Parabola Property: Analyzing Negative Values for X-Axis Vertex Functions

Q: What does it mean for a vertex to be 'on the x-axis'?

When the vertex is on the x-axis, it means the y-coordinate of the vertex equals zero. So the vertex is at point (h, 0), making the parabola touch the x-axis at exactly one point.

Q: Why is the function negative everywhere except at the vertex?

Since $ a (downward opening) and the vertex form is \( y = a(x - h)^2 $, we have negative times positive equals negative everywhere except when $ (x - h)^2 = 0 $.

Q: Can a downward parabola ever have positive y-values?

Yes, normally! Most downward parabolas have positive values near the vertex. But when the vertex is specifically on the x-axis, the highest point the parabola reaches is y = 0.

Q: How do I identify if a parabola bends downward?

Look at the coefficient of the $ x^2 $ term (called 'a'). If a , the parabola opens downward like an upside-down U shape.

Q: What happens if I plug in x = h into the equation?

When you substitute $ x = h $, you get $ y = a(h - h)^2 = a(0) = 0 $. This confirms the vertex is at (h, 0) on the x-axis!

Parabola Functions with X-Axis Vertex Analysis

If the vertex of a parabola is on the x-axis and the parabola is bending downwards, then the function is always negative except at the vertex point.

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If the vertex of a parabola is on the x-axis and the parabola is bending downwards, then the function is always negative except at the vertex point.

Step-by-step solution

To solve this problem, let's analyze the given conditions:

Step 1: We examine the standard form of a downward-opening parabola. The standard vertex form is $y = a(x - h)^2 + k$ . If the vertex lies on the x-axis, $k = 0$ , so the equation simplifies to $y = a(x - h)^2$ .
Step 2: Since the parabola is bending downwards, it implies that $a < 0$ . Therefore, the quadratic term $a(x - h)^2$ is non-positive for all values of $x$ , becoming 0 only when $x = h$ .
Step 3: Evaluate the function for $x \neq h$ . For any $x \neq h$ , $(x - h)^2 > 0$ , so $y = a(x - h)^2 < 0$ because $a < 0$ .

Conclusively, the function value is always negative for all $x \neq h$ , and it is exactly zero at $x = h$ (the vertex). The statement provided corresponds precisely with this behavior.

Therefore, the statement that the function is always negative except at the vertex point is indeed Correct.

Final Answer

Correct

Key Points to Remember

Essential concepts to master this topic

Rule: Downward parabola with vertex on x-axis: y = a(x - h)²
Technique: When a < 0 and vertex at (h, 0), function equals zero only at x = h
Check: Test points left and right of vertex: both give negative y-values ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex position with function sign
Don't assume vertex on x-axis means function is sometimes positive = wrong interpretation! The vertex being on the x-axis just means k = 0, not that the parabola crosses above the axis. Always remember: downward parabola with vertex on x-axis stays at or below zero.

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

What does it mean for a vertex to be 'on the x-axis'?

When the vertex is on the x-axis, it means the y-coordinate of the vertex equals zero. So the vertex is at point (h, 0), making the parabola touch the x-axis at exactly one point.

Why is the function negative everywhere except at the vertex?

Since $a < 0$ (downward opening) and the vertex form is $y = a(x - h)^2$ , we have negative times positive equals negative everywhere except when $(x - h)^2 = 0$ .

Can a downward parabola ever have positive y-values?

Yes, normally! Most downward parabolas have positive values near the vertex. But when the vertex is specifically on the x-axis, the highest point the parabola reaches is y = 0.

How do I identify if a parabola bends downward?

Look at the coefficient of the $x^2$ term (called 'a'). If a < 0, the parabola opens downward like an upside-down U shape.

What happens if I plug in x = h into the equation?

When you substitute $x = h$ , you get $y = a(h - h)^2 = a(0) = 0$ . This confirms the vertex is at (h, 0) on the x-axis!

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