Problem 31
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. $$ f(x)=-14 x-x^{2}-109 $$
Step-by-Step Solution
Verified Answer
The function has a maximum value of -256. Domain: (-∞,∞), Range: (-∞,-256].
1Step 1: Identify the Type of Function
The given function is \( f(x) = -14x - x^2 - 109 \). This is a quadratic function because it includes an \( x^2 \) term.
2Step 2: Determine the Nature of the Quadratic Function
Since the coefficient of \( x^2 \) is \(-1\) (negative), the parabola opens downwards. Therefore, the function has a maximum value.
3Step 3: Find the Vertex of the Parabola
The vertex form of a parabola \( ax^2 + bx + c \) gives the maximum or minimum value. The \( x \)-coordinate of the vertex is found using \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = -14 \), so \( x = -\frac{-14}{2(-1)} = 7 \).
4Step 4: Calculate the Maximum Value
Substitute \( x = 7 \) back into the function to find the maximum value: \[ f(7) = -14(7) - (7)^2 - 109 \] \[ f(7) = -98 - 49 - 109 = -256 \] So the maximum value is \(-256\).
5Step 5: Determine the Domain of the Function
The domain of a quadratic function is all real numbers, because there are no restrictions on \( x \). So the domain is \( (-\infty, \infty) \).
6Step 6: Determine the Range of the Function
The range of the function is all values \( y \leq -256 \) since the maximum value is \(-256\), and the parabola opens downward. Therefore, the range is \((-fty, -256] \).
Key Concepts
Maximum ValueVertex of a ParabolaDomain and Range
Maximum Value
When working with quadratic functions, an important aspect to understand is whether the function has a maximum or a minimum value. For the function \( f(x) = -14x - x^2 - 109 \), the coefficient of \( x^2 \) is \(-1\), which is negative. This tells us that the parabola opens downwards, creating an upside-down U-shape.
In such cases, the function possesses a maximum value rather than a minimum. To find this maximum value, you first need to locate the vertex of the parabola, because the maximum occurs at the vertex when the parabola opens down.
Once the \( x \)-coordinate of the vertex is found (which is 7 in this case), you substitute it back into the function to find the maximum value. For our function, the maximum value comes out to be \(-256\). This value represents the highest point on the graph of the function.
In such cases, the function possesses a maximum value rather than a minimum. To find this maximum value, you first need to locate the vertex of the parabola, because the maximum occurs at the vertex when the parabola opens down.
Once the \( x \)-coordinate of the vertex is found (which is 7 in this case), you substitute it back into the function to find the maximum value. For our function, the maximum value comes out to be \(-256\). This value represents the highest point on the graph of the function.
- Parabola opens downwards
- Maximum value at the vertex
- Calculated maximum value is \(-256\)
Vertex of a Parabola
The vertex of a parabola is a crucial element when analyzing quadratic functions like \( f(x) = -14x - x^2 - 109 \). It provides significant insight into the graph's properties, such as the direction of the parabola and its maximum or minimum value.
For a standard quadratic equation, \( ax^2 + bx + c \), the \( x \)-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). For our function:
With \( x = 7 \), the parabola reaches its maximum value at this point on the horizontal axis. To find the corresponding \( y \)-value, you plug \( x = 7 \) back into the original function, confirming the maximum value of \(-256\). Thus, the vertex of the parabola is \((7, -256)\).
For a standard quadratic equation, \( ax^2 + bx + c \), the \( x \)-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \). For our function:
- \( a = -1 \)
- \( b = -14 \)
With \( x = 7 \), the parabola reaches its maximum value at this point on the horizontal axis. To find the corresponding \( y \)-value, you plug \( x = 7 \) back into the original function, confirming the maximum value of \(-256\). Thus, the vertex of the parabola is \((7, -256)\).
- Vertex formula: \( x = -\frac{b}{2a} \)
- Vertex: \((7, -256)\)
Domain and Range
Understanding the domain and range of a quadratic function is essential in grasping the full picture of what the function can do. The domain of any quadratic function, such as \( f(x) = -14x - x^2 - 109 \), is all real numbers. This means there are no restrictions on what \( x \) can be—it spans from \(-\infty\) to \( \infty \).
The range, however, is determined by the maximum or minimum value of the function. Since our function has a maximum value and the parabola opens downwards, the range includes all values \( y \) that are less than or equal to \(-256\). This reflects the fact that the parabola never goes above this maximum point. Thus, the range of the function is \(( -\infty, -256 ]\).
The range, however, is determined by the maximum or minimum value of the function. Since our function has a maximum value and the parabola opens downwards, the range includes all values \( y \) that are less than or equal to \(-256\). This reflects the fact that the parabola never goes above this maximum point. Thus, the range of the function is \(( -\infty, -256 ]\).
- Domain: all real numbers \((-\infty, \infty)\)
- Range: \( y \leq -256 \)
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