Parabola Properties: Analyzing Negative Values with Downward-Facing Curves and Vertex Position

Q: If a parabola opens downward, isn't it always negative?

No! The direction tells us the parabola eventually goes down, but if the vertex is above the x-axis, the parabola starts positive near its highest point.

Q: How can I tell if a downward parabola has positive values?

Look at the vertex position! If the vertex $ (h,k) $ has $ k > 0 $, then the parabola reaches positive y-values at and near the vertex.

Q: When would a downward parabola be always negative?

Only when its vertex is on or below the x-axis! If $ k ≤ 0 $, then the maximum point is not positive, so the entire parabola stays at or below zero.

Q: Does the parabola formula help me see this?

Yes! In $ y = a(x-h)^2 + k $ with $ a , the k value is your vertex height. If \( k > 0 $, you have positive y-values!

Q: What about the x-intercepts?

If the vertex is above the x-axis, the parabola will cross the x-axis at two points, creating both positive and negative regions on the graph.

Parabola Sign Analysis with Vertex Position

If a parabola is bending downwards and its vertex is above the x-axis, then it is always negative.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If a parabola is bending downwards and its vertex is above the x-axis, then it is always negative.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Understand the form of the parabola.
Step 2: Analyze the nature of values above and below the x-axis.
Step 3: Determine if the parabola can have positive values.

Now, let's work through each step:

Step 1: Given the parabola is opening downwards, the quadratic formula can be defined as:
$y = a(x-h)^2 + k$ , with $a < 0$ and $k > 0$ .

Step 2: Because the vertex is above the x-axis ( $k > 0$ ), the vertex itself is positive when considered as a point ( $h, k$ ).

Step 3: A parabola that opens downward will eventually intersect the x-axis, creating two roots unless it remains above the x-axis—which is not generally the case when $k$ is small enough. Therefore, for values of $x$ surrounding the vertex and large enough in magnitude, $y$ can be negative.

Conclusion: The parabola is not always negative as it can be positive near its vertex.

The statement in the problem is thus Incorrect.

Final Answer

Incorrect

Key Points to Remember

Essential concepts to master this topic

Rule: Downward parabolas with vertex above x-axis can be positive
Technique: Check vertex value: if $k > 0$ , parabola starts positive
Check: Test points near vertex: if $y(h) = k > 0$ , then positive values exist ✓

Common Mistakes

Avoid these frequent errors

Assuming downward parabolas are always negative
Don't think that downward-opening means always negative = missing positive regions! This ignores the vertex position and nearby points. Always check if the vertex is above the x-axis, which creates positive y-values near the maximum point.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

Find all values of $$ x $$ where
$f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

If a parabola opens downward, isn't it always negative?

No! The direction tells us the parabola eventually goes down, but if the vertex is above the x-axis, the parabola starts positive near its highest point.

How can I tell if a downward parabola has positive values?

Look at the vertex position! If the vertex $(h,k)$ has $k > 0$ , then the parabola reaches positive y-values at and near the vertex.

When would a downward parabola be always negative?

Only when its vertex is on or below the x-axis! If $k ≤ 0$ , then the maximum point is not positive, so the entire parabola stays at or below zero.

Does the parabola formula help me see this?

Yes! In $y = a(x-h)^2 + k$ with $a < 0$ , the k value is your vertex height. If $k > 0$ , you have positive y-values!

What about the x-intercepts?

If the vertex is above the x-axis, the parabola will cross the x-axis at two points, creating both positive and negative regions on the graph.

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