Find Where f(x) > 0: Analyzing Function Values Between x = 1 and x = 7

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

Look at the shape of the curve! If it looks like a U, it opens upward. If it looks like an upside-down U, it opens downward. This determines where $ f(x) > 0 $.

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

f(x) > 0 means strictly positive (graph above x-axis), while f(x) ≥ 0 includes the x-intercepts where f(x) = 0. Use open circles for > and closed circles for ≥.

Q: Why isn't the answer 1 < x < 7?

Between the roots (1 below the x-axis, so $ f(x) . We need regions where the graph is above the x-axis for \( f(x) > 0 $.

Q: What if the parabola doesn't cross the x-axis?

If there are no x-intercepts, the parabola is either always positive (opens up, above x-axis) or always negative (opens down, below x-axis) for all real x values.

Quadratic Inequalities with Graphical Analysis

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

Step-by-step solution

To determine the values of $x$ at which the function $f(x) > 0$ , we analyze the graphically provided quadratic function. The function $f(x)$ is represented as a parabola in the given diagram. We find the points where the parabola intersects the x-axis, marking critical points for determining sign changes in the function.

Upon examining the graph, we identify the x-intercepts at $x = 1$ and $x = 7$ . The function changes sign around these x-intercepts as follows:

For $x < 1$ , the graph is above the x-axis, implying $f(x) > 0$ .
For $1 < x < 7$ , the graph is below the x-axis, implying $f(x) < 0$ .
For $x > 7$ , the graph is again above the x-axis, implying $f(x) > 0$ .

Consequently, the solutions where $f(x) > 0$ are at $x < 1$ and $x > 7$ . Comparing these results with the given answer options, option 3, which is $x > 7$ or $x < 1$ , corresponds precisely to our solution.

Therefore, the correct solution to the problem is $x > 7$ or $x < 1$ .

Final Answer

$x > 7$ or $x < 1$

Key Points to Remember

Essential concepts to master this topic

Sign Analysis: Function is positive when graph is above x-axis
Root Identification: Find x-intercepts at x = 1 and x = 7
Interval Testing: Check sign in each region: (-∞,1), (1,7), (7,∞) ✓

Common Mistakes

Avoid these frequent errors

Confusing regions where function is positive vs negative
Don't assume f(x) > 0 between the roots = wrong answer! For upward-opening parabolas, the function is negative between roots and positive outside them. Always identify where the graph sits above the x-axis.

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 shape of the curve! If it looks like a U, it opens upward. If it looks like an upside-down U, it opens downward. This determines where $f(x) > 0$ .

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

f(x) > 0 means strictly positive (graph above x-axis), while f(x) ≥ 0 includes the x-intercepts where f(x) = 0. Use open circles for > and closed circles for ≥.

Why isn't the answer 1 < x < 7?

Between the roots (1 < x < 7), the parabola is below the x-axis, so $f(x) < 0$ . We need regions where the graph is above the x-axis for $f(x) > 0$ .

How can I verify my answer without the graph?

Pick test points from each interval! Try x = 0 (should be positive), x = 4 (should be negative), and x = 8 (should be positive). This confirms the sign pattern.

What if the parabola doesn't cross the x-axis?

If there are no x-intercepts, the parabola is either always positive (opens up, above x-axis) or always negative (opens down, below x-axis) for all real x values.

🌟 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