Positive and Negative Intervals- Multiple Choice Questions
What Positive and Negative Intervals Actually Mean
Before we get into the multiple choice questions, let's be clear about what we're actually dealing with.
A positive interval on a graph means the function's output is above the x-axis — y is greater than zero. A negative interval means the output is below the x-axis — y is less than zero.
That's it. No complicated definitions. No fancy terminology. Just above or below the line.
How to Identify Them From a Graph
When you see a graph in an MCQ, here's what you do:
- Find where the graph crosses the x-axis (the zeros/intercepts)
- Check each section between the intercepts
- If the curve is above the x-axis → positive interval
- If the curve is below the x-axis → negative interval
The sign changes at each x-intercept unless the graph touches and bounces back.
Common MCQ Question Types
Type 1: Identify Positive Intervals
You'll get a graph and be asked "on which interval is f(x) > 0?"
Answer: Where the graph sits above the x-axis. Count the sections carefully — students lose marks here by rushing.
Type 2: Identify Negative Intervals
Same deal, just backwards. "On which interval is f(x) < 0?"
Answer: Where the graph dips below the x-axis.
Type 3: Combined Positive and Negative
These questions ask for both. You need to mark every section correctly or you'll pick a partially wrong answer.
Type 4: Find Intervals From an Equation
No graph provided. You have to factor the polynomial, find the zeros, then test points in each interval. This is where people mess up.
Quick Reference Table
| Question Phrasing | What You're Looking For |
|---|---|
| "f(x) > 0" | Above x-axis sections |
| "f(x) < 0" | Below x-axis sections |
| "f(x) ≥ 0" | Above x-axis + where it touches |
| "f(x) ≤ 0" | Below x-axis + where it touches |
How to Solve: Step-by-Step
From a Graph:
- Locate every x-intercept (where y = 0)
- Draw vertical lines at each intercept to divide the graph into sections
- Pick one test point from each section
- Check if that test point gives a positive or negative y-value
- Match your sections to the answer choices
From an Equation:
- Set the function equal to zero and solve for x
- These solutions are your boundary points
- Plot those points on a number line
- Pick a test point from each created interval
- Plug the test point into the function
- If result > 0, that interval is positive; if < 0, it's negative
Where Students Actually Fail
Confusing the sign of the test point with the sign of the interval. The test point's y-value tells you the interval's sign, not its x-value.
Forgetting about multiplicities. If a factor appears twice (like (x-2)²), the graph touches the x-axis and bounces back — the sign doesn't change.
Not checking the outermost intervals. Students often test the middle sections and miss the far left and far right.
Misreading the inequality direction. "f(x) > 0" and "f(x) ≥ 0" are different. One includes the intercepts, one doesn't.
Practice Tip
For every practice problem, write down the x-intercepts first before looking at the answer choices. This forces you to analyze instead of guess from the options.
When you get a question wrong, figure out if you miscounted the sections or mis-evaluated the sign. Those are the only two ways to get these wrong once you know the method.