A decreasing function is one where the output goes down as the input goes up. In Honors Pre-Calculus, you spot it on graphs that fall from left to right and use it to describe negative rates of change.
A decreasing function in Honors Pre-Calculus is a function where larger input values produce smaller output values. If x goes up and f(x) goes down, the function is decreasing on that interval. On a graph, you usually see the curve or line fall from left to right.
That idea sounds simple, but in this course you use it in a more precise way. A function does not have to be decreasing everywhere to count as decreasing. Many functions only decrease on part of their domain, then increase later. So when you describe a function, you often name the interval where it decreases, not just the whole graph.
For linear functions, decreasing means the slope is negative. That gives you a constant downward trend, which is why lines with negative slope look like they tilt down as you move right. Example: f(x) = -2x + 5 is decreasing everywhere, because every time x increases by 1, the output drops by 2.
The same idea shows up in non-linear functions too. A quadratic might decrease until it reaches its vertex, then increase. An exponential decay function decreases as x increases because the values keep shrinking by a constant factor. In Honors Pre-Calculus, this matters because you are not just naming a shape, you are reading how the function behaves over time or across inputs.
A quick way to check is to compare two x-values. If x1 < x2 and f(x1) > f(x2), the function is decreasing on that interval. On a graph, that means the y-values are moving downward as you move to the right. In function notation, the input is increasing, but the output is falling.
Decreasing functions show up anytime Honors Pre-Calculus asks you to describe change instead of just compute it. That includes reading graphs, finding intervals of increase or decrease, interpreting slope, and deciding whether a model fits real data.
This term connects directly to rates of change. A decreasing linear function has negative slope, so the sign of the slope tells you the direction of change right away. In a table or graph, you can also tell when values are dropping from one x-value to the next, which is a faster way to describe behavior than rewriting the whole function.
It also matters for modeling. If a quantity goes down as another quantity goes up, a decreasing function is usually the right shape. Think about cost dropping as discount quantity increases, or temperature falling over time in a cooling process. In those situations, the graph is not just a picture, it is the story of the relationship.
You also need this term when comparing functions. A graph might be decreasing on one interval and increasing on another, which helps you find turning points, local behavior, and key features like maxima or minima. That makes decreasing functions a building block for later topics in the course, especially when you start analyzing more complicated graphs.
Keep studying Honors Pre-Calculus Unit 1
Visual cheatsheet
view galleryIncreasing Function
This is the opposite behavior. An increasing function goes up as x goes up, while a decreasing function goes down as x goes up. Comparing the two helps you describe intervals of change on graphs, especially when a function switches direction at a turning point. You will often name where a graph is increasing and where it is decreasing in the same problem.
Constant Function
A constant function is neither increasing nor decreasing because the output stays the same as the input changes. That makes it a useful contrast when you are looking at slope and rate of change. If the graph is flat, there is no downward trend, so it cannot be called decreasing.
Linear Function
Linear functions make decreasing behavior easy to spot because their graphs are straight lines. If the slope is negative, the function decreases everywhere. This is one of the cleanest places to connect the algebraic form y = mx + b to graph behavior, since the sign of m tells you the direction of the line.
Global Minimum
A function that decreases and then turns around may reach a low point called a global minimum. That point is the smallest output on the entire domain. When you study decreasing intervals, you are often getting closer to the minimum or trying to explain where the function stops falling.
A quiz or problem-set question will usually ask you to identify where a graph is decreasing, tell whether a function is decreasing from its equation, or match a table to a downward trend. You might also need to explain the meaning of a negative slope in words.
For a linear function, you look at the sign of the slope. For a graph, you scan from left to right and see whether the y-values are going down. For a table, you compare outputs as inputs increase. If the outputs get smaller, you have a decreasing interval.
A common mistake is saying a function is decreasing just because it has negative y-values. Negative outputs and decreasing behavior are not the same thing. Another common mistake is forgetting that a function can increase on one interval and decrease on another, so always check the interval named in the question.
These two are easy to mix up because both describe how outputs change as inputs change. Increasing means the function rises as x rises, while decreasing means it falls as x rises. On a graph, increasing slopes upward from left to right, and decreasing slopes downward from left to right. The sign of the slope or the direction of the graph tells you which one you have.
A decreasing function has outputs that get smaller as the inputs get larger.
On a graph, a decreasing function moves downward from left to right.
A linear decreasing function has a negative slope.
A function can decrease on one interval and increase on another, so always check the interval named in the problem.
Negative outputs do not automatically mean the function is decreasing.
A decreasing function is a function where bigger x-values give smaller y-values. In Honors Pre-Calculus, you use that idea to describe graph behavior and rate of change. If the graph falls from left to right, the function is decreasing on that interval.
Read the graph from left to right. If the y-values go down as x-values go up, the function is decreasing. For a line, that usually means the slope is negative. For a curve, the function may only be decreasing over part of the graph.
No. A function can have negative outputs and still be increasing, or have positive outputs and still be decreasing. Decreasing describes how the outputs change as x increases, not whether the outputs are above or below zero.
For linear functions, decreasing behavior means the slope is negative. That tells you the line drops as you move to the right. In more advanced graphs, the same idea shows up as negative average rate of change over an interval.