Problem 46 - College Algebra

Question

Give the domain and the range of each quadratic function whose graph is described. The vertex is (- 3, - 4) and the parabola opens down.

Step-by-Step Solution

Answer

The domain of the function is \( (-\infty, +\infty) \) and the range is \( (-\infty, -4] \).

Step 1: Identify Vertex of the Parabola

Given that the vertex of the parabola is \( (-3, -4) \), this means that in the vertex form \( f(x) = a(x-h)^2 + k \), the values of \( h \) and \( k \) are -3 and -4 respectively.

Step 2: Determine the direction of the Parabola's opening

The problem states that the parabola opens downwards. This indicates that the coefficient \( a \) in the vertex form is negative. However, to find the domain and range, we don't need the exact value of \( a \).

Step 3: Determine the Domain of the Function

For any real valued function, and especially for any quadratic function, the domain is the set of all real numbers, because you can substitute any real number for \( x \) in the function. Thus the domain is \( (-\infty, +\infty) \).

Step 4: Determine the Range of the Function

Since the parabola opens downwards and the vertex is the maximum point of the parabola, the range consists of all values less than or equal to the y-coordinate of the vertex. Therefore, the range is \( (-\infty, -4] \).

Key Concepts

Vertex Form of a Quadratic

In algebra, the vertex form of a quadratic equation is a way of expressing the equation that makes it simple to identify its vertex. The basic form is given as \[ f(x) = a(x-h)^2 + k \]where

• \( a \) determines the shape and direction of the parabola.
• \( (h, k) \) is the vertex of the parabola.

This form is particularly useful because it allows us to immediately read off the vertex from the equation without needing extra calculations.
For example, in the equation given in the original exercise, \( f(x) = a(x+3)^2 - 4 \)indicates a vertex of \((-3, -4)\).
This information is crucial for finding both the range of the function and understanding its maximum or minimum values.

Parabola Opening Direction

When analyzing a quadratic function, understanding the parabola's opening direction tells us how the graph behaves. A parabola can open either upwards or downwards depending on the coefficient \( a \) in the vertex form \[ f(x) = a(x-h)^2 + k \].

• If \( a > 0 \), the parabola opens upwards. It forms a 'U' shape, and the vertex is the minimum point.
• If \( a < 0 \), the parabola opens downwards, forming an 'n' shape. Here, the vertex is the maximum point.

In this exercise, because the parabola opens downwards, the equation implies \( a \) is negative, though the exact value isn't necessary for determining domain and range. Identifying the opening direction helps in understanding if the vertex represents the highest point or the lowest point of the graph.

Real Numbers Domain

In mathematics, when we talk about the domain of a function, we refer to all the possible inputs (or \( x \) values) for which the function is defined. For quadratic functions, the domain is always all real numbers.

• This is because quadratic functions are polynomials, and polynomials are defined for all real numbers.
• No matter what value \( x \) takes, a polynomial function will have an output value \( y \).

Thus, for any quadratic function written in vertex form like \[ f(x) = a(x-h)^2 + k \], the domain is \((-\infty, +\infty)\). This means you can substitute any real number into the function, and you'll always get a valid output.

Inequalities in Mathematics

Inequalities describe the relationship between two values that are not equal. When dealing with quadratic functions and their ranges, inequalities can tell us how the output values are distributed.
For a parabola that opens downwards, the range is described using inequality because the values of the function never exceed the y-coordinate of the vertex.

• This particular function has its vertex at \((-3, -4)\), so the function's outputs will always be less than or equal to -4.
• The inequality describing the range would be \( y \leq -4 \).

Inequalities help us understand the boundaries of the range in a precise manner. They are vital for interpreting situations where variables can take on a continuous set of values within a specific range.