Increasing and Decreasing Intervals

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A function is increasing on an interval when larger inputs always give larger outputs. This is about change as you move right, not about whether the output itself is positive.

Method

Strict increase means $x_1<x_2\Rightarrow f(x_1)<f(x_2)$ for every pair in the interval. Strict decrease reverses the output inequality. Later, derivative signs provide an efficient test.

Worked example

For $f(x)=(x-1)^2$ , the graph decreases toward its vertex on $(-\infty,1]$ and increases away from it on $[1,\infty)$ . The values on both intervals are nonnegative, illustrating why positive output does not mean increasing.

Common mistake

A flat tangent at one point does not automatically mean a change in monotonicity. For example, $x^3$ increases through zero.

Check your understanding

Give the increasing and decreasing intervals of $f(x)=-x^2$ .

Show answer

Increasing on $(-\infty,0]$ , decreasing on $[0,\infty)$ .

Explore

Move and scale a graph

Try this. Change one slider at a time. Positive h moves the vertex right; k moves it up; negative a reflects the curve and a = 0 makes it constant.

Vertical multiplier a1Horizontal shift h0Vertical shift k0

y = 1(x − (0))² + (0). Vertex (0, 0); opens up. Range y ≥ 0.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

On which interval is f(x)=x² decreasing?

All of ℝ

(−∞,0]

[0,∞)

Hint 1 · Find a starting point

Compare outputs while moving from left to right.

Hint 2 · Take the next step

From x=−3 to −2 to −1, the outputs go 9, 4, 1.

Show the reasoning

Answer: (−∞,0]

The square decreases as a negative input approaches zero, then increases to the right.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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