Calculate Negative Area: Finding Area Below y=(x-1)²-4

Q: What does 'negative area' mean for a parabola?

The negative area refers to the interval where the parabola dips below the x-axis (where y

Q: Why don't we include the endpoints -1 and 3?

At x = -1 and x = 3, the function equals zero, not negative! Since we want strictly negative values, we use open intervals: \( -1 .

Q: How do I find where a parabola is negative?

Step 1: Find the roots by setting y = 0Step 2: Since the parabola opens upward, it's negative between the rootsStep 3: Use strict inequalities for the interval

Q: What if I can't factor the quadratic easily?

For $ (x-1)^2 - 4 = 0 $, rearrange to $ (x-1)^2 = 4 $, then take the square root: $ x-1 = \pm 2 $. This gives x = 3 or x = -1!

Q: How can I verify my interval is correct?

Pick any test point inside your interval. For example, try x = 0: $ y = (0-1)^2 - 4 = 1 - 4 = -3 $. Since -3 \( -1 is correct!

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

Find the negative area of the function

$y=(x-1)^2-4$

2

Step-by-step solution

To solve this problem, we'll consider the steps as follows:

Step 1: Recognize the given function as a standard parabola centered at $(h, k) = (1, -4)$ that opens upwards.
Step 2: The formula $y = (x-1)^2 - 4$ represents the parabola with vertex at $(1, -4)$ .
Step 3: Determine roots by solving the equation $(x-1)^2 - 4 = 0$ .
Step 4: Rearrange to $(x-1)^2 = 4$ leading to $x - 1 = \pm 2$ .
Step 5: Solve these to find roots: $x = 3$ and $x = -1$ .
Step 6: Calculate regions where the function is negative by testing intervals between the roots.

Now, we apply these steps:

Step 1: The function, expressed in its vertex form, indicates a parabola opening upwards (since the coefficient of $(x-1)^2$ is positive).

Step 2: The equation for determining where the parabola touches the x-axis is $(x−1)^2−4=0$ . Solving this, we rearrange it to $(x-1)^2 = 4$ .

Step 3: Solving for $(x-1)^2 = 4$ , gives $x - 1 = 2$ or $x - 1 = -2$ , leading to solutions $x = 3$ and $x = -1$ .

Step 4: The parabola is negative between these roots. So, the inequality $(x-1)^2 < 4$ holds true for $-1 < x < 3$ .

Therefore, the negative area of the function lies in the interval $-1 < x < 3$ .

Conclusively, the negative domain of the function is $-1 < x < 3$ .

3

Final Answer

$-1 < x < 3$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Parabola $y=(x-1)^2-4$ has vertex at (1, -4)
Find Roots: Set equation equal to zero: $(x-1)^2-4=0$ gives x = -1, 3
Test Interval: Check x = 0: $y=(0-1)^2-4=-3 < 0$ ✓

Common Mistakes

Avoid these frequent errors

Including endpoints in the negative interval
Don't write $-1 \le x \le 3$ for the negative area! At x = -1 and x = 3, the function equals zero, not negative. The parabola crosses the x-axis at these points. Always use strict inequalities $-1 < x < 3$ for where the function is truly negative.

Practice Quiz

Test your knowledge with interactive questions

Find the corresponding algebraic representation of the drawing:

FAQ

Everything you need to know about this question

What does 'negative area' mean for a parabola?

+

The negative area refers to the interval where the parabola dips below the x-axis (where y < 0). It's the region between the roots where the function values are negative.

Why don't we include the endpoints -1 and 3?

+

At x = -1 and x = 3, the function equals zero, not negative! Since we want strictly negative values, we use open intervals: $-1 < x < 3$ .

How do I find where a parabola is negative?

+

Step 1: Find the roots by setting y = 0
Step 2: Since the parabola opens upward, it's negative between the roots
Step 3: Use strict inequalities for the interval

What if I can't factor the quadratic easily?

+

For $(x-1)^2 - 4 = 0$ , rearrange to $(x-1)^2 = 4$ , then take the square root: $x-1 = \pm 2$ . This gives x = 3 or x = -1!

How can I verify my interval is correct?

+

Pick any test point inside your interval. For example, try x = 0: $y = (0-1)^2 - 4 = 1 - 4 = -3$ . Since -3 < 0, the interval $-1 < x < 3$ is correct!

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Calculate Negative Area: Finding Area Below y=(x-1)²-4

Parabola Analysis with Negative Region Identification

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