Problem 36
Question
\(29-38=\) Find the maximum or minimum value of the function. $$ f(x)=-\frac{x^{2}}{3}+2 x+7 $$
Step-by-Step Solution
Verified Answer
The maximum value of the function is 10.
1Step 1: Identify the form of the quadratic function
The function given is a quadratic function of the form \( f(x) = ax^2 + bx + c \). In this case, \( a = -\frac{1}{3} \), \( b = 2 \), and \( c = 7 \). Since \( a < 0 \), the parabola opens downwards, indicating a maximum value.
2Step 2: Calculate the vertex x-coordinate
The x-coordinate of the vertex of a parabola \( ax^2 + bx + c \) is given by the formula \( x = -\frac{b}{2a} \). Substitute the values of \( b \) and \( a \) into the formula: \( x = -\frac{2}{2 \times -\frac{1}{3}} = 3 \).
3Step 3: Find the maximum value of the function
Substitute the x-coordinate back into the function to find the maximum value: \( f(3) = -\frac{(3)^2}{3} + 2 \times 3 + 7 = -3 + 6 + 7 = 10 \). The maximum value of the function is 10.
Key Concepts
Vertex of a ParabolaMaximum Value of a FunctionDownward Opening Parabola
Vertex of a Parabola
In a quadratic function, the vertex represents an essential point on a parabola. This point is either the highest or lowest point on the curve, depending on the parabola's orientation. When we have the quadratic function in the form \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula for the x-coordinate: \( x = -\frac{b}{2a} \). For the function \( f(x) = -\frac{x^2}{3} + 2x + 7 \),
Identify the coefficients as follows: \( a = -\frac{1}{3} \), \( b = 2 \), and \( c = 7 \). Substituting these values into our vertex formula, we obtain \( x = 3 \).
Since the vertex is a point, the y-coordinate can be found by substituting \( x = 3 \) back into the function: \( f(3) = 10 \). Therefore, the vertex is \((3, 10)\), marking the significant point of the parabola.
Identify the coefficients as follows: \( a = -\frac{1}{3} \), \( b = 2 \), and \( c = 7 \). Substituting these values into our vertex formula, we obtain \( x = 3 \).
Since the vertex is a point, the y-coordinate can be found by substituting \( x = 3 \) back into the function: \( f(3) = 10 \). Therefore, the vertex is \((3, 10)\), marking the significant point of the parabola.
Maximum Value of a Function
The maximum value of a parabola is the greatest y-value the function attains. For a downward opening parabola, the function achieves its peak at the vertex. To find this, determine the x-coordinate of the vertex using the formula \( x = -\frac{b}{2a} \). In this exercise, with coefficients \(a = -\frac{1}{3}\) and \(b = 2\), this calculation gives us \( x = 3 \).
Once the x-coordinate is found, substitute it back into the function to calculate the maximum y-value: \( f(3) = 10 \).
Thus, the maximum value of the function \( f(x) = -\frac{x^{2}}{3}+2x+7 \) is 10.
Once the x-coordinate is found, substitute it back into the function to calculate the maximum y-value: \( f(3) = 10 \).
Thus, the maximum value of the function \( f(x) = -\frac{x^{2}}{3}+2x+7 \) is 10.
Downward Opening Parabola
A downward opening parabola is characterized by its concave shape that resembles a n shape. This is determined by the sign of the leading coefficient \(a\) in the quadratic function \( f(x) = ax^2 + bx + c \).
If \(a < 0\), as with our function \( f(x) = -\frac{x^{2}}{3}+2x+7 \), the parabola opens downwards.
This indicates that the parabola has a maximum point at the vertex rather than a minimum. The downward orientation plays a vital role when analyzing the function, as it affects the domain and range. In the case of a downward opening parabola like ours, the range comprises all values less than or equal to the maximum value, which is found at the vertex, here being 10.
If \(a < 0\), as with our function \( f(x) = -\frac{x^{2}}{3}+2x+7 \), the parabola opens downwards.
This indicates that the parabola has a maximum point at the vertex rather than a minimum. The downward orientation plays a vital role when analyzing the function, as it affects the domain and range. In the case of a downward opening parabola like ours, the range comprises all values less than or equal to the maximum value, which is found at the vertex, here being 10.
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Problem 35
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