a teacher asked her students to think of a number, multiply by 5, and subtract the product from 20 , then she asked them to multiply the resulting difference by the number they first thougt of

a) use function notation to express the final answer in terms of the orginal number

b)determine the outputs for the input numbers 1, -1, & 7

c) determine the maximum result possible

Start with n. Follow the steps:

x5 = 5n
from 20 = 20 - 5n
xn = n(20-5n)

a) 20n - 5n^2
b) f(1) = 15
f(-1) = -25
f(7) = -105

c) 20n-5n^2 = 5n(4-n)
This is zero at n=0,4

It thus attains its maximum at x=2
f(2) = 20

a) Let's call the original number "x". The final answer can be expressed as: f(x) = x(20 - 5x).

b) To find the outputs for the input numbers 1, -1, and 7, we substitute these values into our function f(x):

For x = 1:
f(1) = 1(20 - 5(1)) = 1(20 - 5) = 1(15) = 15.

For x = -1:
f(-1) = -1(20 - 5(-1)) = -1(20 + 5) = -1(25) = -25.

For x = 7:
f(7) = 7(20 - 5(7)) = 7(20 - 35) = 7(-15) = -105.

c) To determine the maximum result possible, we need to find the vertex of the quadratic function f(x) = x(20 - 5x). The vertex is given by the formula: x = -b / (2a).

In this case, a = -5 and b = 20. Substituting these values into the formula:
x = -20 / (2(-5)) = -20 / (-10) = 2.

So, the maximum result possible occurs when the original number "x" is equal to 2. Substituting this value into the function:
f(2) = 2(20 - 5(2)) = 2(20 - 10) = 2(10) = 20.

Therefore, the maximum result possible is 20.

a) Let's use the variable "x" to represent the original number.

The first operation is to multiply the original number by 5, which can be written as 5x.

The second operation is to subtract the product from 20. So, the difference is (20 - 5x).

Finally, the third operation is to multiply the resulting difference by the original number. This can be expressed as x * (20 - 5x).

Therefore, the final answer in terms of the original number (x) is: f(x) = x * (20 - 5x).

b) To determine the outputs for the input numbers 1, -1, and 7, we substitute these numbers into the function f(x).

For the input number 1:
f(1) = 1 * (20 - 5(1))
= 1 * (20 - 5)
= 1 * 15
= 15

For the input number -1:
f(-1) = -1 * (20 - 5(-1))
= -1 * (20 + 5)
= -1 * 25
= -25

For the input number 7:
f(7) = 7 * (20 - 5(7))
= 7 * (20 - 35)
= 7 * (-15)
= -105

Therefore, the outputs for the input numbers 1, -1, and 7 are 15, -25, and -105 respectively.

c) To determine the maximum result possible, we can analyze the function f(x) = x * (20 - 5x).

Since the coefficient of the x^2 term (-5) is negative, the function represents a downward-opening parabola.

To find the maximum result, we can find the x-coordinate of the vertex of this quadratic function.

The x-coordinate of the vertex can be found using the formula: x = -b / 2a, where the quadratic function is in the form ax^2 + bx + c.

For our function f(x) = x * (20 - 5x), we have a = -5 and b = 20.

x = -20 / (2 * -5)
= -20 / -10
= 2

Therefore, the x-coordinate of the vertex is 2.

To find the maximum result, we substitute this x-coordinate back into the function:
f(2) = 2 * (20 - 5(2))
= 2 * (20 - 10)
= 2 * 10
= 20

Hence, the maximum result possible is 20.

To solve this problem, let's break it down step by step:

Step 1: Think of a number.
Let's use "x" to represent the number the students think of.

Step 2: Multiply the number by 5.
The product of the number "x" and 5 can be written as 5x.

Step 3: Subtract the product from 20.
The difference between 20 and 5x can be written as 20 - 5x.

Step 4: Multiply the resulting difference by the number they first thought of.
The product of (20 - 5x) and "x" can be written as x(20 - 5x).

a) Use function notation to express the final answer in terms of the original number.
Let's define a function "f(x)" to represent the final answer. So, f(x) = x(20 - 5x).

b) Determine the outputs for the input numbers 1, -1, and 7.
To find the outputs for these input numbers, we can substitute each value into the function and calculate the results:

For x = 1:
f(1) = 1(20 - 5(1)) = 1(20 - 5) = 15

For x = -1:
f(-1) = -1(20 - 5(-1)) = -1(20 + 5) = -25

For x = 7:
f(7) = 7(20 - 5(7)) = 7(20 - 35) = -105

Therefore, the outputs for the input numbers 1, -1, and 7 are 15, -25, and -105, respectively.

c) Determine the maximum result possible.
To find the maximum result, we need to find the vertex of the quadratic function f(x) = x(20 - 5x). The vertex of a quadratic function is obtained by evaluating -b/2a, where the quadratic form is ax^2 + bx + c.

Here, a = -5, b = 20, and c = 0. Evaluating -b/2a, we get -20/(2(-5)) = -20/(-10) = 2.

So, the maximum result is obtained when x = 2. Substituting this value back into the function:
f(2) = 2(20 - 5(2)) = 2(20 - 10) = 2(10) = 20

Therefore, the maximum result possible is 20.