Write an exponential function whose graph passes through the given points.
1. (0, 3) and (1, 15)
2. (0, -5) and (-3, -135)
3. (0, -0.3) and (5, -9.6)
y = 4^(x+1) - 1 ??
To write an exponential function that passes through two given points (x₁, y₁) and (x₂, y₂), we can use the general form of an exponential function:
y = a * b^x
where 'a' is the initial value or y-intercept, 'b' is the base, and 'x' is the input (independent variable).
Let's find the values of 'a' and 'b' for each set of given points:
1. (0, 3) and (1, 15):
Substituting these values into the general form, we get two equations:
3 = a * b^0 → a * 1 = 3 → a = 3
15 = a * b^1 → 3 * b = 15 → b = 5
Therefore, the exponential function that passes through the given points is:
y = 3 * 5^x
2. (0, -5) and (-3, -135):
Similarly, substituting these values into the general form, we get:
-5 = a * b^0 → a * 1 = -5 → a = -5
-135 = a * b^(-3) → -5 * (1/b^3) = -135
Simplifying the equation, we get:
1/b^3 = 27 → b^3 = 1/27 → b = 1/3
Therefore, the exponential function that passes through the given points is:
y = -5 * (1/3)^x
3. (0, -0.3) and (5, -9.6):
Using the same process, we have:
-0.3 = a * b^0 → a * 1 = -0.3 → a = -0.3
-9.6 = a * b^5 → -0.3 * b^5 = -9.6
To simplify this equation, divide both sides by -0.3:
b^5 = (-9.6) / (-0.3) → b^5 = 32
Taking the fifth root of both sides gives us:
b = ∛(32) → b = 2
Therefore, the exponential function that passes through the given points is:
y = -0.3 * 2^x