Quadratic Function Analysis: Finding the Increasing Interval Between Points A and B

Q: Why doesn't the function increase between A and B?

Between A and B, the parabola goes down first, then up. The function decreases from A to the vertex C, then increases from C to B. So the entire interval A to B includes both decreasing and increasing parts!

Q: How can I tell if a parabola opens upward or downward?

Look at the vertex! If the vertex is the lowest point (like a smile), the parabola opens upward. If it's the highest point (like a frown), it opens downward.

Q: What's the difference between the vertex and the x-intercepts?

The x-intercepts (points A and B) are where the parabola crosses the x-axis. The vertex (point C) is the turning point - either the highest or lowest point of the parabola.

Quadratic Functions with Vertex Analysis

Note that the graph of the function intersects the x-axis at points A and B

Moreover the vertex of the parabola is marked at point C

Identify the segment below where the function increases:

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Understand the problem

Note that the graph of the function intersects the x-axis at points A and B

Moreover the vertex of the parabola is marked at point C

Identify the segment below where the function increases:

Step-by-step solution

From the graph we can see that the parabola is a smiling parabola,

meaning that its extreme point is a minimum point.

If we describe it in words, until the extreme point the function decreases,

after the extreme point it increases.

Since we measure the progress using X,

we can say that the function increases whenever X is greater than point C, the extreme point.

Mathematically we can write:

X>C

As we already said, as long as X is greater than C, the function increases.

Final Answer

$x > C$

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: For upward parabolas, function increases after the vertex
Technique: Identify vertex C as minimum, function increases when $x > C$
Check: Trace the curve from left to right after vertex point ✓

Common Mistakes

Avoid these frequent errors

Confusing the interval between x-intercepts with increasing intervals
Don't assume the function increases between points A and B = wrong answer! Between A and B, the parabola actually decreases then increases. Always identify the vertex first and remember that upward parabolas increase only after the vertex.

Practice Quiz

Test your knowledge with interactive questions

Note that the graph of the function shown below does not intersect the x-axis

The parabola's vertex is A

Identify the interval where the function is decreasing:

FAQ

Everything you need to know about this question

Why doesn't the function increase between A and B?

Between A and B, the parabola goes down first, then up. The function decreases from A to the vertex C, then increases from C to B. So the entire interval A to B includes both decreasing and increasing parts!

How can I tell if a parabola opens upward or downward?

Look at the vertex! If the vertex is the lowest point (like a smile), the parabola opens upward. If it's the highest point (like a frown), it opens downward.

What's the difference between the vertex and the x-intercepts?

The x-intercepts (points A and B) are where the parabola crosses the x-axis. The vertex (point C) is the turning point - either the highest or lowest point of the parabola.

Does the function ever increase before the vertex?

For an upward-opening parabola, no! The function decreases as you approach the vertex from the left, then increases as you move away from the vertex to the right.

How do I write the answer mathematically?

Since the function increases when x is greater than the vertex point C, write it as $x > C$ . This means all x-values to the right of point C.

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