Given an exponential graph that are concave up and the points (1,4) and (-1, 16), give a possible formula of the function.
I plugged the variables into the f(x)=b times a^x and got:
4= b times a^1 and
16=b times a^-1
I am not sure where to go from here.
huh? a^1 = 1 and a^-1 = 1/a, so
4 = ba
16 = b/a
now divide:
4/16 = ba/(b/a)
1/4 = a^2
a = 1/2
so, b=8
f(x) = 8*2^-x
Thank you! I forgot that you could divide the two!
To find the formula of an exponential function given two points on the graph, you need to solve the system of equations formed by plugging in the given points into the general equation of the form
f(x) = b * a^x.
Let's start by substituting the coordinates (1,4) into the equation:
4 = b * a^1
And substituting the coordinates (-1,16):
16 = b * a^(-1)
Now, we can solve this system of equations to find the values of b and a.
To eliminate b, let's divide the second equation by the first equation:
16/4 = (b * a^(-1)) / (b * a^1)
Simplifying:
4 = (a^(-1))/a
Using the rule of exponents that states a^(-1) = 1/a, we get:
4 = (1/a) / a
Next, we can cross multiply:
4 * a = 1/a
Multiply both sides by a:
4 * a^2 = 1
Simplify:
a^2 = 1/4
Taking the square root of both sides:
a = ±√(1/4)
a = ±1/2
Now, let's substitute the value of a back into one of the original equations to find b. Let's use the equation:
4 = b * a^1
Substituting a = 1/2:
4 = b * (1/2)^1
4 = b * (1/2)
Simplifying:
4 = b/2
Multiply both sides by 2:
8 = b
Now we have found the values of both a and b. Therefore, the possible formula for the function is:
f(x) = 8 * (1/2)^x
Alternatively, you could also use the value a = -1/2 (negative because the graph is concave up) and solve for b similarly to obtain:
f(x) = -8 * (1/2)^x