We want to write an exponential function so that the slope from the point (0, f(0)) to the point (2, f(2)) is equal to 12. Let's move the points so that the slope between the points is 12 and that the x-coordinates of the points are 0 and 12.
Now, we have an idea about how the graph changes as the points change. Using the Slope Formula, we can write the following equation.
m=y_2-y_1/x_2-x_1 ⇒ 12=f(2)-f(0)/2-0
Since f(x) is an exponential function, it has the form f(x)=a(b)^x. Then, f(2)=a(b)^2 and f(0)=a(b)^0.
f(x)=a(b)^x ⇒ lf(0)= a(b)^0 f(2)= a(b)^2
Let's substitute the equivalent expressions.
If we let a be equal to, for example, 1, then we have the following equation.
24 =a(b+1)(b-1)
⇓
24 =(b+1)(b-1)
We need to find two factors whose difference is 2 and product is 24. For example, 6 and 4.
24 = (b+1)_6 (b-1)_4 ⇒ b=5
Therefore, b is 5. We can now substitute a=1 and b=5 into the function form.
f(x) = a (b)^2 ⇒ f(x) = 1(5)^x
This exponential function satisfies the desired condition. Please note that we can find different functions for this exercise.
Checking Our Answer
Checking Our Answer
Let's first find the points on the exponential function when x=0 and x=2.
f(x)= 5^x
Operation
x=0
x=2
Substitution
f( 0)= 5^0
f( 2)= 5^2
Point
(0,1)
(2,25)
Now, we can calculate the slope from (0,1) to (2,25).
