Determine X Values for Which f(x) > 0 Using a Graph

Q: How do I know if the function is positive or negative from a graph?

Look at the y-values! When the graph is above the x-axis, f(x) > 0 (positive). When it's below the x-axis, f(x)

Q: What if the parabola opened downward instead?

If the parabola opened downward, it would be positive between the x-intercepts and negative outside them - exactly the opposite! Always check which direction the parabola opens.

Q: Do I include the x-intercepts -3 and 3 in my answer?

No! At x = -3 and x = 3, the function equals zero, not positive. Since we want f(x) > 0 (strictly greater), we use \( -3 , not ≤.

Q: How can I tell this is a parabola from the graph?

The smooth, U-shaped curve with exactly two x-intercepts is the classic shape of a quadratic function (parabola). It has a lowest point (vertex) and extends infinitely upward on both sides.

Q: What if I can't see the exact x-intercepts clearly?

Look for the points where the curve crosses the x-axis - these are clearly marked as -3 and 3 in this graph. The function changes from negative to positive (or vice versa) at these crossing points.

Quadratic Inequalities with Graph Analysis

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

Step-by-step solution

First, we examine the provided graph of the quadratic function $f(x)$ . The graph clearly shows the x-intercepts (where the function crosses the x-axis) at $x = -3$ and $x = 3$ .

Since the quadratic function appears to be a standard parabola opening upwards, the portion of the graph between these two x-intercepts will be above the x-axis, which means that $f(x) > 0$ in this interval.

The intervals to the left of $x = -3$ and to the right of $x = 3$ will be where the graph lies below the x-axis, meaning $f(x) < 0$ in those regions.

Thus, the graph shows that the function $f(x)$ is positive between $x = -3$ and $x = 3$ . Therefore, the solution to the problem is:

$-3 < x < 3$

Final Answer

$-3 < x < 3$

Key Points to Remember

Essential concepts to master this topic

Graph Reading: Function is positive when graph lies above x-axis
Technique: Identify x-intercepts at -3 and 3, then find interval between
Check: Verify parabola opens upward and is above axis between intercepts ✓

Common Mistakes

Avoid these frequent errors

Reading inequality direction incorrectly
Don't confuse f(x) > 0 with f(x) < 0 when reading graphs = opposite answer! Students often identify where the graph is below the x-axis instead of above it. Always remember: f(x) > 0 means the graph is ABOVE the x-axis.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

How do I know if the function is positive or negative from a graph?

Look at the y-values! When the graph is above the x-axis, f(x) > 0 (positive). When it's below the x-axis, f(x) < 0 (negative). The x-axis is where f(x) = 0.

What if the parabola opened downward instead?

If the parabola opened downward, it would be positive between the x-intercepts and negative outside them - exactly the opposite! Always check which direction the parabola opens.

Do I include the x-intercepts -3 and 3 in my answer?

No! At x = -3 and x = 3, the function equals zero, not positive. Since we want f(x) > 0 (strictly greater), we use $-3 < x < 3$ , not ≤.

How can I tell this is a parabola from the graph?

The smooth, U-shaped curve with exactly two x-intercepts is the classic shape of a quadratic function (parabola). It has a lowest point (vertex) and extends infinitely upward on both sides.

What if I can't see the exact x-intercepts clearly?

Look for the points where the curve crosses the x-axis - these are clearly marked as -3 and 3 in this graph. The function changes from negative to positive (or vice versa) at these crossing points.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 The Quadratic Function questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime