The Graph Of Which Function Has A Minimum Located At \[$(4, -3)\$\]?A. \[$f(x) = -\frac{1}{2} X^2 + 4x - 11\$\]B. \[$f(x) = -2x^2 + 16x - 35\$\]C. \[$f(x) = \frac{1}{2} X^2 - 4x + 5\$\]D. \[$f(x) = 2x^2 - 16x +
**The Graph of Which Function Has a Minimum Located at (4, -3)?**
Understanding the Problem
To determine which function has a minimum located at (4, -3), we need to analyze each given function and find the vertex of its graph. The vertex of a parabola is the point where the function reaches its minimum or maximum value. In this case, we are looking for the function with a minimum at (4, -3).
Analyzing the Functions
Let's analyze each function given in the options:
A. f(x) = -\frac{1}{2} x^2 + 4x - 11
To find the vertex of this function, we can use the formula:
x = -b / 2a
where a = -\frac{1}{2} and b = 4
x = -4 / (2 * -\frac{1}{2}) x = -4 / -1 x = 4
Now, we need to find the y-coordinate of the vertex. We can plug x = 4 into the function:
f(4) = -\frac{1}{2} (4)^2 + 4(4) - 11 f(4) = -\frac{1}{2} (16) + 16 - 11 f(4) = -8 + 16 - 11 f(4) = -3
So, the vertex of this function is (4, -3).
B. f(x) = -2x^2 + 16x - 35
Using the same formula, we get:
x = -b / 2a
where a = -2 and b = 16
x = -16 / (2 * -2) x = -16 / -4 x = 4
Now, we need to find the y-coordinate of the vertex. We can plug x = 4 into the function:
f(4) = -2(4)^2 + 16(4) - 35 f(4) = -2(16) + 64 - 35 f(4) = -32 + 64 - 35 f(4) = -3
So, the vertex of this function is (4, -3).
C. f(x) = \frac{1}{2} x^2 - 4x + 5
Using the same formula, we get:
x = -b / 2a
where a = \frac{1}{2} and b = -4
x = -(-4) / (2 * \frac{1}{2}) x = 4 / 1 x = 4
Now, we need to find the y-coordinate of the vertex. We can plug x = 4 into the function:
f(4) = \frac{1}{2} (4)^2 - 4(4) + 5 f(4) = \frac{1}{2} (16) - 16 + 5 f(4) = 8 - 16 + 5 f(4) = -3
So, the vertex of this function is (4, -3).
D. f(x) = 2x^2 - 16x + 35
Using the same formula, we get:
x = -b / 2a
where a = 2 and b = -16
x = -(-16) / (2 * 2) x = 16 / 4 x = 4
Now, we need to find the y-coordinate of the vertex. We can plug x = 4 into the function:
f(4) = 2(4)^2 - 16(4) + 35 f(4) = 2(16) - 64 + 35 f(4) = 32 - 64 + 35 f(4) = -3
So, the vertex of this function is (4, -3).
Conclusion
All four functions have a minimum located at (4, -3). Therefore, the correct answer is:
A, B, C, and D
However, if we are looking for a single function, we can choose any one of the above functions as the correct answer.
Q&A
Q: What is the vertex of a parabola?
A: The vertex of a parabola is the point where the function reaches its minimum or maximum value.
Q: How do we find the vertex of a parabola?
A: We can use the formula x = -b / 2a to find the x-coordinate of the vertex, and then plug this value into the function to find the y-coordinate.
Q: What is the significance of the vertex of a parabola?
A: The vertex of a parabola represents the minimum or maximum value of the function.
Q: Can a parabola have more than one vertex?
A: No, a parabola can have only one vertex.
Q: How do we determine if a parabola is a minimum or maximum?
A: We can use the coefficient of the x^2 term to determine if the parabola is a minimum or maximum. If the coefficient is positive, the parabola is a minimum. If the coefficient is negative, the parabola is a maximum.
Q: Can a parabola have a vertex at (4, -3)?
A: Yes, a parabola can have a vertex at (4, -3). In fact, all four functions given in the options have a vertex at (4, -3).