Calculate the Ascending Area: Finding Area Under y=(x+4)²-5

Q: Why does the parabola increase after the vertex?

Since the coefficient of $(x+4)^2$ is positive, the parabola opens upward. This means it goes down until it reaches the vertex, then goes up forever after that point.

Q: How do I find the vertex from vertex form?

In $y = (x+h)^2 + k$, the vertex is at $(-h, k)$. For $y = (x+4)^2 - 5$, we have h = 4, so the vertex x-coordinate is -4.

Q: What's the difference between ascending and increasing?

Ascending and increasing mean the same thing in mathematics - the function values get larger as x increases. We're looking for where the graph goes upward from left to right.

Q: Do I need to consider the -5 in the equation?

The -5 shifts the parabola up or down but doesn't change where it increases or decreases. Only the $(x+4)^2$ part determines the vertex's x-coordinate and the increasing interval.

Q: How can I double-check my answer?

Pick a value greater than -4 (like x = 0) and one less than -4 (like x = -5)Calculate the derivative or slope at both pointsThe slope should be positive for x > -4 and negative for x

Parabola Increasing Intervals with Vertex Form

Find the ascending area of the function

$y=(x+4)^2-5$

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Understand the problem

Find the ascending area of the function

$y=(x+4)^2-5$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the vertex of the function.
Step 2: Determine the direction in which the parabola opens.
Step 3: Use the vertex to determine where the function is increasing.

Now, let's work through each step:
Step 1: The function $y = (x+4)^2 - 5$ is in vertex form, with the vertex at $x = -4$ .
Step 2: The expression $(x+4)^2$ has a positive coefficient, indicating the parabola opens upwards.
Step 3: For an upwards opening parabola, the function increases for values of $x$ greater than the vertex. Therefore, the function is increasing for $x > -4$ .

Therefore, the solution to the problem is $-4 < x$ .

Final Answer

$-4 < x$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: In y = (x+h)² + k, vertex is at x = -h
Opening Direction: Positive coefficient means parabola opens upward from vertex
Verification: Test x-values: at x = -3, slope is positive confirming increase ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex x-coordinate sign
Don't think the vertex of (x+4)² is at x = +4 = wrong increasing interval! The vertex formula uses opposite signs, so (x+4)² has vertex at x = -4. Always remember: (x+h)² has vertex at x = -h.

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

Why does the parabola increase after the vertex?

Since the coefficient of $(x+4)^2$ is positive, the parabola opens upward. This means it goes down until it reaches the vertex, then goes up forever after that point.

How do I find the vertex from vertex form?

In $y = (x+h)^2 + k$ , the vertex is at $(-h, k)$ . For $y = (x+4)^2 - 5$ , we have h = 4, so the vertex x-coordinate is -4.

What's the difference between ascending and increasing?

Ascending and increasing mean the same thing in mathematics - the function values get larger as x increases. We're looking for where the graph goes upward from left to right.

Do I need to consider the -5 in the equation?

The -5 shifts the parabola up or down but doesn't change where it increases or decreases. Only the $(x+4)^2$ part determines the vertex's x-coordinate and the increasing interval.

How can I double-check my answer?

Pick a value greater than -4 (like x = 0) and one less than -4 (like x = -5)
Calculate the derivative or slope at both points
The slope should be positive for x > -4 and negative for x < -4

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