Please help me plot this
2 answers:
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Answer:
f(t) = -t + 21
g(t) = 18*e^( - t / 12 + 1/4 )
Step-by-step explanation:
Given:
- The graphs for the similar question is attached.
- The same graph would be used as reference but with different coordinates for point of intersection of f(t) and g(t) @ ( 3 , 18 ) & ( 15 , 6 ).
Find:
- The formulas for functions f(t) and g(t).
Solution:
- First we will determine f(t) the blue graph which is a "linear" function. The general equation for the linear function is given as:
f(t) = m*t + c
Where, m: is the gradient ( constant )
c: The f(t) intercept. ( constant )
- The gradient m can be determined by the given points that lie on the graph:
m = ( f(t2) - f(t1) ) / ( t2 - t1 )
m = ( 6 - 18 ) / ( 15 - 3 )
m = -12 / 12 = -1
- The constant c can be evaluated by using any one point and m substituted back into the linear expression as follows:
f(t) = -t + c
18 = -(3) + c
c = 21
- The function f(t) is as follows:
f(t) = -t + 21
- The general expression for an exponential function can be written as:
g(t) = a*e^(b*t)
Where, a and b are constants to be evaluated.
- We will develop two expressions for g(t) using two given points that lie on the curve as follows:
18 = a*e^(3*b)
6 = a*e^(15*b)
- Divide the two expressions we have:
3 = e^( 3b - 15b )
Ln(3) = -12*b
b = - Ln(3) / 12
- Then the expression 1 becomes:
18 = a*e^( - Ln(3)*3 / 12)
18 = 3*a*e^(-1/4)
6 = a / e^(0.25)
a = 6*e^( 1 / 4 )
- The function g(t) can be expressed as:
g(t) = 18*e^( - t / 12 + 1/4 )
The formula for the Pythagorean theorem is,
Answer: The correct line is
Step-by-step explanation: We are given the following two sets of quadratic expressions in various forms:
We are to select one of the lines from above that represent three equivalent expressions.
We can see that there are three different forms of a quadratic expression in each of the lines:
First one is the simplified form, second is the factorised form and third one is the vertex form.
So, to check which line is correct, we need to calculate the factorised form and the vertex form from the simplified form.
We have
and
So,
Thus, Line 1 contains three equivalent expressions.
Now,
and
So,
Thus, Line 2 does not contain three equivalent expressions.
Hence, Line 1 is correct.