Solve y = -x² + 1: Finding Values Where Function is Positive

Q: Why isn't the answer x > 1 or x < -1?

That would be the solution to $ x^2 > 1 $, not \( x^2 ! Remember: when x² is small (less than 1), x must be close to zero, meaning \( -1 .

Q: How do I visualize this on a parabola?

The function $ y = -x^2 + 1 $ is an upside-down parabola with vertex at (0,1). It's positive (above x-axis) between its roots at x = -1 and x = 1.

Q: What happens at x = 1 and x = -1?

At both points, $ f(x) = 0 $, so the function touches the x-axis. Since we want $ f(x) > 0 $ (strictly greater), we use open intervals: \( -1 .

Q: Can I solve this by factoring?

Yes! Factor as $ -(x^2 - 1) = -(x-1)(x+1) > 0 $. This means \( (x-1)(x+1) , which happens when factors have opposite signs: between the roots.

Q: Why is my graphing calculator showing a curve?

That's the parabola $ y = -x^2 + 1 $! The solution is the x-values where this curve is above the x-axis (positive y-values), which is between x = -1 and x = 1.

Quadratic Inequalities with Parabola Analysis

Look at the following function:

$y=-x^2+1$

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

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

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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:

$y=-x^2+1$

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

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

Step-by-step solution

To solve the problem of finding the values of $x$ for which $f(x) = -x^2 + 1 > 0$ , we'll follow these steps:

Step 1: Begin with the inequality $-x^2 + 1 > 0$ .
Step 2: Rearrange it to $1 > x^2$ .
Step 3: Recognize this can also be expressed as $x^2 < 1$ .
Step 4: Since $x^2 < 1$ , the solutions for $x$ are those values between $-1$ and $1$ : $-1 < x < 1$ .

Therefore, the values of $x$ for which $f(x) > 0$ are $-1 < x < 1$ .

Final Answer

$-1 < x < 1$

Key Points to Remember

Essential concepts to master this topic

Rule: Set quadratic expression greater than zero and solve inequality
Technique: Rearrange $-x^2 + 1 > 0$ to $x^2 < 1$
Check: Test x = 0: $-(0)^2 + 1 = 1 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Solving x² < 1 as x < 1 only
Don't forget the negative values when solving $x^2 < 1$ = missing half the solution! This ignores that both positive and negative numbers can have squares less than 1. Always remember $x^2 < 1$ means $-1 < x < 1$ .

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 isn't the answer x > 1 or x < -1?

That would be the solution to $x^2 > 1$ , not $x^2 < 1$ ! Remember: when x² is small (less than 1), x must be close to zero, meaning $-1 < x < 1$ .

How do I visualize this on a parabola?

The function $y = -x^2 + 1$ is an upside-down parabola with vertex at (0,1). It's positive (above x-axis) between its roots at x = -1 and x = 1.

What happens at x = 1 and x = -1?

At both points, $f(x) = 0$ , so the function touches the x-axis. Since we want $f(x) > 0$ (strictly greater), we use open intervals: $-1 < x < 1$ .

Can I solve this by factoring?

Yes! Factor as $-(x^2 - 1) = -(x-1)(x+1) > 0$ . This means $(x-1)(x+1) < 0$ , which happens when factors have opposite signs: between the roots.

Why is my graphing calculator showing a curve?

That's the parabola $y = -x^2 + 1$ ! The solution is the x-values where this curve is above the x-axis (positive y-values), which is between x = -1 and x = 1.

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