Problem 26 - 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. $$ f(x)=4 x-x^{2}+1 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Function Type

The given function is a quadratic function of the form \( f(x) = ax^2 + bx + c \). Here, \( a = -1 \), \( b = 4 \), and \( c = 1 \). This is a downward-opening parabola since \( a < 0 \).

Step 2: Determine if the Function has a Maximum or Minimum

For a quadratic function \( ax^2 + bx + c \), if \( a > 0 \), the parabola opens upward, indicating a minimum point. If \( a < 0 \), the parabola opens downward, indicating a maximum point. Since \( a = -1 < 0 \), the function has a maximum value.

Step 3: Find the Vertex of the Parabola

The vertex form of a quadratic function provides either the minimum or maximum point. The \( x \)-coordinate of the vertex is given by \( x = -\frac{b}{2a} \). Substitute \( b = 4 \) and \( a = -1 \) to get \( x = -\frac{4}{2(-1)} = 2 \).

Step 4: Calculate the Maximum Value of the Function

Substitute \( x = 2 \) back into the function to find the maximum value: \( f(2) = 4(2) - 2^2 + 1 = 8 - 4 + 1 = 5 \). Thus, the maximum value is 5.

Step 5: State the Domain of the Function

The domain of any quadratic function is all real numbers, expressed as \( (-\infty, \infty) \).

Step 6: State the Range of the Function

Since the function has a maximum value of 5 and it opens downwards, the range consists of values less than or equal to 5. The range is \(( -\infty, 5 ]\).

Key Concepts

Understanding Parabolas

A parabola is a symmetrical, U-shaped graph that represents quadratic functions. In our exercise, the equation is defined as \( f(x) = 4x - x^2 + 1 \). This function graphs into a parabola because it's a quadratic equation.
Parabolas can open upwards or downwards, which depends on the coefficient of the \( x^2 \) term (denoted as \( a \)). Here, since \( a = -1 \) (which is less than zero), the parabola opens downwards.
A downward-opening parabola means that it showcases a maximum point at its vertex. This is because the arms of the parabola extend infinitely downward from this point, making the vertex the topmost point in the graph.
This knowledge helps us understand where the function achieves its highest value, which is particularly useful in solving optimization problems.

The Vertex: Key Point on a Parabola

The vertex of a parabola is a critical point that tells us the maximum or minimum value of the quadratic function. It's essentially the "tip" of the parabola.
For a quadratic equation \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is calculated using the formula \( x = -\frac{b}{2a} \). In our exercise, with \( b = 4 \) and \( a = -1 \), we find \( x = 2 \).
Once we know the x-coordinate, we substitute it back into the original equation to find the y-coordinate. So, we compute \( f(2) = 4(2) - 2^2 + 1 = 5 \). This tells us that the vertex is the point \( (2, 5) \).
In this scenario, the vertex represents the maximum point since the parabola opens downward. This means \( f(2) = 5 \) is the highest value the function can achieve.

Exploring Domain and Range of Quadratic Functions

Understanding domain and range is essential when analyzing functions.
The domain of a function refers to all possible input values (x-values) for which the function is defined. For quadratic functions like \( f(x) = 4x - x^2 + 1 \), the domain is all real numbers. This is because you can substitute any real number for \( x \), and the function will yield a corresponding output.
Mathematically, the domain is expressed as \( (-\infty, \infty) \), indicating the function accepts all real x-values.

The range, however, depends on the direction in which the parabola opens. Since our parabola opens downward and has a maximum point at \( (2, 5) \), the range covers all y-values from negative infinity up to and including 5. We express this range as \( (-\infty, 5] \), meaning that the highest output value is 5 and it can be any real number less than or equal to this.