The sketch shows a curve with equation y=ab^x where a and b are constants and b > 0
The curve passes through the points (2,50), (5,6250) and (-1,p)
Calculate the value of p.
6250 = a b^5
50 = a b^2
dividing equations (to eliminate a) ... 125 = b^3 ... b = 5
substituting ... 50 = a * 25 ... a = 2
p = 2 / 5
To calculate the value of p, we need to substitute the given points into the equation of the curve and solve for the corresponding values.
Let's start with the first point (2, 50). We can substitute the x-coordinate, 2, and the y-coordinate, 50, into the equation y = ab^x:
50 = ab^2 -------(1)
Next, we'll use the second point (5, 6250) in the equation:
6250 = ab^5 -------(2)
Lastly, we'll use the third point (-1, p) in the equation:
p = ab^(-1) -------(3)
Now, we have three equations (1), (2), and (3) with three variables: a, b, and p. To solve for these variables, we can use a system of equations:
From equation (1), we can express a in terms of b: a = 50 / b^2
From equation (3), we can express a in terms of p and b: a = p * b^(-1)
Setting these two expressions of a equal to each other, we get:
50 / b^2 = p * b^(-1)
Rearranging the equation, we can multiply both sides by b^3 to get:
50 * b = p
Now, we can substitute this value of p back into equation (3):
p = ab^(-1) becomes 50 * b = ab^(-1)
Substituting a = 50 / b^2, we have:
50 * b = (50 / b^2) * b^(-1)
Simplifying, we get:
50 * b = 50 / b
Now, multiply both sides by b to eliminate the denominator:
50 * b^2 = 50
Divide both sides by 50:
b^2 = 1
Taking the square root of both sides, we find two possible values for b: b = 1 and b = -1.
Let's examine these two options:
For b = 1, we substitute it into equation (1):
50 = a * 1^2
50 = a
Thus, if b = 1, then a = 50, and our equation y = ab^x becomes:
y = 50 * 1^x => y = 50
For b = -1, we substitute it into equation (1):
50 = a * (-1)^2
50 = a * 1
50 = a
Similarly, if b = -1, then a = 50, and the equation becomes:
y = 50 * (-1)^x => y = -50
Since the given point (-1, p) lies on the curve, we can rule out the y-value being -50, as it is not possible for p to be negative.
Therefore, the value of p must be 50.