Shapes of Graphs
Before you can compute anything about a function, you should be able to describe what its graph is doing: where it rises, where it falls, where it sits above the x-axis, and where it turns around. These are called the key features of a graph, and every one of them is read the same way — left to right, like a sentence.
The vocabulary comes in two pairs plus turning points. Increasing and decreasing describe the direction the graph is heading. Positive and negative describe which side of the x-axis it is on. Relative maximums and minimums are the points where it changes direction.
Increasing and decreasing
A graph is increasing wherever it rises as you move left to right, and decreasing wherever it falls. The answer is always a set of -values — you are reporting which inputs make the graph rise, not how high it gets. The parabola below rises until and falls afterward, so it is increasing for and decreasing for .
Positive and negative
A graph is positive wherever it lies above the x-axis — the outputs are positive numbers there — and negative wherever it lies below. The boundary points are the -intercepts, where the output is exactly zero.
Positive is not the same as increasing. A graph can be above the axis while falling, or below the axis while rising. Increasing describes direction of travel; positive describes location above or below the axis. Mixing these up is the single most common error on this topic.
Relative maximums and minimums
A relative maximum is a point higher than the points immediately around it — the top of a hill where the graph switches from increasing to decreasing. A relative minimum is the bottom of a valley, where it switches from decreasing to increasing.
The word relative matters: a relative maximum is only the highest point in its neighborhood, not necessarily the highest point of the whole graph. A wavy graph can have several hills of different heights, and each hilltop is a relative maximum.
Worked examples
Example 1: increasing and decreasing intervals
A parabola opens downward with vertex . Where is it increasing and where is it decreasing?
Answer: Increasing for , decreasing for
Example 2: where a graph is positive
A downward parabola crosses the x-axis at and . Where is the function positive? Where is it negative?
Answer: Positive for ; negative for or
Example 3: reading a turning point in context
A ball's height over time has a relative maximum at . What does that tell you?
Answer: The ball reaches its peak height of feet at seconds
Try one yourself
Common questions
Why are increasing and decreasing answers written with x, not y?
Because you are reporting where the behavior happens, and location along the graph is tracked by the input . "Increasing for " means: for every input less than , the graph is rising.
What's the difference between positive and increasing?
Positive means the graph is above the x-axis — the outputs are greater than zero. Increasing means the graph is rising left to right. They are independent: a falling graph high above the axis is positive and decreasing at the same time.
How is a relative maximum different from the maximum?
The maximum (sometimes called the absolute maximum) is the single highest point of the entire graph. A relative maximum only has to beat its neighbors — the top of any hill counts, even a short one.
Where does a graph switch between positive and negative?
At its -intercepts. The output is zero exactly there, so crossing the axis is the only way for outputs to change sign. That is why finding the zeros first makes positive/negative questions quick.
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