Use the equation of the exponential function whose graph passes through the points (0,-2) and (2,-50) to find the value of y when x = -2.
I know the formula is y=ab^x but I don't know what numbers to plug in.
Any help would be great.
Thanks.
first sub in (0,-2) to get
-2 = ab^0, but b^0 = 1
so a=-2
so far we have
y = -2b^x
now use the other point (2,-50)
-50 = -2(b^2)
b^2 = 25
b= ±5
however in exponentional equations the base is usually positve, so now you have
y = -2(5^x)
sub in x=-2 and you got it.
To find the equation of the exponential function that passes through the points (0, -2) and (2, -50), we can use the formula:
y = ab^x
Let's plug in the values of the points into this equation and solve for a and b.
Using the point (0, -2):
-2 = ab^0
-2 = a
Now, using the other point (2, -50):
-50 = (-2)b^2
-50 = -2b^2
Divide both sides of the equation by -2 to isolate b^2:
25 = b^2
Take the square root of both sides:
b = ±5
Since b represents the base of the exponential function, we will choose the positive value, b = 5, as exponential functions have a positive base.
Now that we have the value of b, we can substitute it back into the equation to find the value of a.
Using the point (0, -2):
-2 = a(5)^0
-2 = a(1)
-2 = a
So, the equation of the exponential function is:
y = -2(5)^x
To find the value of y when x = -2, we can substitute x = -2 into the equation:
y = -2(5)^(-2)
Simplify the expression:
y = -2(1/25)
y = -2/25
Therefore, when x = -2, the value of y is -2/25.
To find the equation of an exponential function in the form of y = ab^x, we need to determine the values of a and b.
Given the points (0, -2) and (2, -50), we can substitute these x and y values into the equation to create a system of equations. Let's start by substituting the coordinates of the first point (0, -2):
-2 = ab^0
Since any number raised to the power of 0 is equal to 1, we can simplify the equation to:
-2 = a(1)
-2 = a
Now, we substitute the coordinates of the second point (2, -50):
-50 = ab^2
We already found the value of a, which is -2, so we can substitute it into the equation:
-50 = -2b^2
Divide both sides of the equation by -2:
25 = b^2
To find the value of b, we take the square root of both sides:
b = ±√25
This gives us two possible values for b: b = 5 or b = -5.
Now, we can substitute the values of a and b into the equation y = ab^x:
Using b = 5:
y = (-2)(5^x)
Using b = -5:
y = (-2)(-5^x)
To find the value of y when x = -2, we substitute x = -2 into the equation:
Using b = 5:
y = (-2)(5^-2)
Simplifying this equation:
y = (-2)(1/25)
y = -2/25
Using b = -5:
y = (-2)(-5^-2)
Simplifying this equation:
y = (-2)(1/25)
y = -2/25
So, regardless of the value of b, when x = -2, the value of y is -2/25.