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
a) Let's start by defining a function to express the final answer in terms of the original number.
Let x be the original number.
The first step is to multiply x by 5, which gives 5x.
Then, subtract the product from 20, resulting in 20 - 5x.
Finally, multiply the resulting difference by the original number x, which gives x * (20 - 5x).
Therefore, the function to express the final answer in terms of the original number is:
f(x) = x * (20 - 5x)
b) To determine the outputs for the input numbers 1, -1, and 7, we need to substitute these values into the function we defined in part a.
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
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 need to find the maximum value of the function f(x) = x * (20 - 5x).
First, let's find the critical points of the function by setting the derivative equal to zero:
f'(x) = 0
20 - 10x = 0
10x = 20
x = 2
Next, we need to evaluate the function at the critical point and the endpoints of the possible domain, which is all real numbers:
f(2) = 2 * (20 - 5 * 2) = 2 * (20 - 10) = 2 * 10 = 20
f(-∞) = (∞) * (-∞) = -∞
f(∞) = (∞) * (∞) = ∞
From the calculations, we find that f(2) = 20 is the maximum result possible.
Therefore, the maximum result possible is 20.
a) To express the final answer in terms of the original number, we can use function notation. Let's define the original number as "x".
First, the students would think of a number and multiply it by 5, which gives us 5x.
Then, they subtract the product from 20, giving us 20 - 5x.
Finally, they multiply the resulting difference by the original number, resulting in x(20 - 5x).
So, the function can be expressed as f(x) = x(20 - 5x).
b) To determine the outputs for the input numbers 1, -1, and 7, we substitute these values into the function:
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
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 need to find the vertex of the quadratic function represented by f(x) = x(20 - 5x).
We can start by rewriting the equation in standard form:
f(x) = -5x^2 + 20x
The vertex of the quadratic function can be found using the formula:
x = -b/2a
In this case, a = -5 and b = 20. Plugging these values into the formula, we get:
x = -20/(2*(-5)) = -20/(-10) = 2
Substituting x = 2 back into the function, we get:
f(2) = 2(20 - 5*2) = 2(20 - 10) = 2(10) = 20
Therefore, the maximum result possible is 20.