Find a power function of the form y=ax^b whose graph passes throught the points (2, 5) and (8, 12).
How do i solve this??
sub in the two points to get two equations
5 = a(2)^b
12=a(8)^b
divide one by the other
12/5 = 8^b/2^b
2^3b / 2^b = 2.4
2^2b = 2.4
log 2^2b = log 2.4
2b = log2.4/log2
b = .631517..
sub back into
5 = a(2^.631517)
a = 3.2275
so y = 3.2275(x^.631517)
check:
if x = 2 , y = 3.2275(2^.631517) = 5
if x=8 , y = 3.2275(8^.631517) = 1.9999998
not bad!
To find a power function of the form y = ax^b that passes through the points (2, 5) and (8, 12), you can follow these steps:
Step 1: Substitute the x and y values from the first point (2, 5) into the equation to get an equation in terms of the unknowns a and b. This gives you the equation 5 = a(2^b).
Step 2: Substitute the x and y values from the second point (8, 12) into the equation to get another equation in terms of a and b. This gives you the equation 12 = a(8^b).
Step 3: You now have a system of two equations:
Equation 1: 5 = a(2^b)
Equation 2: 12 = a(8^b)
Step 4: Simplify both equations as much as possible. In this case, you can't simplify them further.
Step 5: Solve the system of equations to find the values of a and b. There are different methods to solve systems of equations, such as substitution or elimination. In this case, substitution is a convenient method.
From Equation 1, solve for a:
a = 5 / (2^b)
Substitute this expression for a in Equation 2:
12 = (5 / (2^b)) * (8^b)
Simplify the expression and solve for b:
12 = (5 * 2^(3b)) / 2^b
12 = 5 * 2^(2b)
2^(2b) = 12 / 5
2^(2b) = 2.4
Take the logarithm of both sides:
log(2^(2b)) = log(2.4)
2b * log(2) = log(2.4)
2b = log(2.4) / log(2)
b = (log(2.4) / log(2)) / 2
Step 6: Substitute the value of b back into either Equation 1 or Equation 2 to solve for a. Let's substitute it into Equation 1:
5 = a * (2^b)
5 = a * (2^((log(2.4) / log(2)) / 2))
Simplify the expression and solve for a:
a = 5 / (2^((log(2.4) / log(2)) / 2))
Step 7: Finally, substitute the values of a and b back into the original power function equation: y = ax^b
Therefore, the power function of the form y = ax^b that passes through the points (2, 5) and (8, 12) is y = (5 / (2^((log(2.4) / log(2)) / 2))) * x^((log(2.4) / log(2)) / 2).
To find a power function of the form y = ax^b that passes through the points (2, 5) and (8, 12), you will need to use the process of solving a system of equations.
Let's start by substituting the x and y values of the first point, (2, 5), into the equation y = ax^b:
5 = a(2^b) (Equation 1)
Next, substitute the x and y values of the second point, (8, 12), into the equation:
12 = a(8^b) (Equation 2)
Now, we have a system of equations with two variables, a and b. To solve for these variables, we can use the method of substitution or elimination.
Let's use the method of substitution. Solve Equation 1 for a:
a = 5 / (2^b) (Equation 3)
Now, substitute Equation 3 into Equation 2:
12 = (5 / (2^b)) * (8^b)
To simplify this equation, multiply across by (2^b):
12 * (2^b) = 5 * (8^b) (Note: The multiplication is associative, so you can multiply 5 and 8 before raising it to the power of b)
Simplify further:
12 * (2^b) = 5 * (2^3)^b (Since 8 = 2^3)
12 * (2^b) = 5 * (2^(3b))
Apply the property of exponents. When you raise a power to another power, multiply the exponents:
12 * (2^b) = 5 * (2^3b)
Now, the bases are the same (2), so we can equate the exponents:
b = 3b
Simplify:
2b = 0
Divide both sides by 2 to solve for b:
b = 0
Now that we have found the value of b, we can substitute it back into Equation 3 to solve for a:
a = 5 / (2^0)
Since any number raised to the power of 0 is equal to 1, we have:
a = 5
Therefore, the power function of the form y = ax^b that passes through the points (2, 5) and (8, 12) is:
y = 5x^0
Simplifying further:
y = 5
So, the answer is y = 5.