Given the function f(x) = 5x^2-40x+87 and the quadratic formula g(x) represented in the table below, which function has the smaller minimum value?
x {1} {2} {3} {4} {5}
g(x) {34} {19} {10} {7} {10}
That means both functions have the same minimum value.
To compare the minimum values of the functions f(x) and g(x), we need to find the minimum value of each function.
For f(x), we can find the minimum using calculus. The function f(x) is a quadratic function in the form of ax^2+bx+c, where a = 5, b = -40, and c = 87. The vertex of a quadratic function is given by (-b/2a, f(-b/2a)).
Using this formula, we can find the x-coordinate of the vertex:
x = -b/2a = -(-40)/(2*5) = 40/10 = 4
To find the y-coordinate of the vertex, substitute this x-value back into the function:
f(4) = 5(4)^2 - 40(4) + 87
= 5(16) - 160 + 87
= 80 - 160 + 87
= -80 + 87
= 7
So, the minimum value of f(x) is 7.
For g(x), we can see from the table that the minimum value is 7.
Comparing the minimum values, we can conclude that both f(x) and g(x) have the same minimum value of 7.
well, f(x) has its vertex at (4,7) so ... trick question
To determine which function has the smaller minimum value, we need to analyze the minimum values of both functions.
The function f(x) = 5x^2 - 40x + 87 is a quadratic function in the form of f(x) = ax^2 + bx + c, where a = 5, b = -40, and c = 87. The minimum value of a quadratic function occurs at the vertex of the parabola, which is given by the formula x = -b / (2a). Substituting the values of a and b, we can find the x-coordinate of the vertex:
x = -(-40) / (2 * 5)
x = 40 / 10
x = 4
To find the minimum value, substitute x = 4 into the function:
f(4) = 5(4)^2 - 40(4) + 87
f(4) = 80 - 160 + 87
f(4) = 7
So f(x) has a minimum value of 7.
The quadratic formula g(x) is given in a tabular form with different values of x and their corresponding values of g(x). We can observe that the minimum value of g(x) is 7, which occurs at x = 4.
Comparing the minimum values of f(x) and g(x), both functions have the same minimum value of 7. Therefore, neither function has a smaller minimum value.