Analyze the Graph: Finding Intervals Where f(x) < 0

Q: How do I know which parts of the graph are negative?

Look for where the curve is below the x-axis. The x-axis represents y = 0, so anything below it means \( f(x) .

Q: What do the x-intercepts tell me?

X-intercepts are where $ f(x) = 0 $. They act as boundary points - the function changes from positive to negative (or vice versa) at these points.

Q: Why is the answer 'x > 3 or x < -3' and not '-3 < x < 3'?

Look carefully at the graph! This parabola opens upward, so it's above the x-axis between the intercepts and below the x-axis outside the intercepts.

Q: How can I test if my answer is correct?

Pick a test point from your interval and see if it makes sense. For example, try $ x = 5 $: the graph shows $ f(5) > 0 $, which contradicts our need for negative values.

Q: What if the parabola opened downward instead?

If it opened downward, then \( f(x) would occur between the x-intercepts. Always check the parabola's orientation first!

Graph Analysis with Negative Function Values

Look at the function graphed below.

Find all values of $x$

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

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Understand the problem

Look at the function graphed below.

Find all values of $x$

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

Step-by-step solution

To solve this problem, we need to determine the values of $x$ where $f(x) < 0$ . Given the graph, observe that this condition occurs between the x-intercepts.

The provided graph shows that $f(x) = 0$ at $x = -3$ and $x = 3$ , which are the intercepts. To find where $f(x)$ is negative, observe where the parabola dips below the x-axis. This happens between the points:

From $x = -3$ the graph dips below the x-axis until $x = 3$ .

Thus, the function $f(x) < 0$ within the interval $-3 < x < 3$ .

Based on this analysis, we identify the intervals where $f(x)$ is below the x-axis:

Since we need $f(x) < 0$ , we observe it happens outside the interval of the roots, specifically:

$x < -3$ and $x > 3$ .

Therefore, the solution to the problem is $x > 3$ or $x < -3$ .

Final Answer

$x > 3$ or $x < -3$

Key Points to Remember

Essential concepts to master this topic

Sign Analysis: Function is negative when graph lies below x-axis
Technique: Identify x-intercepts at -3 and 3, then determine regions
Check: Test point like x = 4: if f(4) > 0, then x > 3 gives negative values ✓

Common Mistakes

Avoid these frequent errors

Confusing where function is positive vs negative
Don't assume f(x) < 0 means the region between x-intercepts = wrong intervals! This happens when students misread the graph orientation. Always identify which side of the x-axis represents negative y-values.

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 which parts of the graph are negative?

Look for where the curve is below the x-axis. The x-axis represents y = 0, so anything below it means $f(x) < 0$ .

What do the x-intercepts tell me?

X-intercepts are where $f(x) = 0$ . They act as boundary points - the function changes from positive to negative (or vice versa) at these points.

Why is the answer 'x > 3 or x < -3' and not '-3 < x < 3'?

Look carefully at the graph! This parabola opens upward, so it's above the x-axis between the intercepts and below the x-axis outside the intercepts.

How can I test if my answer is correct?

Pick a test point from your interval and see if it makes sense. For example, try $x = 5$ : the graph shows $f(5) > 0$ , which contradicts our need for negative values.

What if the parabola opened downward instead?

If it opened downward, then $f(x) < 0$ would occur between the x-intercepts. Always check the parabola's orientation first!

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