Problem 29 - 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. $$ f(x)=x^{2}+8 x+15 $$

Step-by-Step Solution

Answer

The function has a minimum value of -1. Domain: \((-\infty, \infty)\). Range: \([-1, \infty)\).

Step 1: Identify Function Type

The function given is a quadratic function: \( f(x) = x^2 + 8x + 15 \). Quadratic functions are represented by the formula \( ax^2 + bx + c \), where \( a = 1 \), \( b = 8 \), and \( c = 15 \). Since the coefficient \( a > 0 \), this parabola opens upwards, indicating that the function has a minimum value.

Step 2: Calculate Vertex

The vertex of a quadratic function in the form \( ax^2 + bx + c \) is at \( x = -\frac{b}{2a} \). Substituting \( a = 1 \) and \( b = 8 \), we find: \( x = -\frac{8}{2 imes 1} = -4 \). We substitute \( x = -4 \) into the function to find the minimum value.

Step 3: Find Function Value at Vertex

Substitute \( x = -4 \) into \( f(x) = x^2 + 8x + 15 \): \[ f(-4) = (-4)^2 + 8(-4) + 15 \] \[ = 16 - 32 + 15 \] \[ = -1 \]. Thus, the minimum value of the function is \( f(-4) = -1 \).

Step 4: Identify Domain and Range

For any quadratic function \( f(x) = ax^2 + bx + c \), the domain is all real numbers, i.e., \( (-\infty, \infty) \). Since the minimum value of the function is \( -1 \) and the parabola opens upwards, the range is \( [-1, \infty) \).

Key Concepts

Parabolas

A parabola is a U-shaped curve that represents the graph of a quadratic function. Quadratic functions have the general form \( y = ax^2 + bx + c \).
The shape of a parabola is determined by the sign of the coefficient \( a \) in this expression.

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

The point where the parabola changes direction is known as the vertex. It represents the highest or lowest point on the graph, depending on whether the parabola opens downwards or upwards.
Parabolas are symmetric along a vertical line through the vertex, called the axis of symmetry.

Vertex Form

The vertex form of a quadratic function provides a clear view of the parabola's vertex. It is expressed as \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola.
This form is quite useful for quickly determining both the vertex and the direction in which the parabola opens. It is especially helpful for graphing and understanding the transformation of the parabola.

• To convert a quadratic from standard form, \( y = ax^2 + bx + c \), to vertex form, you often complete the square.
• The vertex (\(h, k\)) can tell us the maximum or minimum value directly.

Understanding this can simplify many tasks related to quadratic functions, such as determining the graph's key features.

Domain and Range

The domain of a quadratic function is quite straightforward. It refers to all the possible input values (\(x\)-values) for which the function is defined.
For any quadratic function, the domain is all real numbers, expressed as \((-\infty, \infty)\).
The range, however, depends on the direction the parabola opens and its vertex.

• If the parabola opens upwards, the range is from the minimum value upwards, \([k, \infty)\), where \(k\) is the \(y\)-coordinate of the vertex.
• Conversely, if the parabola opens downwards, the range extends from negative infinity to the maximum value, \((-\infty, k]\).

Therefore, knowing the vertex's position is crucial for determining the function's range.

Minimum Value

The minimum value of a quadratic function occurs at the vertex when the parabola opens upwards. Since the quadratic function \(f(x)=x^{2}+8x+15\) opens upwards (because \(a > 0\)), it has a minimum value.
To find this minimum value, calculate the \(x\)-coordinate of the vertex using the formula \(x = -\frac{b}{2a}\). For this function:

• \(b = 8\) and \(a = 1\) give us \(x = -\frac{8}{2 \times 1} = -4\).

Substitute \(x = -4\) back into the function to find \(f(-4)\): \[ f(-4) = (-4)^2 + 8(-4) + 15 = 16 - 32 + 15 = -1 \] Thus, the minimum value is \(-1\), occurring at the vertex.