Find the Ascending Region of y=-(x-6)²: Quadratic Function Analysis

Q: Why does the negative sign matter for finding increasing intervals?

The negative sign tells us the parabola opens downward like an upside-down U. This means the function increases as you move toward the vertex from the left, then decreases as you move away from the vertex to the right.

Q: How do I find the vertex from y = -(x-6)²?

In vertex form $ y = a(x-h)^2 + k $, the vertex is at (h, k). Here, $ h = 6 $ and $ k = 0 $, so the vertex is (6, 0).

Q: What's the difference between ascending and descending regions?

Ascending (increasing) means the y-values get larger as x increases. Descending (decreasing) means y-values get smaller as x increases. For this downward parabola: increasing when $ x , decreasing when \( x > 6 $.

Q: Can I use the derivative to check my answer?

Yes! The derivative of $ y = -(x-6)^2 $ is $ y' = -2(x-6) $. When $ y' > 0 $ (function increasing), we get $ -2(x-6) > 0 $, which gives us \( x ✓

Q: What if the vertex wasn't at y = 0?

The y-coordinate of the vertex doesn't affect where the function increases or decreases. Only the x-coordinate of the vertex matters for finding increasing/decreasing intervals!

Quadratic Functions with Downward Parabolas

Find the ascending area of the function

$y=-(x-6)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find where the function is increasing.

00:09 First, use the shortened multiplication formulas to open the parentheses.

00:15 Remember, a negative times a positive is always negative.

00:20 Look at the coefficient of X squared. If it's negative, the parabola is a sad face.

00:28 Check all the function's coefficients to understand its shape.

00:33 Use the formula to find the vertex of the parabola.

00:38 Substitute the values in the formula, and solve to get the vertex.

00:52 This will give us the X value at the vertex.

00:57 For a maximum parabola, it increases before the vertex.

01:03 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the ascending area of the function

$y=-(x-6)^2$

Step-by-step solution

To solve this problem, follow these steps:

The function given is $y = -(x-6)^2$ . This indicates that the parabola opens downwards because of the negative sign.

Step 1: Identify the vertex of the parabola:

The vertex form of a parabola is $y = a(x-h)^2 + k$ . Here, the vertex is at $(6, 0)$ .

Step 2: Determine the interval where the function is increasing:

Since the parabola opens downwards, the function is increasing as it approaches the vertex from the left.

Thus, the function is increasing for $x < 6$ .

Therefore, the correct answer is $x < 6$ .

Final Answer

$x < 6$

Key Points to Remember

Essential concepts to master this topic

Direction Rule: Negative coefficient makes parabola open downward
Vertex Method: From y = -(x-6)² identify vertex at (6,0)
Increasing Check: Left of vertex means x < 6 for upward slope ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing direction with parabola opening
Don't think 'negative coefficient = decreasing everywhere' = wrong intervals! The negative sign only shows the parabola opens down, not that it's always decreasing. Always remember: downward parabolas increase LEFT of vertex, decrease RIGHT of vertex.

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

Why does the negative sign matter for finding increasing intervals?

The negative sign tells us the parabola opens downward like an upside-down U. This means the function increases as you move toward the vertex from the left, then decreases as you move away from the vertex to the right.

How do I find the vertex from y = -(x-6)²?

In vertex form $y = a(x-h)^2 + k$ , the vertex is at (h, k). Here, $h = 6$ and $k = 0$ , so the vertex is (6, 0).

What's the difference between ascending and descending regions?

Ascending (increasing) means the y-values get larger as x increases. Descending (decreasing) means y-values get smaller as x increases. For this downward parabola: increasing when $x < 6$ , decreasing when $x > 6$ .

Can I use the derivative to check my answer?

Yes! The derivative of $y = -(x-6)^2$ is $y' = -2(x-6)$ . When $y' > 0$ (function increasing), we get $-2(x-6) > 0$ , which gives us $x < 6$ ✓

What if the vertex wasn't at y = 0?

The y-coordinate of the vertex doesn't affect where the function increases or decreases. Only the x-coordinate of the vertex matters for finding increasing/decreasing intervals!

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