola opens and its vertex point.
Since the parabola opens upwards and has a vertex of \( (1, -12) \), it indicates that the function's minimum output value is \( -12 \) and increases thereafter.
Thus, the range for this quadratic function is:

• \( [-12, \infty) \)

The range starts at \( -12 \), which is the lowest point on the graph, and extends to positive infinity, including every value greater than \( -12 \).
This means the function's outputs can cover an entire half-plane of y-values above \( -12 \).