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

00:06 Let's find where our function is getting bigger. This is called the d omain of increase.

00:13 We'll start with the formula for a parabola. Are you ready?

00:18 We need to find where it crosses the X-axis. Think of this as the poi nt, P comma K.

00:25 Here, P is three. Yes, P equals three.

00:30 And K, that's zero. So, K equals zero.

00:34 These points help us find where the graph hits the X-axis.

00:39 Next, we'll use some quick tricks to handle multiplication. Get read y!

00:44 Here's a tip: Negative times positive always becomes negative .

00:50 And negative times negative? That always gives us a positive.

00:55 Let's check out the X squared part. If it's negative, the shape is c oncave. Like a frown.

01:02 Where the graph hits the X-axis is our vertex too. Important spot!

01:08 In this upside-down parabola, things grow before reaching the v ertex.

01:13 And that's how we solve this problem! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

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

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

With the X

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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