Find the Domain of (x-8)²-1: Complete Function Analysis

Quadratic Functions with Sign Analysis

Find the positive and negative domains of the function below:

y=(x−8)2−1 y=\left(x-8\right)^2-1

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the positive and negative domains of the function below:

y=(x−8)2−1 y=\left(x-8\right)^2-1

2

Step-by-step solution

To solve for the positive and negative domains of y=(x−8)2−1 y = (x-8)^2 - 1 :

  • Identify when y y equals zero. Set the equation (x−8)2−1=0 (x-8)^2 - 1 = 0 .
  • Add 1 to both sides: (x−8)2=1 (x-8)^2 = 1 .
  • Take the square root: x−8=±1 x-8 = \pm 1 .
  • Solve for x x : x=8±1→x=7 x = 8 \pm 1 \rightarrow x = 7 or x=9 x = 9 .

So, the function y=(x−8)2−1 y = (x-8)^2 - 1 is zero at x=7 x = 7 and x=9 x = 9 . These points divide the x-axis into intervals:

  • Interval 1: x<7 x < 7 , test x=6 x = 6 . (6−8)2−1=3 (6-8)^2 - 1 = 3 , which is positive.
  • Interval 2: 7<x<9 7 < x < 9 , test x=8 x = 8 . (8−8)2−1=−1 (8-8)^2 - 1 = -1 , which is negative.
  • Interval 3: x>9 x > 9 , test x=10 x = 10 . (10−8)2−1=3 (10-8)^2 - 1 = 3 , which is positive.

From this analysis:

The negative domain (where y<0 y < 0 ) is 7<x<9 7 < x < 9 .

The positive domain (where y>0 y > 0 ) consists of x<7 x < 7 and x>9 x > 9 .

Therefore, the correct answer from the provided choices is:

x<0:7<x<9 x < 0 : 7 < x < 9

x>9 x > 9 or x>0:x<7 x > 0 : x < 7

3

Final Answer

x<0:7<x<9 x < 0 : 7 < x < 9

x>9 x > 9 or x>0:x<7 x > 0 : x < 7

Key Points to Remember

Essential concepts to master this topic
  • Zeros First: Find where function equals zero by solving equation
  • Test Points: Check sign in each interval: (6−8)2−1=3>0 (6-8)^2 - 1 = 3 > 0
  • Verify: Substitute test values back into original function ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to test intervals between zeros
    Don't just find x = 7 and x = 9 then guess the signs = wrong domains! The parabola changes from positive to negative at these points. Always test a value in each interval: x < 7, 7 < x < 9, and x > 9.

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

AAAX

FAQ

Everything you need to know about this question

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

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The zeros (where y = 0) are the boundary points where the function changes from positive to negative or vice versa. These points divide the x-axis into intervals you need to test separately.

How do I know which values to test in each interval?

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Pick any convenient number within each interval! For example, if your interval is x<7 x < 7 , you could test x = 0, x = 6, or x = -5. The specific value doesn't matter - just make sure it's in the right interval.

What does 'positive domain' and 'negative domain' mean?

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Positive domain: All x-values where y > 0 (function output is positive)
Negative domain: All x-values where y < 0 (function output is negative)

Why is the answer written in that confusing format?

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The notation separates positive and negative domains clearly:

  • x<0:7<x<9 x < 0: 7 < x < 9 means "negative values occur when 7 < x < 9"
  • x>0:x<7 x > 0: x < 7 or x>9 x > 9 means "positive values occur when x < 7 or x > 9"

Can I solve this by graphing instead?

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Yes! The graph of y=(x−8)2−1 y = (x-8)^2 - 1 is a parabola opening upward with vertex at (8, -1). You can see where it's above the x-axis (positive) and below the x-axis (negative).

What if I made an algebra mistake when finding the zeros?

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Double-check by substituting back: If x = 7, then (7−8)2−1=1−1=0✓ (7-8)^2 - 1 = 1 - 1 = 0 ✓ . If x = 9, then (9−8)2−1=1−1=0✓ (9-8)^2 - 1 = 1 - 1 = 0 ✓ . Both should give zero!

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