Parabola Function Analysis: Finding Points Where f(x) < 0

Q: Why is the answer both x A?

Since the parabola opens downward and only touches the x-axis at point A, the function is negative everywhere except at A itself. This means both sides of A have negative values.

Q: What does it mean that A is the vertex?

The vertex is the highest or lowest point of a parabola. Here, A is the highest point (maximum) because the parabola opens downward, and it's exactly on the x-axis where $ f(x) = 0 $.

Q: Could f(x) = 0 be part of the solution?

No! The question asks for where $ f(x) (strictly less than zero). At point A, \( f(x) = 0 $, so A itself is not included in the solution.

Q: How can I visualize this on the graph?

Imagine the parabola as an upside-down U with its peak touching the x-axis at A. Everything to the left and right of A dips below the x-axis, making f(x) negative in those regions.

Parabola Inequalities with Vertex Intersection

The graph of the function below intersects the X-axis at point A (the vertex of the parabola).

Find all values of $x$ where

$f\left(x\right) < 0$ .

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The graph of the function below intersects the X-axis at point A (the vertex of the parabola).

Find all values of $x$ where

$f\left(x\right) < 0$ .

Step-by-step solution

To determine where the function $f(x) < 0$ , it's given that the parabola intersects the X-axis exactly at point A, the vertex, indicating the function has its maximum (if it opens downwards) or minimum (if it opens upwards) at this point.

Since it intersects (not crosses) the X-axis at one point, this must mean the parabola opens downwards, having its vertex at the X-axis. Thus, it tests negative to the left and right of point A, except for the vertex A itself, where $f(x) \geq 0$ .

Here's the solution approach:

Step 1: Determine parabolic orientation (vertex as max or min point).
Step 2: Verify negative function values f(x) outside vertex.
Step 3: Solution: If the parabola opens downwards and A is the only intersection, intervals are $x < A$ and $x > A$ .

The analysis shows negative regions surrounding the vertex for downwards opening, consistent with options (b) and (c).

Therefore, the solutions are Answers (b) + (c) are correct.

Final Answer

Answers (b) + (c) are correct.

Key Points to Remember

Essential concepts to master this topic

Vertex Analysis: Parabola touches x-axis at one point means downward opening
Sign Analysis: For downward parabola, f(x) < 0 everywhere except at vertex
Verification: Check regions: left of A is negative, right of A is negative ✓

Common Mistakes

Avoid these frequent errors

Assuming parabola opens upward when vertex touches x-axis
Don't think touching the x-axis means the parabola opens upward = completely wrong regions! When a parabola intersects the x-axis at exactly one point (the vertex), it must open downward. Always remember: one intersection point means vertex is the maximum, so the parabola opens downward.

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

How do I know if the parabola opens upward or downward?

Look at the number of x-axis intersections! If it touches at exactly one point (the vertex), it opens downward. If it crosses at two points, it opens upward with vertex below the x-axis.

Why is the answer both x < A and x > A?

Since the parabola opens downward and only touches the x-axis at point A, the function is negative everywhere except at A itself. This means both sides of A have negative values.

What does it mean that A is the vertex?

The vertex is the highest or lowest point of a parabola. Here, A is the highest point (maximum) because the parabola opens downward, and it's exactly on the x-axis where $f(x) = 0$ .

Could f(x) = 0 be part of the solution?

No! The question asks for where $f(x) < 0$ (strictly less than zero). At point A, $f(x) = 0$ , so A itself is not included in the solution.

How can I visualize this on the graph?

Imagine the parabola as an upside-down U with its peak touching the x-axis at A. Everything to the left and right of A dips below the x-axis, making f(x) negative in those regions.

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