Solve: When is -(1/3x + 15)² Greater Than Zero?

Q: Why can't this function ever be positive?

Because any squared expression is non-negative, and when you multiply a non-negative number by -1, you get a non-positive result. So $ y = -(\text{something})^2 $ can never be greater than zero!

Q: What does the graph of this function look like?

It's a downward-opening parabola with its highest point (vertex) at $ x = -45 $. The vertex touches the x-axis at $ y = 0 $, and everywhere else the function is negative.

Q: How is this different from a regular parabola?

A regular parabola like $ y = x^2 $ opens upward and has positive values. The negative sign flips this parabola upside down, so it opens downward and has only negative values (except at the vertex).

Q: Could any answer choice with 'x >' or 'x <' be correct?

No! Since this function is never positive for any value of x, inequalities like 'x > -45' or 'x

Quadratic Functions with Negative Leading Coefficients

Look at the function below:

$y=-\left(\frac{1}{3}x+15\right)^2$

Then 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 function below:

$y=-\left(\frac{1}{3}x+15\right)^2$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

Step-by-step solution

To solve the problem, we must analyze the given function $y = -\left(\frac{1}{3}x + 15\right)^2$ .

Firstly, as a general rule of algebra, the square of any real number is non-negative. Therefore, $\left(\frac{1}{3}x + 15\right)^2 \geq 0$ for all real values of $x$ .

Secondly, the function is $y = -\left(\frac{1}{3}x + 15\right)^2$ . The negative sign in front affects the entire expression, making the range of $y$ non-positive ( $y \leq 0$ ) since the expression within the square is always non-negative. This implies every $y$ is either zero or negative.

Thus, the function $y$ will never be greater than zero because multiplying any non-negative number by $(-1)$ results in a non-positive number.

Conclusion: The function $f(x) > 0$ is true for no values of $x$ .

Therefore, the correct answer choice is: True for no values of $x$ .

Final Answer

True for no values of $x$

Key Points to Remember

Essential concepts to master this topic

Rule: Any squared expression is always non-negative or zero
Technique: Negative sign flips $(\frac{1}{3}x + 15)^2 \geq 0$ to $y \leq 0$
Check: Test any x-value: when x = 0, y = -225 which is negative ✓

Common Mistakes

Avoid these frequent errors

Forgetting that the negative sign affects the entire squared expression
Don't think $-(\frac{1}{3}x + 15)^2$ can be positive just because there's a square = wrong conclusion! The negative sign makes the entire function non-positive. Always remember that multiplying any non-negative value by -1 gives a non-positive result.

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

Why can't this function ever be positive?

Because any squared expression is non-negative, and when you multiply a non-negative number by -1, you get a non-positive result. So $y = -(\text{something})^2$ can never be greater than zero!

When does this function equal exactly zero?

The function equals zero when the squared part equals zero. This happens when $\frac{1}{3}x + 15 = 0$ , which gives us $x = -45$ .

What does the graph of this function look like?

It's a downward-opening parabola with its highest point (vertex) at $x = -45$ . The vertex touches the x-axis at $y = 0$ , and everywhere else the function is negative.

How is this different from a regular parabola?

A regular parabola like $y = x^2$ opens upward and has positive values. The negative sign flips this parabola upside down, so it opens downward and has only negative values (except at the vertex).

Could any answer choice with 'x >' or 'x <' be correct?

No! Since this function is never positive for any value of x, inequalities like 'x > -45' or 'x < -45' cannot be correct. The function is always ≤ 0.

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Solve: When is -(1/3x + 15)² Greater Than Zero?

Quadratic Functions with Negative Leading Coefficients

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't this function ever be positive?

When does this function equal exactly zero?

What does the graph of this function look like?

How is this different from a regular parabola?

Could any answer choice with 'x >' or 'x <' be correct?

🌟 Unlock Your Math Potential

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