Problem 48 - 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. $$ f(x)=-20 x+5 x^{2}+9 $$

Step-by-Step Solution

Answer

The function has a minimum value of -11, with domain \((-\infty, \infty)\) and range \([-11, \infty)\).

Step 1: Identify the function type

The given function is a quadratic function, which can generally be represented as \( f(x) = ax^2 + bx + c \). Here, \( f(x) = 5x^2 - 20x + 9 \).

Step 2: Determine the direction of the parabola

To determine whether the quadratic function opens upwards or downwards, we check the sign of the coefficient of \( x^2 \), which is \( 5 \) in this case. Since it's positive, the parabola opens upwards, indicating that the function has a minimum value.

Step 3: Find the vertex of the parabola

The vertex of the parabolic function \( f(x) = ax^2 + bx + c \) occurs at the point \( x = -\frac{b}{2a} \). Substituting \( a = 5 \) and \( b = -20 \), we find the vertex at \( x = -\frac{-20}{2 \times 5} = 2 \).

Step 4: Calculate the minimum value

Substitute \( x = 2 \) back into the function to find the minimum value:\[f(2) = 5(2)^2 - 20(2) + 9 = 20 - 40 + 9 = -11\]Thus, the minimum value is \(-11\).

Step 5: State the domain of the function

As a quadratic function defined for all real numbers, the domain of \( f(x) = 5x^2 - 20x + 9 \) is all real numbers, denoted as \( (-\infty, \infty) \).

Step 6: State the range of the function

Since the parabola opens upwards and the minimum value is \(-11\), the range of the function is \([-11, \infty)\).

Key Concepts

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function. Quadratic functions are mathematical expressions of the form \( ax^2 + bx + c \). The shape and direction in which the parabola opens can tell us many things about the properties of the function.

• If the coefficient \(a\) (in front of \(x^2\)) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

Understanding how the parabola opens helps determine whether the function has a maximum or a minimum value. In our function \(f(x) = 5x^2 - 20x + 9\), since \(5\) is positive, the parabola opens upwards.

Vertex

The vertex of a parabola is a significant point, as it represents the highest or lowest point of the curve. For quadratic functions like \(f(x) = ax^2 + bx + c\), the vertex can be found using the formula for the x-coordinate: \(-\frac{b}{2a}\).
In the case of our function, where \(a = 5\) and \(b = -20\), we calculate the x-coordinate of the vertex:\[x = -\frac{-20}{2 \times 5} = 2\]
Substitute this back into the function to find the y-coordinate or the function value at this x, which gives us the vertex point \((2, -11)\). This vertex is key in finding the extremum of the function.

Minimum Value

The minimum value of a quadratic function is the lowest point the parabola reaches. When the parabola opens upwards, this minimum occurs at the vertex.
For our function \(f(x) = 5x^2 - 20x + 9\), we've already calculated the vertex at \(x = 2\). To find the minimum value, plug this x-value back into the function to get:\[f(2) = 5(2)^2 - 20(2) + 9 = -11\]
Thus, the minimum value of the function is \(-11\), providing the smallest y-value on the graph of the parabola.

Domain and Range

The domain of a quadratic function is the complete set of possible x-values that the function can accept. Quadratic functions can take any real number as an input, hence the domain is \((-\infty, \infty)\).
The range refers to the set of possible output values (y-values). Since the given parabola opens upwards and has a minimum point at \(-11\), the range starts at this minimum value and includes all numbers greater than or equal to \(-11\).
Therefore, the range of \(f(x) = 5x^2 - 20x + 9\) is \([-11, \infty)\). This denotes that as x assumes any real number, the function's y-value will always be \(-11\) or higher.