Problem 5 - Algebra and Trigonometry

Question

The graph of a quadratic function is given. (a) Find the coordinates of the vertex. (b) Find the maximum or minimum value of f. (c) Find the domain and range of f. $$ f(x)=-x^{2}+6 x-5 $$

Step-by-Step Solution

Answer

The vertex is (3, 4), maximum value is 4, domain is all real numbers, and range is (-∞, 4].

Step 1: Transform the Equation

Rewrite the function in vertex form by completing the square. Start with the given quadratic function: \[ f(x) = -x^2 + 6x - 5 \] Factor out the negative from the first two terms: \[ f(x) = -(x^2 - 6x) - 5 \] To complete the square, take half of the coefficient of \(x\), square it, and add inside the parentheses and subtract it outside: \[ f(x) = -((x^2 - 6x + 9) - 9) - 5 \] \[ f(x) = -(x - 3)^2 + 9 - 5 \] Finally, write it in vertex form: \[ f(x) = -(x - 3)^2 + 4 \].

Step 2: Identify the Vertex

From the vertex form equation, \[ f(x) = -(x - 3)^2 + 4 \], the vertex \((h, k)\) is easily identified as \((3, 4)\). Therefore, the coordinates of the vertex are \((3, 4)\).

Step 3: Determine Maximum or Minimum

Since the quadratic coefficient \(-1\) is negative, the parabola opens downward. Thus, the function has a maximum value. The maximum value of \(f(x)\) corresponds to the \(y\)-coordinate of the vertex, which is \(4\).

Step 4: Find the Domain

For any quadratic function, the domain is all real numbers \(x\). Therefore, the domain of \(f(x) = -x^2 + 6x - 5\) is \((-\infty, \infty)\).

Step 5: Find the Range

Since the function opens downward and has a maximum value of \(4\), we have the range as all values below or equal to 4. Thus, the range of \(f(x)\) is \(( -\infty, 4 ]\).

Key Concepts

Vertex Form

Transforming a quadratic function into its vertex form makes it easier to identify key features such as the vertex, which indicates the highest or lowest point on the graph. The vertex form of a quadratic function is expressed as \[ f(x) = a(x - h)^2 + k \]where

• \(a\) determines the opening direction and width of the parabola,
• \((h, k)\) are the coordinates of the vertex.

To convert a standard quadratic function to vertex form, completing the square is a crucial step. This transformation not only reveals the vertex but also the maximum or minimum value of the function.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that makes it easier to graph. It's especially useful for converting the standard quadratic form\[ ax^2 + bx + c \]into vertex form. Here's how you complete the square:1. **Factor out** the coefficient of \(x^2\) from the first two terms if necessary.2. **Identify** the coefficient of \(x\) and divide it by 2, then square the result.3. **Add and subtract** this square inside the quadratic, adjusting the equation accordingly.For example, with the function \[ f(x) = -x^2 + 6x - 5 \],we factor out \(-1\) from the first two terms, and proceed to complete the square:

• Take half of \(6\), which is \(3\), and square it to get \(9\)
• Insert this inside the parentheses and adjust outside: \[ f(x) = -(x^2 - 6x + 9 - 9) - 5 \]then simplify and write as: \[ f(x) = -(x - 3)^2 + 4 \]

This process unveils the vertex of the parabola, making graphing and analysis straightforward.

Maximum and Minimum Values

Quadratic functions exhibit parabolic graphs, which inherently either have a highest or a lowest point, known as a vertex. Whether a function has a maximum or minimum depends on the coefficient \(a\) in the equation\[ f(x) = a(x - h)^2 + k \].

• If \(a > 0\), the parabola opens upwards, and the vertex is a minimum point.
• If \(a < 0\), the parabola opens downwards, indicating a maximum point at the vertex.

Let's consider the function \(f(x) = -(x - 3)^2 + 4\).Here, \(a = -1\), which means the parabola opens downwards. Thus, there is a maximum value, which can be found at the \(y\)-coordinate of the vertex. For our function, this is \(4\).

Domain and Range

Understanding the domain and range of a quadratic function is key in knowing what \(x\) and \(y\) values are feasible. **Domain:**

• For quadratic functions, the domain is all real numbers since you can substitute any real number into \(x\) and obtain a valid output. Thus, the domain is expressed as \[ (-\infty, \infty) \].

**Range:**

• The range depends on whether the quadratic has a maximum or minimum value. If the parabola opens upwards, all outputs are above the vertex's \(y\)-coordinate. Conversely, if it opens downwards, outputs are below the vertex level.
• For \(f(x) = -(x - 3)^2 + 4\),the parabola opens downwards,delineating the range as \[ (-\infty, 4 ] \].

Consider both aspects when analyzing quadratic functions to fully grasp their input and output behaviors.