Question

The function f(x) = −x2 + 50x − 264 models the profit, in dollars, a yoga studio makes for selling monthly memberships, where x is the number of memberships sold and f(x) is the amount of profit.

Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points)

Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)

Answer 1

The given function is
`f(x) = -x^2 + 50x - 264`. This is a quadratic function because the degree of the polynomial, the variable with the highest exponent, is 2.

All quadratic functions can be expressed in the form
`f(x) = ax^2 + bx + c`.
In your equation,
a = -1
b = 50
c = -264

The equation for the x-coordinate vertex of a quadratic function is
`x = (-b)/(2a)`.
Applying this to the given function,

`x = (-(50))/(2(-1)) = (50)/(2) = 25`.

Now plug the value of x into f(x) to find f(x) (aka the y-coordinate).

`f(x) = -x^2 + 50x - 264 \\ f(25) = -(25)^2 + 50(25) - 264 \\ f(25) = 361`

The vertex is (25, 361)

Part B
The x-intercept is where the graph crosses the x-axis. Therefore, f(x) (aka the y-coordinate) will be 0. So the x-intercept will be in the form of (x, 0).

To find the x-intercept, set f(x) = 0.

`0 = -x^2 + 50x - 264 \\ 0 = (x-6)(x-44) \\ x = 6, x = 44`.

Therefore the x-intercepts will be at (6, 0) and (44, 0)

Answer 2

The vertex is (25,361), the x intercepts are (6,0) and (44,0).

Explanation:

Consider the provided function,

`f(x) = -x^2 + 50x - 264`

The standard equation of the quadratic equation is
`f(x) = ax^2 + bx + c`

And the vertex of the quadratic equation can be calculated as:

`h=(-b)/(2a)` and k = f(h)

Part A:

By comparing the provided equation with the standard equation we can concluded that a = -1, b = 50 and c = -264

Now use the vertex formula and substitute the respective value in the formula that will give us the x coordinate of the vertex.

`h=(-50)/(2(-1))`

`h=(50)/(2)`

`h=25`

Now substitute h = 25 in the provided equation as shown:

`f(h) = -(25)^2 + 50(25) - 264`

`f(h) = -625 + 1250 - 264`

`f(h) =361`

Hence, the vertex is (25,361).

The provided equation has a negative x² coefficient that means it is a downward parabola having maximum at vertex (25,361) but there exist no minimum point.

In context of the problem: The profit will be maximum when the number of membership sold is 25.

Part B:

To determine the x intercept simply substitute f(x)=0 and solve for x.

`-x^2 + 50x - 264=0`

`x^2 -50x + 264=0`

`x^2 -6x-44x +264=0`

`x(x-6)-44(x-6)=0`

`(x-6)(x-44)=0`

`x=6\ or\ x=44`

Hence, the x intercepts are (6,0) and (44,0).

That means, there will be zero profit when the number of memberships sold are 6 or 44.