Question

Approximate the intervals where each function is increasing and decreasing.

Answer 1

Increasing: (-∞, 3)

Decreasing: (3, ∞)

Explanation:

A function is increasing when the slope is positive (y gets bigger as x gets bigger).

A function is decreasing when the slope is negative (y gets smaller as x gets bigger).

From inspection of the given parabola, the turning point is (3, 4).

The function is increasing as the curve approaches the turning point.

The function is decreasing as the curve approaches infinity after the turning point.

Therefore, the intervals where the function is increasing and decreasing are:

• Increasing: (-∞, 3)
• Decreasing: (3, ∞)

Note: x = 3 is not included in the intervals, as the curve is not increasing or decreasing when x = 3. There is a turning point (stationary point) at x = 3 which means the slope is zero as this point. Therefore, the function is not increasing or decreasing when x = 3.

Answer 2

Given is the parabola opening down.

It increases up to vertex and then decreases in value.

The vertex is at (3, 4) so the intervals are:

• Increase: ( - ∞, 3],
• Decrease: [3, ∞).