Question 424108
The general exponential function has the form:
{{{y = a*b^x}}} (with b < 0)
For the specific exponential function whose graph passes through the two given points we have to find the specific values of "a" and "b".<br>
So we have 2 unknowns to find. For this we need two equations. Since the given points are supposed to be on the graph, they must fit the equation. So
{{{-5 = a*b^(0)]]]
and
{{{-4 = a*b^(-3)}}}
With these two equations we should be able to find a and b.<br>
In the first equation, since b is not zero, {{{b^0 = 1}}}. So it simplifies to:
-5 = a
Already we have found one of the two values we need. To find b we just use the newly found value for a and the second equation:
{{{-4 = (-5)*b^(-3)}}}
Since it is easier to work with positive exponents we will replace the right side with its positive exponent equivalent:
{{{-4 = (-5)*(1/b^3)}}}
which simplifies to:
{{{-4 = (-5)/b^3}}}
Next we will eliminate the fraction by multiplying both sides by {{{b^3}}}:
{{{-4b^3 = -5}}}
Next we isolate the base and its exponent. Dividing by -4 we get:
{{{b^3 = 5/4}}}
Now we just find the cube root of each side:
{{{b = root(3, 4/5)}}}
Last of all we rationalize the denominator:
{{{b = root(3, (4/5)(5^2/5^2))}}}
{{{b = root(3, (4*5^2)/5^3)}}}
{{{b = root(3, 4*5^2)/root(3, 5^3)}}}
{{{b = root(3, 100)/5}}}<br>
With the values of a and b that we have found we can finally write the specific exponential equation that passes through the two given points:
{{{y = (-5)(root(3, 100)/5)^x}}}