Analyzing Function Graphs: Determine Positive X Intervals for f(x)

Q: How do I know which intervals to choose from the graph?

Look for where the parabola is above the x-axis. Since this parabola opens downward, it's positive between the two x-intercepts at 2 and 8.

Q: What's the difference between f(x) > 0 and f(x) = 0?

f(x) = 0 gives you the x-intercepts (where graph crosses x-axis). f(x) > 0 gives you intervals where the graph is above the x-axis.

Q: Why do I use open interval notation like (2, 8)?

We use \( 2 because the question asks for f(x) > 0, not f(x) ≥ 0. At x = 2 and x = 8, the function equals zero, not greater than zero.

Q: How can I tell if a parabola opens up or down from the graph?

Look at the shape! This parabola has its maximum point at the top and curves downward on both sides, so it opens downward. If it had a minimum at the bottom, it would open upward.

Q: What if I can't see the exact intercept values?

Look for the labeled points on the axes. This graph clearly shows intercepts at x = 2 and x = 8. Always read the scale and labels carefully.

Quadratic Inequalities with Graph Analysis

Based on the data in the sketch, find for which X values the graph of the function $f\left(x\right) > 0$

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Understand the problem

Based on the data in the sketch, find for which X values the graph of the function $f\left(x\right) > 0$

Step-by-step solution

To solve this problem effectively, follow these steps:

Step 1: Identify where the parabola crosses the x-axis. These are the x-intercepts or roots. Let’s call these intercepts $x_1$ and $x_2$ .
Step 2: Since the question asks for $f(x) > 0$ , we need to find where the graph is above the x-axis.
Step 3: Examine the sketch: The parabola dips below the x-axis after the first root and re-emerges above it until the second root, which is characteristic of a quadratic function.
Step 4: Thus, the function is positive between the two intercepts.

From the sketch, it is clear that the x-values where the function $f(x) > 0$ are between the two intercepts, specifically $(x_1, x_2)$ , which look like $2$ and $8$ on the graph.

Therefore, for this problem, the interval where $f(x) > 0$ is $2 < x < 8$ .

The solution to this problem is $2 < x < 8$ .

Final Answer

$2 < x < 8$

Key Points to Remember

Essential concepts to master this topic

Graph Reading: Find where parabola is above x-axis for positive values
Technique: Identify x-intercepts at 2 and 8, function positive between them
Check: Test x = 5: function value is above x-axis ✓

Common Mistakes

Avoid these frequent errors

Reading where function equals zero instead of greater than zero
Don't find x-intercepts and think those are your answer = wrong intervals! The intercepts show where f(x) = 0, not f(x) > 0. Always look for where the graph is above the x-axis between intercepts.

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 intervals to choose from the graph?

Look for where the parabola is above the x-axis. Since this parabola opens downward, it's positive between the two x-intercepts at 2 and 8.

What's the difference between f(x) > 0 and f(x) = 0?

f(x) = 0 gives you the x-intercepts (where graph crosses x-axis). f(x) > 0 gives you intervals where the graph is above the x-axis.

Why do I use open interval notation like (2, 8)?

We use $2 < x < 8$ because the question asks for f(x) > 0, not f(x) ≥ 0. At x = 2 and x = 8, the function equals zero, not greater than zero.

How can I tell if a parabola opens up or down from the graph?

Look at the shape! This parabola has its maximum point at the top and curves downward on both sides, so it opens downward. If it had a minimum at the bottom, it would open upward.

What if I can't see the exact intercept values?

Look for the labeled points on the axes. This graph clearly shows intercepts at x = 2 and x = 8. Always read the scale and labels carefully.

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