Calculate Area: Finding Descending Region of y=-(x+3)²-10

Q: How do I know which direction the parabola opens?

Look at the coefficient of the squared term. In $ y = -(x+3)^2 - 10 $, the coefficient is -1, which is negative, so it opens downward!

Q: Why does the parabola decrease to the right of the vertex?

For downward-opening parabolas, the vertex is the highest point. As you move right from the vertex, the function values get smaller (decrease). As you move left, they also get smaller.

Q: What's the difference between x > -3 and -3 < x?

They mean exactly the same thing! Both say that x is greater than -3. The correct answer \( -3 just writes it in a different order.

Q: How can I check if x = 0 is in the decreasing region?

Since 0 > -3, yes! At x = 0: $ y = -(0+3)^2 - 10 = -19 $. At x = -1: $ y = -(-1+3)^2 - 10 = -14 $. The function decreased from -14 to -19. ✓

Q: What if the parabola opened upward instead?

If the coefficient were positive, the parabola would open upward, and it would increase to the right of the vertex instead of decrease.

Parabolic Functions with Decreasing Intervals

Find the descending area of the function

$y=-(x+3)^2-10$

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Understand the problem

Find the descending area of the function

$y=-(x+3)^2-10$

Step-by-step solution

To solve this problem, we'll identify the vertex of the given parabolic function and determine on which side of the vertex the function decreases.

Step 1: Identify the Vertex
The function $y = -(x+3)^2 - 10$ is in the vertex form $y = a(x-h)^2 + k$ . Here, $a = -1$ , $h = -3$ , and $k = -10$ . Thus, the vertex of the parabola is at $(-3, -10)$ .

Step 2: Analyze Parabola's Direction
Since $a = -1$ (a negative value), the parabola opens downwards. For downward-opening parabolas, the function decreases to the right of its vertex.

Step 3: Determine the Decreasing Domain
Since the parabola decreases for values of $x$ greater than the x-coordinate of the vertex, it decreases when $x > -3$ .

Therefore, the descending area of the function is $-3 < x$ .

The correct answer, corresponding to the choices given, is choice 3: $-3 < x$ .

Final Answer

$-3 < x$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Identify vertex from $y = a(x-h)^2 + k$
Direction Rule: If a < 0, parabola opens down and decreases right of vertex
Check: Test values: f(-2) = -11, f(-4) = -11, decreasing when x > -3 ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex coordinates with decreasing intervals
Don't think vertex (-3, -10) means decreasing at x < -3! The vertex only tells you where the turning point is. Always check if the parabola opens up or down, then determine which side of the vertex shows decreasing behavior.

Practice Quiz

Test your knowledge with interactive questions

Which equation represents the function:

$$ y=x^2 $$

moved 2 spaces to the right

and 5 spaces upwards.

FAQ

Everything you need to know about this question

How do I know which direction the parabola opens?

Look at the coefficient of the squared term. In $y = -(x+3)^2 - 10$ , the coefficient is -1, which is negative, so it opens downward!

Why does the parabola decrease to the right of the vertex?

For downward-opening parabolas, the vertex is the highest point. As you move right from the vertex, the function values get smaller (decrease). As you move left, they also get smaller.

What's the difference between x > -3 and -3 < x?

They mean exactly the same thing! Both say that x is greater than -3. The correct answer $-3 < x$ just writes it in a different order.

How can I check if x = 0 is in the decreasing region?

Since 0 > -3, yes! At x = 0: $y = -(0+3)^2 - 10 = -19$ . At x = -1: $y = -(-1+3)^2 - 10 = -14$ . The function decreased from -14 to -19. ✓

What if the parabola opened upward instead?

If the coefficient were positive, the parabola would open upward, and it would increase to the right of the vertex instead of decrease.

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