Question

Given f(x) = -x^2 + 10x + n, where n is a constant.

a) Express f(x) in the form of a vertex.
b) If the maximum value of the quadratic function is 7, find the value of n.
a) (a) 7; (b) n = 0
b) (a) 5; (b) n = 3
c) (a) 5; (b) n = 2
d) (a) 7; (b) n = 3

Answer 1

The quadratic function f(x) = -x^2 + 10x + n in vertex form is f(x) = -1(x - 5)^2 + 7, and the value of n is -18, given that the maximum value of f(x) is 7.

Step-by-step explanation:

To express the quadratic function f(x) = -x^2 + 10x + n in vertex form, we need to complete the square. The standard form of a quadratic function is f(x) = ax^2 + bx + c, but to get the vertex form, we use f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.

First, we factor out the coefficient of x^2, which in this case is -1:

f(x) = -1(x^2 - 10x) + n

Next, we find the value that completes the square for x^2 - 10x. This value is ±(±b/(2a))^2 and substituting a = -1 and b = 10, we get (10/2)^2 = 25. We add and subtract this value inside the parenthesis:

f(x) = -1(x^2 - 10x + 25 - 25) + n
f(x) = -1((x - 5)^2 - 25) + n

Finally we can write the function in vertex form:

f(x) = -1(x - 5)^2 + 25 + n

To find the value of n given that the maximum value of f(x) is 7, we set the vertex form equal to 7 at the vertex (h, k). Since the vertex here is (5, 25 + n), we have:

7 = -1(5 - 5)^2 + 25 + n
7 = 25 + n
n = 7 - 25
n = -18

Thus, the original quadratic function in vertex form is f(x) = -1(x - 5)^2 + 7, and the value of n is –18.