Quadratic Function Analysis: Find When y = (1/2)x² - 8 is Negative

Q: Why do I need to find where the function equals zero first?

The roots (where y = 0) are the boundary points where the function changes from positive to negative or vice versa. These critical values divide the number line into intervals with consistent signs.

Q: How do I know which interval to test?

Test one point in each interval created by the roots. For $ x = \pm 4 $, test points like x = -5, x = 0, and x = 5 to determine the sign in each region.

Q: Why is the parabola negative between the roots?

Since a = 1/2 > 0, this parabola opens upward like a U-shape. The function dips below the x-axis between the two roots, making it negative in that interval.

Q: Do the endpoints count in my answer?

For strict inequalities like f(x) not included. Use open interval notation: \( -4

Quadratic Inequalities with Sign Analysis

Find the positive and negative domains of the function below:

$y=\frac{1}{2}x^2-8$

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

Find the positive and negative domains of the function below:

$y=\frac{1}{2}x^2-8$

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

$f(x) < 0$

Step-by-step solution

Let's solve the problem by following the outlined steps:

Step 1: Solve the Quadratic Equation.
First, solve the equation $\frac{1}{2}x^2 - 8 = 0$ to find the critical values:

$\frac{1}{2}x^2 - 8 = 0$

$\frac{1}{2}x^2 = 8$

$x^2 = 16$

$x = \pm 4$

Thus, the roots are $x = 4$ and $x = -4$ .

Step 2: Determine Intervals and Test Sign of Function.
The roots divide the number line into three intervals: $x < -4$ , $-4 < x < 4$ , and $x > 4$ .

For $x < -4$ , choose a test point like $x = -5$ :

$y = \frac{1}{2}(-5)^2 - 8 = \frac{1}{2} \times 25 - 8 = 12.5 - 8 = 4.5$ (Positive)

For $-4 < x < 4$ , choose a test point like $x = 0$ :

$y = \frac{1}{2}(0)^2 - 8 = -8$ (Negative)

For $x > 4$ , choose a test point like $x = 5$ :

$y = \frac{1}{2}(5)^2 - 8 = \frac{1}{2} \times 25 - 8 = 12.5 - 8 = 4.5$ (Positive)

Conclusion:
Therefore, the function is negative in the interval where $-4 < x < 4$ .

Thus, the solution for the inequality $\frac{1}{2}x^2 - 8 < 0$ is $-4 < x < 4$ .

Final Answer

$-4 < x < 4$

Key Points to Remember

Essential concepts to master this topic

Roots: Set equation equal to zero to find critical values
Test Points: Check sign in each interval: y = (1/2)(0)² - 8 = -8
Verify: Function changes sign at roots x = -4 and x = 4 ✓

Common Mistakes

Avoid these frequent errors

Confusing when function is positive vs negative
Don't assume the parabola is negative everywhere just because it has a negative constant term = wrong intervals! The parabola opens upward (positive leading coefficient), so it's negative only between the roots. Always test points in each interval to determine the actual sign.

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 do I need to find where the function equals zero first?

The roots (where y = 0) are the boundary points where the function changes from positive to negative or vice versa. These critical values divide the number line into intervals with consistent signs.

How do I know which interval to test?

Test one point in each interval created by the roots. For $x = \pm 4$ , test points like x = -5, x = 0, and x = 5 to determine the sign in each region.

Why is the parabola negative between the roots?

Since a = 1/2 > 0, this parabola opens upward like a U-shape. The function dips below the x-axis between the two roots, making it negative in that interval.

What if I get the inequality backwards?

Always double-check by substituting a test point from your answer interval. If $-4 < x < 4$ , try x = 0: $y = -8 < 0$ ✓

Do the endpoints count in my answer?

For strict inequalities like f(x) < 0, the endpoints where f(x) = 0 are not included. Use open interval notation: $-4 < x < 4$

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