Calculate Ascending Area of y=-(x+3)² : Quadratic Function Analysis

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

Look at the coefficient of the squared term. In $ y = -(x+3)^2 $, the negative sign means it opens downward, like an upside-down U.

Q: Where exactly is the vertex of this parabola?

The vertex is at (-3, 0). In the form $ y = a(x - h)^2 + k $, rewrite as $ y = -1(x - (-3))^2 + 0 $, so h = -3 and k = 0.

Q: Why does the function increase for x < -3?

Since the parabola opens downward, values increase as you approach the vertex from either side. Moving from left toward x = -3, the function climbs up to its maximum at the vertex.

Q: How can I verify this is the increasing interval?

Test values! Pick x = -4: $ y = -(-4+3)^2 = -1 $. Pick x = -5: $ y = -(-5+3)^2 = -4 $. Since -1 > -4, the function increases as x moves from -5 to -4.

Q: What happens for x > -3?

For x > -3, the function decreases. After reaching the maximum at the vertex, the downward parabola falls as you move right from x = -3.

Parabola Intervals with Downward Opening

Find the ascending area of the function

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the domain of increase of the function

00:03 We'll use the formula for representing a parabola function

00:07 The formula for the intersection point with X-axis (P,K)

00:12 The member P equals (3)

00:16 The member K equals (0)

00:19 Intersection point with X-axis according to the members

00:22 We'll use abbreviated multiplication formulas to open the parentheses

00:25 Negative times positive always equals negative

00:28 Negative times negative always equals positive

00:31 We'll examine the coefficient of X squared, negative - concave function

00:35 The intersection point with X-axis is also the vertex point

00:41 In a maximum parabola, the domain of increase is before the vertex point

00:45 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Find the ascending area of the function

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

Step-by-step solution

We start by recognizing the function $y = -(x+3)^2$ , which is a parabola that opens downwards with its vertex at $(-3, 0)$ . In the vertex form of a parabola $y = a(x - h)^2 + k$ , the values increase until reaching the vertex if the parabola opens downwards.

The expression $y = -(x+3)^2$ indicates that as $x$ moves towards $-3$ from the left, the function's values increase until reaching the vertex since the parabola is opening downwards.

Therefore, the interval over which the function is increasing corresponds to $x < -3$ .

Thus, the ascending area of the parabola described by the function $y = -(x+3)^2$ is for $x < -3$ .

Final Answer

$x<-3$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Identify vertex location from $y = a(x - h)^2 + k$
Direction Rule: Negative coefficient means downward opening parabola at vertex (-3,0)
Interval Check: Test values like x = -4: increasing left of vertex ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing direction with parabola opening
Don't assume downward parabola decreases everywhere = wrong intervals! A downward parabola increases until the vertex, then decreases. Always identify the vertex first, then determine which side increases toward it.

Practice Quiz

Test your knowledge with interactive questions

Find the intersection of the function

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

With the Y

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 $y = -(x+3)^2$ , the negative sign means it opens downward, like an upside-down U.

Where exactly is the vertex of this parabola?

The vertex is at (-3, 0). In the form $y = a(x - h)^2 + k$ , rewrite as $y = -1(x - (-3))^2 + 0$ , so h = -3 and k = 0.

Why does the function increase for x < -3?

Since the parabola opens downward, values increase as you approach the vertex from either side. Moving from left toward x = -3, the function climbs up to its maximum at the vertex.

How can I verify this is the increasing interval?

Test values! Pick x = -4: $y = -(-4+3)^2 = -1$ . Pick x = -5: $y = -(-5+3)^2 = -4$ . Since -1 > -4, the function increases as x moves from -5 to -4.

What happens for x > -3?

For x > -3, the function decreases. After reaching the maximum at the vertex, the downward parabola falls as you move right from x = -3.

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