Analyze the Function: Where Is -1/9x² + 1 2/3x Greater Than Zero?

Q: Why does the negative leading coefficient matter for determining where f(x) > 0?

The negative leading coefficient $ -\frac{1}{9} $ tells us the parabola opens downward. This means the function is positive between the roots and negative outside them.

Q: How do I convert the mixed number 1 2/3 to an improper fraction?

Convert by multiplying: $ 1\frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3} $. Always convert mixed numbers before solving!

Q: What if I get confused about which interval to choose?

Test a point! Pick any number in each interval and substitute it into the original function. If the result is positive, that interval is part of your answer.

Q: Why do we factor out x instead of using the quadratic formula?

When you can factor easily, it's faster! Here, $ x $ is a common factor, so we get $ x(-\frac{1}{9}x + \frac{5}{3}) = 0 $ immediately.

Q: How do I know the function is positive between the roots?

Since the leading coefficient is negative, the parabola opens downward. This means it's above the x-axis (positive) between its roots and below the x-axis (negative) outside them.

Quadratic Inequalities with Sign Analysis

Look at the following function:

$y=-\frac{1}{9}x^2+1\frac{2}{3}x$

Determine for which values of $x$ the following is true:

$f(x) > 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$y=-\frac{1}{9}x^2+1\frac{2}{3}x$

Determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To determine where the function $f(x) = -\frac{1}{9}x^2 + \frac{5}{3}x$ is positive, we follow these steps:

Step 1: Convert the mixed fraction: The term $1\frac{2}{3}x$ can be written as $\frac{5}{3}x$ .
Step 2: The function can be expressed as $f(x) = -\frac{1}{9}x^2 + \frac{5}{3}x$ .
Step 3: Set the function equal to zero to find the roots: $-\frac{1}{9}x^2 + \frac{5}{3}x = 0$ .
Step 4: Factor out $x$ : $x\left(-\frac{1}{9}x + \frac{5}{3}\right) = 0$ .
Step 5: Solve for $x$ : From $x = 0$ and $-\frac{1}{9}x + \frac{5}{3} = 0$ , find the second root:

Solving the second equation:

$\frac{5}{3} = \frac{1}{9}x$ , which simplifies to:

$x = \frac{5}{3} \times 9 = 15$ .

The roots are $x = 0$ and $x = 15$ .

Since the parabola opens downwards (as indicated by the negative leading coefficient $-\frac{1}{9}$ ), the function will be positive between the roots.

Thus, $f(x) > 0$ for $0 < x < 15$ .

Therefore, the values of $x$ such that $f(x) > 0$ are given by:

$0 < x < 15$ .

Final Answer

$0 < x < 15$

Key Points to Remember

Essential concepts to master this topic

Rule: Set quadratic equal to zero to find critical points
Technique: Factor $-\frac{1}{9}x^2 + \frac{5}{3}x = x(-\frac{1}{9}x + \frac{5}{3})$ to get roots
Check: Test x = 5 in interval (0, 15): negative coefficient makes parabola positive between roots ✓

Common Mistakes

Avoid these frequent errors

Assuming function is positive everywhere the parabola is above x-axis
Don't just look at where y > 0 on a graph without considering the inequality direction! A downward parabola can mislead you about which intervals satisfy f(x) > 0. Always find the roots first, then use the leading coefficient to determine where the function is positive between or outside 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

Why does the negative leading coefficient matter for determining where f(x) > 0?

The negative leading coefficient $-\frac{1}{9}$ tells us the parabola opens downward. This means the function is positive between the roots and negative outside them.

How do I convert the mixed number 1 2/3 to an improper fraction?

Convert by multiplying: $1\frac{2}{3} = \frac{3 \times 1 + 2}{3} = \frac{5}{3}$ . Always convert mixed numbers before solving!

What if I get confused about which interval to choose?

Test a point! Pick any number in each interval and substitute it into the original function. If the result is positive, that interval is part of your answer.

Why do we factor out x instead of using the quadratic formula?

When you can factor easily, it's faster! Here, $x$ is a common factor, so we get $x(-\frac{1}{9}x + \frac{5}{3}) = 0$ immediately.

How do I know the function is positive between the roots?

Since the leading coefficient is negative, the parabola opens downward. This means it's above the x-axis (positive) between its roots and below the x-axis (negative) outside them.

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