Mathematics, 30.01.2020 01:00 anna8352

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)

Answers: 3

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Answer from: shariaharper

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 $x = \frac{-b}{2a}$ .
Applying this to the given function,
$x = \frac{-(50)}{2(-1)} = \frac{50}{2} = 25$ .

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 from: jfleming733

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

Step-by-step 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:

$h=\frac{-b}{2a}$ 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.

$h=\frac{-50}{2(-1)}$

$h=\frac{50}{2}$

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