Parabola Intersection Analysis: Find X Where f(x) < 0

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

Look at the vertex position! In this graph, vertex C is below the x-axis, and the parabola curves upward from there. An upward-opening parabola has its vertex at the lowest point.

Q: Why is f(x) < 0 between points A and B?

Because the parabola is below the x-axis between the roots! When a graph is below the x-axis, the y-values (function values) are negative. Between A and B, the parabola dips down below zero.

Q: What if I pick a point outside A and B to test?

Great thinking! If you test a point like x = 0 (assuming it's outside A and B), you'll find f(0) > 0, confirming the function is positive outside the roots and negative between them.

Q: Does the vertex matter for finding where f(x) < 0?

The vertex helps you visualize the parabola's shape, but what matters most are the roots A and B. The function is negative between the roots regardless of where exactly the vertex is located.

Q: Can I write the answer as an inequality with actual numbers?

Yes! If you can determine the x-coordinates of points A and B from the graph, you can write something like \( 2 instead of \( A .

Step-by-step video solution

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Step-by-step written solution

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1

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

2

Step-by-step solution

To solve this problem, let's analyze the graph of the quadratic function around points A and B where it intersects the $x$ -axis.

Step 1: Identify the nature of the quadratic. From the graph, it is clear that the parabola intersects the $x$ -axis, suggesting $f(x) = 0$ at these points.
Step 2: Since the problem indicates points A and B as interceptions, we can conclude the parabola crosses or touches the $x$ -axis at these points.
Step 3: Determine where $f(x) < 0$ . Since A and B are roots, the parabola's graph will be below the $x$ -axis between A and B if the parabola opens upwards, given by $A < x < B$ . If it opens downwards, the parabola would be negative outside A and B. Based on typical quadratic behavior with a vertex below the $x$ -axis, the parabola likely opens upwards.

The solution, therefore, is found within the interval between the intercepts on an upward-opening parabola. This conclusion is consistent with the graphical representation of most standard quadratics.

Thus, the values of $x$ where $f(x) < 0$ are precisely in the interval $A < x < B$ .

3

Final Answer

$A < x < B$

Key Points to Remember

Essential concepts to master this topic

Zero Rule: Parabola crosses x-axis where function equals zero
Technique: Between roots A and B, upward parabola is negative
Check: Test point between A and B: if f(test) < 0, answer confirmed ✓

Common Mistakes

Avoid these frequent errors

Confusing where function is positive versus negative
Don't assume f(x) < 0 means above the x-axis = wrong regions! The graph shows function values, not position. Always remember: negative function values mean the parabola is below the x-axis between the roots.

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 vertex position! In this graph, vertex C is below the x-axis, and the parabola curves upward from there. An upward-opening parabola has its vertex at the lowest point.

Why is f(x) < 0 between points A and B?

+

Because the parabola is below the x-axis between the roots! When a graph is below the x-axis, the y-values (function values) are negative. Between A and B, the parabola dips down below zero.

What if I pick a point outside A and B to test?

+

Great thinking! If you test a point like x = 0 (assuming it's outside A and B), you'll find f(0) > 0, confirming the function is positive outside the roots and negative between them.

Does the vertex matter for finding where f(x) < 0?

+

The vertex helps you visualize the parabola's shape, but what matters most are the roots A and B. The function is negative between the roots regardless of where exactly the vertex is located.

Can I write the answer as an inequality with actual numbers?

+

Yes! If you can determine the x-coordinates of points A and B from the graph, you can write something like $2 < x < 6$ instead of $A < x < B$ .

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