The k = 0 or k = 14. Since the graph passes through the point (-1, k), and k cannot be negative, we can conclude that k = 0 .
Since the graph of y = ab^x passes through the point (6, 448), we can write the equation:
448 = ab^6
Similarly, since the graph also passes through the point (-1, k), we can write the equation:
k = ab^-1
Dividing the first equation by the second equation, we get:
448 / k = ab^6 / ab^-1
Simplifying, we get:
448 / k = b^7
Since b > 0, we can take the square root of both sides to get:
b = sqrt(448 / k)
Substituting this value of b into the equation y = ab^x, we get:
y = ab^x = sqrt(448 / k) * ab^x
Since the graph passes through the point (2, 28), we can substitute x = 2 and y = 28 into this equation to get:
28 = sqrt(448 / k) * ab^2
Dividing both sides by sqrt(448 / k), we get:
28 / sqrt(448 / k) = ab^2
Simplifying, we get:
28 sqrt(k / 448) = ab^2
Since ab^2 is a positive number, we can take the square root of both sides to get:
sqrt(28 sqrt(k / 448)) = ab
Simplifying, we get:
sqrt(14k) = ab
Substituting this value of ab into the equation k = ab^-1, we get:
k = sqrt(14k) * -1
Squaring both sides, we get:
k^2 = 14k
Subtracting 14k from both sides, we get:
k^2 - 14k = 0
Factoring, we get:
k(k - 14) = 0
Therefore, k = 0 or k = 14. Since the graph passes through the point (-1, k), and k cannot be negative, we can conclude that k = 0.