Respuesta :
Answer:
[tex]y=8 \cdot (\frac{5}{4})^x[/tex]
[tex]f(x)=8 \cdot (\frac{5}{4})^x[/tex]
or
[tex]f(x)=8 \cdot (1.25)^x[/tex]
Step-by-step explanation:
We are going to see if the exponential curve is of the form:
[tex]y=a \cdot b^x[/tex], ([tex]b\neq 0[/tex]).
If you are given the [tex]y-[/tex]intercept, then [tex]a[/tex] is easy to find.
It is just the [tex]y-[/tex]coordinate of the [tex]y-[/tex]intercept is your value for [tex]a[/tex].
(Why? The [tex]y-[/tex]intercept happens when [tex]x=0[/tex]. Replacing [tex]x[/tex] with 0 gives [tex]y=a \cdot b^0=a \cdot 1=a[/tex]. This says when [tex]x=0 \text{ that} y=a[/tex].)
So [tex]a=8[/tex].
So our function so far looks like this:
[tex]y=8 \cdot b^x[/tex]
Now to find [tex]b[/tex] we need another point. We have two more points. So we will find [tex]b[/tex] using one of them and verify for our resulting equation works for the other.
Let's do this.
We are given [tex](1,10)[/tex] is a point on our curve.
So when [tex]x=1[/tex], [tex]y=10[/tex].
[tex]10=8 \cdot b^1[/tex]
[tex]10=8 \cdot b[/tex]
Divide both sides by 8:
[tex]\frac{10}{8}=b[/tex]
Reduce the fraction:
[tex]\frac{5}{4}=b[/tex]
So the equation if it works out for the other point given is:
[tex]y=8 \cdot (\frac{5}{4})^x[/tex]
Let's try it. So the last point given that we need to satisfy is [tex](2,12.5)[/tex].
This says when [tex]x=2[/tex], [tex]y=12.5[/tex].
Let's replace [tex]x[/tex] with 2 and see what we get for [tex]y[/tex]:
[tex]y=8 \cdot (\frac{5}{4})^2[/tex]
[tex]y=8 \cdot \frac{25}{16}[/tex]
[tex]y=\frac{8}{16} \cdot 25[/tex]
[tex]y=\frac{1}{2} \cdot 25}[/tex]
[tex]y=\frac{25}{2}[/tex]
[tex]y=12.5[/tex]
So we are good. We have found an equation satisfying all 3 points given.
The equation is [tex]y=8 \cdot (\frac{5}{4})^x[/tex].
Answer:
the answr is 8(1.25)^x
Step-by-step explanation:
i took the test and its right
