Question

Draw the graph of an absolute value function that has these key features:

vertex: (2,-3)
increasing: (2, [infinity])
decreasing: (-[infinity], 2)
domain: all real numbers
range: [-3,[infinity])
end behavior: As x approaches negative infinity, f(x) approaches positive infinity. As x approaches positive infinity, f(x) approaches positive infinity.

Answer 1

The absolute value function has a vertex at (2,-3), is increasing for x > 2, decreasing for x < 2, and approaches positive infinity at both ends.

For an absolute value function with a vertex at (2, -3), the general form is f(x) = a|x - h| + k, where (h, k) is the vertex.

In this case:

- h = 2

- k = -3

Since the function is increasing to the right of the vertex and decreasing to the left, the coefficient "a" is positive. The end behavior indicates that the graph approaches positive infinity as x goes to both negative and positive infinity.

Putting it all together, the function can be represented as
f(x) = a|x - 2| - 3, where a is a positive constant. You can choose a specific value for a based on the steepness of the graph you desire.