The absolute value function can be written as:
f(x) = |x - a| + b
where a is the x-coordinate of the vertex and b is the y-coordinate of the vertex.
To find the equation of the absolute value function that passes through the points (-3, 9), (-1, 1), and (2, 5), we need to find the vertex and the value of b.
The vertex of an absolute value function that opens upwards is the point where the absolute value function changes direction. This occurs at the point where the argument of the absolute value function equals zero. In this case, the argument of the absolute value function is x - a, so we need to find a such that:
-3 - a = 0 or a = -3
-1 - a = 0 or a = -1
2 - a = 0 or a = 2
So the vertex of the absolute value function is at (-1, b), where b is the y-coordinate of the vertex. To find the value of b, we can substitute one of the points into the equation of the absolute value function:
f(-3) = |(-3) - (-1)| + b = 2 + b = 9
f(-1) = |(-1) - (-1)| + b = b = 1
f(2) = |2 - (-1)| + b = 3 + b = 5
Solving these equations, we get:
b = 7/2
So the equation of the absolute value function that passes through the points (-3, 9), (-1, 1), and (2, 5) is:
f(x) = |x + 1| + 7/2
The graph of f is translated 3 units up and 6 units to the left from the graph of g(x) = 4|x + 5| - 3.