Problem 43
Question
An equation of a quadratic function is given. a. Determine, without graphing, whether the function has a minimum value or a maximum value. b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. $$f(x)-5 x^{2}-5 x$$
Step-by-Step Solution
Verified Answer
The function has a maximum value. This maximum value is -2.5, and it occurs at x = 0.5. The domain of the function is all real numbers, and the range is (-∞, -2.5].
1Step 1: Determining Minimum or Maximum
The given function is a quadratic function of the form \(f(x) = ax^2 + bx + c\). In this exercise, the function is \(f(x) = -5x^2 - 5x\). Its coefficient 'a' is negative (-5). Typically, if 'a' is positive, the function has a minimum value. If 'a' is negative, the function has a maximum value. Therefore, this function has a maximum value.
2Step 2: Find Maximum Value
The maximum or minimum value of a quadratic function \(f(x) = ax^2 + bx + c\) is given at \(x = -b/2a\). Substitute b = -5 and a = -5 into the formula: \(x = -(-5)/(2*-5)\) = 0.5. Now substitute x = 0.5 into the function to find the maximum value: \(f(0.5) = -5*(0.5)^2 - 5*0.5 = -2.5\). So, the maximum value is -2.5 and it occurs at x = 0.5.
3Step 3: Identifying Domain and Range
The domain of a quadratic function is all real numbers. The range of a quadratic function with a maximum value is all values less than or equal to the maximum value. Therefore, the domain is (-∞, ∞) and the range is (-∞, -2.5].
Key Concepts
Understanding Minimum and Maximum ValuesDomain and Range of a FunctionQuadratic Equation Analysis
Understanding Minimum and Maximum Values
When studying quadratic functions, an essential aspect to consider is whether these functions have minimum or maximum values. This characteristic comes from the function's shape, which is a parabola. A parabola can either open upwards or downwards, influenced by the coefficient before the quadratic term, symbolized as 'a' in the general equation f(x) = ax^2 + bx + c.
For example, when 'a' is negative, our parabola opens downwards, indicating that the graph has a highest point, called the maximum value. Conversely, if 'a' is positive, the parabola opens upwards, denoting that the graph possesses a lowest point, known as the minimum value. This foundational understanding helps in identifying the extremal values without even graphing the function.
For example, when 'a' is negative, our parabola opens downwards, indicating that the graph has a highest point, called the maximum value. Conversely, if 'a' is positive, the parabola opens upwards, denoting that the graph possesses a lowest point, known as the minimum value. This foundational understanding helps in identifying the extremal values without even graphing the function.
Domain and Range of a Function
Every function possesses a domain and range, crucial concepts in understanding its behavior. The domain of a function refers to all possible input values, or 'x' values, that the function can accept. For a quadratic function, the domain is always all real numbers, symbolized by (-∞, ∞), because you can plug any real number into the quadratic formula and receive a result.
On the other side, the range corresponds to all the possible outputs or 'y' values that a function can produce. The nature of the range depends on whether the parabola opens upwards or downwards. For our equation f(x) = -5x^2 - 5x, which opens downwards, the range is limited to all real numbers less than or equal to the maximum value we calculated, and in this case, it's (-∞, -2.5].
On the other side, the range corresponds to all the possible outputs or 'y' values that a function can produce. The nature of the range depends on whether the parabola opens upwards or downwards. For our equation f(x) = -5x^2 - 5x, which opens downwards, the range is limited to all real numbers less than or equal to the maximum value we calculated, and in this case, it's (-∞, -2.5].
Quadratic Equation Analysis
Conducting a thorough analysis of quadratic equations helps to grasp their behavior fully. This process involves finding the roots, vertex, axis of symmetry, and intercepts. The roots are the solutions to the equation set to zero and reflect the points where the parabola crosses the x-axis.
The vertex of the parabola, found using the formula x = -b/2a, provides the parabola's peak, which is either the maximum or minimum value. The axis of symmetry can be derived from this location, being a vertical line that divides the parabola into two mirror images. In the given example of the equation f(x) = -5x^2 - 5x, by finding that x equals 0.5 for the vertex, we also understand that the axis of symmetry is the line x = 0.5.
Lastly, the y-intercept occurs where the function crosses the y-axis, which is simply found by evaluating f(0). This comprehensive analysis aids in sketching the graph and better understanding the quadratic function's overall properties.
The vertex of the parabola, found using the formula x = -b/2a, provides the parabola's peak, which is either the maximum or minimum value. The axis of symmetry can be derived from this location, being a vertical line that divides the parabola into two mirror images. In the given example of the equation f(x) = -5x^2 - 5x, by finding that x equals 0.5 for the vertex, we also understand that the axis of symmetry is the line x = 0.5.
Lastly, the y-intercept occurs where the function crosses the y-axis, which is simply found by evaluating f(0). This comprehensive analysis aids in sketching the graph and better understanding the quadratic function's overall properties.
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