Vertex Points and Function Domain: Determining Ascending vs Descending Behavior

Q: What does it mean for a function to be ascending or descending?

A function is ascending (increasing) when y-values get larger as x increases. It's descending (decreasing) when y-values get smaller as x increases. The vertex shows where this behavior changes!

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

The vertex is simply (p, 0)! The value inside the parentheses (with opposite sign) gives you the x-coordinate, and since there's no constant added, the y-coordinate is 0.

Q: Does the vertex always divide ascending and descending behavior?

For parabolas that open upward (positive coefficient), yes! The vertex is the lowest point, so the function decreases to the left and increases to the right of the vertex.

Q: What if the parabola opens downward?

If the coefficient is negative, like $ y = -(x-p)^2 $, the parabola opens downward. Then it increases to the left of the vertex and decreases to the right.

Q: Can I use this method for any quadratic function?

Yes! Convert any quadratic to vertex form $ y = a(x-h)^2 + k $ first. The vertex is (h, k), and the sign of 'a' tells you if it opens up (positive) or down (negative).

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 What should we know about increasing and decreasing intervals?

00:09 Let's draw a graph with a vertex point. This is key.

00:15 Notice that the vertex is where it shifts from decreasing to increasing.

00:31 Now, let's draw a graph with an upside-down vertex and see what happens.

00:39 Here too, the function shifts from increasing to decreasing.

00:43 So, the vertex helps us determine when it increases or decreases.

00:49 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

To know whether the domain of a function is ascending or descending, you need to know the vertex point.

2

Step-by-step solution

To solve this problem, we need to determine whether knowing the vertex of a function $y = (x-p)^2$ allows us to conclude if the function's domain is ascending or descending.

Consider the vertex form of the given parabola: $y = (x-p)^2$ .

The parabolic function opens upward since the coefficient of $(x-p)^2$ is positive (1).
The vertex of the parabola is $(p, 0)$ .
A parabola's opening direction is determined by the sign of the coefficient of the squared term. By understanding this, we ascertain that the function decreases towards the vertex (left side of vertex) and increases away (right side of vertex).
Thus, the left side of $p$ , as $x$ approaches the vertex, is descending, and to the right is ascending, confirming the vertex as a divider of these behaviors.

Overall, knowing the vertex allows us to describe the behavior of the function's graph as ascending or descending at and away from the vertex point, verifying the statement.

The conclusion is that the statement is True.

3

Final Answer

True.

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: Parabola vertex divides function into ascending and descending regions
Technique: For $y = (x-p)^2$ , vertex at (p, 0) shows decreasing left, increasing right
Check: Test points on both sides of vertex to confirm behavior changes ✓

Common Mistakes

Avoid these frequent errors

Confusing domain with function behavior
Don't say 'domain is ascending or descending' = incorrect terminology! The domain is all possible x-values (usually all real numbers). Always say 'the function is ascending or descending on intervals of the domain'.

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

What does it mean for a function to be ascending or descending?

+

A function is ascending (increasing) when y-values get larger as x increases. It's descending (decreasing) when y-values get smaller as x increases. The vertex shows where this behavior changes!

How do I find the vertex of y = (x-p)²?

+

The vertex is simply (p, 0)! The value inside the parentheses (with opposite sign) gives you the x-coordinate, and since there's no constant added, the y-coordinate is 0.

Does the vertex always divide ascending and descending behavior?

+

For parabolas that open upward (positive coefficient), yes! The vertex is the lowest point, so the function decreases to the left and increases to the right of the vertex.

What if the parabola opens downward?

+

If the coefficient is negative, like $y = -(x-p)^2$ , the parabola opens downward. Then it increases to the left of the vertex and decreases to the right.

Can I use this method for any quadratic function?

+

Yes! Convert any quadratic to vertex form $y = a(x-h)^2 + k$ first. The vertex is (h, k), and the sign of 'a' tells you if it opens up (positive) or down (negative).

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Vertex Points and Function Domain: Determining Ascending vs Descending Behavior

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