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'm lost!!
Thanks.
To find the equation of the exponential function, we can use the general form:
y = ab^x, where "a" is the initial value and "b" is the base.
Given the points (0, -2) and (2, -50), we can substitute these values into the equation to form a system of equations:
For point (0, -2):
-2 = ab^0
-2 = a
For point (2, -50):
-50 = ab^2
Now that we have the value of "a", we can substitute it into the second equation:
-50 = (-2)b^2
Divide both sides of the equation by -2:
25 = b^2
Taking the square root of both sides, we have two possible values for "b":
b = ±5
We can now find the value of "y" when x = -2.
Using the first possible value of "b", b = 5:
y = ab^x
y = (-2)(5)^-2
y = (-2)(1/25)
y = -2/25
Using the second possible value of "b", b = -5:
y = ab^x
y = (-2)(-5)^-2
y = (-2)(1/25)
y = -2/25
Therefore, when x = -2, y can be either -2/25 or -2/25.
No problem! I'll walk you through the process step by step.
To find the equation of an exponential function, we need to find the values of the base and the exponent. Once we have those values, we can use the equation to find the value of y when x is a given number.
Step 1: Let's start by finding the base of the exponential function.
Since the graph passes through the point (0,-2), we can substitute x = 0 and y = -2 into the equation to get:
-2 = a(b^0)
Any number raised to the power of 0 is always 1, so we can simplify the equation:
-2 = a(1)
-2 = a
Therefore, the value of a is -2.
Step 2: Now let's find the exponent, b.
Using the point (2,-50), we substitute x = 2 and y = -50 into the equation:
-50 = -2(b^2)
Divide both sides of the equation by -2:
-50 / -2 = b^2
25 = b^2
Take the square root of both sides of the equation:
sqrt(25) = sqrt(b^2)
5 = b
So the base, b, is 5.
Step 3: Now that we have the values of a and b, we can write the equation of the exponential function:
y = ab^x
Substituting the values of a = -2 and b = 5 into the equation, we get:
y = -2(5^x)
Now we can substitute x = -2 into the equation to find the value of y:
y = -2(5^-2)
To simplify this expression, we need to recall the rule for negative exponents.
A negative exponent in the numerator can be rewritten as a positive exponent in the denominator. So we can rewrite 5^-2 as 1/5^2.
Therefore, the equation becomes:
y = -2(1/5^2)
y = -2(1/25)
y = -2/25
y = -0.08
So when x = -2, the value of y is -0.08.
I hope this helps! Let me know if you have any further questions.