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
I need help with this question i am so confused!!!!
think of a number -- x
multiply by 5 ---> 5x
subtract that product from 20 ---> 20 - 5x
multiply that by original number --> x(20-5x)
= -5x^2 + 20x
b) just put in each of the given values
e.g. for x=7
result = -5(49) + 20(7) = -105
c) let y = -5x^2 + 20x
this is represented by a parabola which opens downwards, so the y value of the vertex is the maximim
Since you are dealing with this type of question, I assume that you have been taught how to find the vertex of a parabola.
No worries! I'll walk you through the steps to solve this problem.
First, let's break down the steps given in the question:
1. Think of a number
2. Multiply the number by 5
3. Subtract the product from 20
4. Multiply the resulting difference by the number you first thought of
Now, let's go through each part of the question:
a) To express the final answer in terms of the original number using function notation, we need to come up with a formula. Let's say the original number is "x". We can then express the steps given in the question as follows:
Step 1: Multiply the number by 5: 5x
Step 2: Subtract the product from 20: 20 - 5x
Step 3: Multiply the resulting difference by the number you first thought of: x(20 - 5x)
Therefore, the final answer in terms of the original number using function notation would be f(x) = x(20 - 5x).
b) To determine the outputs for the input numbers 1, -1, and 7, simply substitute each number into the function we derived in part (a):
For x = 1:
f(1) = 1(20 - 5(1))
f(1) = 1(20 - 5)
f(1) = 1(15)
f(1) = 15
For x = -1:
f(-1) = -1(20 - 5(-1))
f(-1) = -1(20 + 5)
f(-1) = -1(25)
f(-1) = -25
For x = 7:
f(7) = 7(20 - 5(7))
f(7) = 7(20 - 35)
f(7) = 7(-15)
f(7) = -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). This can be done by finding the x-coordinate of the vertex of the parabola represented by the function.
The vertex, or maximum point, of a quadratic function with the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a). In our case, a = -5 and b = 20.
x = -20 / (2(-5))
x = -20 / -10
x = 2
So, the maximum result possible occurs at x = 2.
To find the corresponding y-value or output, substitute x = 2 back into the function:
f(2) = 2(20 - 5(2))
f(2) = 2(20 - 10)
f(2) = 2(10)
f(2) = 20
Therefore, the maximum result possible is 20.
I hope this helps! Let me know if you have any further questions.