Finding Negative Values of f(x) in a Quadratic Function with Given X-Axis Intersections

Q: How do I know where the function is negative just by looking at the graph?

Look for where the parabola is below the x-axis! Since the parabola opens upward and crosses at points A and B, it's negative (below the axis) when $ x or \( x > B $.

Q: Why isn't the answer between A and B?

Between A and B, the parabola is above the x-axis, so $ f(x) > 0 $ there! The vertex C is the lowest point, but it's still the maximum negative value, not where the function changes sign.

Q: How can I double-check my answer?

Pick test values! Choose any $ x and any \( x > B $, then check that \( f(x) at those points. The graph should be below the x-axis there.

Q: What if I can't see the exact coordinates of A and B?

That's okay! The question asks for the relationship, not exact numbers. You can still identify that the solution is $ x or \( x > B $ from the graph's shape.

Quadratic Inequalities with Graphical Analysis

The graph of the function below intersects the $x$ -axis at points A and B.

The vertex of the parabola is marked at point C.

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 points A and B.

The vertex of the parabola is marked at point C.

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

Step-by-step solution

To solve this problem, let's analyze the graph of this quadratic function:

The graph intersects the $x$ -axis at points A and B, indicating $f(x) = 0$ at these points.
The function exhibits a parabolic shape with a vertex located at point C, below the $x$ -axis, suggesting that the parabola opens upwards.

For a typical upward opening parabola that intersects the $x$ -axis at A and B, the function $f(x)$ is below the $x$ -axis (i.e., $f(x) < 0$ ) outside the interval between A and B.

Therefore, the solution set for which $f(x) < 0$ is $x < A$ or $x > B$ . This represents where the parabola lies beneath the $x$ -axis.

This corresponds to choice 2: $x > B$ or $x < A$ .

Final Answer

$x>B$ or $x < A$

Key Points to Remember

Essential concepts to master this topic

Sign Analysis: Function is negative when parabola lies below x-axis
Technique: Find where f(x) < 0 by examining graph outside the roots
Check: Verify by testing values: pick x < A and x > B, both should be negative ✓

Common Mistakes

Avoid these frequent errors

Reading where the function is positive instead of negative
Don't identify where f(x) > 0 when asked for f(x) < 0 = wrong regions selected! Students often confuse above/below the x-axis with positive/negative values. Always remember: negative means below the x-axis, which occurs outside the roots for upward-opening parabolas.

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 where the function is negative just by looking at the graph?

Look for where the parabola is below the x-axis! Since the parabola opens upward and crosses at points A and B, it's negative (below the axis) when $x < A$ or $x > B$ .

Why isn't the answer between A and B?

Between A and B, the parabola is above the x-axis, so $f(x) > 0$ there! The vertex C is the lowest point, but it's still the maximum negative value, not where the function changes sign.

Does it matter that the vertex C is below the x-axis?

Yes! The vertex being below the x-axis confirms the parabola opens upward. This means the function is negative outside the roots (A and B) and positive between them.

How can I double-check my answer?

Pick test values! Choose any $x < A$ and any $x > B$ , then check that $f(x) < 0$ at those points. The graph should be below the x-axis there.

What if I can't see the exact coordinates of A and B?

That's okay! The question asks for the relationship, not exact numbers. You can still identify that the solution is $x < A$ or $x > B$ from the graph's shape.

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