which function has a vertex at 2 9

Guri Bee

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18-10-2024

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Finding the Function with a Vertex at (2, 9)

Understanding the relationship between a function's equation and its vertex is crucial in algebra. In this article, we'll explore how to identify the function that has a vertex at (2, 9).

Let's start by defining the key concept:

What is a Vertex?

The vertex of a parabola, which is the graph of a quadratic function, is the point where the parabola changes direction. It's either the highest point (maximum) or the lowest point (minimum) of the parabola.

The Standard Form of a Quadratic Function:

The standard form of a quadratic function is:

f(x) = ax² + bx + c

where 'a', 'b', and 'c' are constants.

The Vertex Formula:

We can find the x-coordinate of the vertex using the following formula:

x = -b / 2a

Identifying the Function:

Now, let's apply this knowledge to find the function with a vertex at (2, 9). We know that the x-coordinate of the vertex is 2. Therefore, we can use the vertex formula to find a relationship between 'a' and 'b':

2 = -b / 2a

Solving for 'b', we get:

b = -4a 

This equation tells us that 'b' is directly proportional to 'a' with a factor of -4.

Important Note: There are infinitely many quadratic functions that can have a vertex at (2, 9). This is because the constant 'c' in the standard form influences the vertical shift of the parabola, and it doesn't affect the vertex's x-coordinate.

Example:

Let's assume 'a' = 1. Substituting this value in the equation for 'b', we get:

b = -4 * 1 = -4

Now, we can write a possible quadratic function:

f(x) = x² - 4x + c

To find the value of 'c', we can use the fact that the vertex is at (2, 9). This means that when x = 2, f(x) = 9. Substituting these values in the function, we get:

9 = 2² - 4 * 2 + c

Solving for 'c', we get:

c = 13

Therefore, one possible quadratic function with a vertex at (2, 9) is:

f(x) = x² - 4x + 13

Conclusion:

While there are infinitely many quadratic functions with a vertex at (2, 9), by using the vertex formula and some basic algebraic manipulations, we can find a specific function by choosing a value for 'a' and then solving for 'b' and 'c'. This example demonstrates the key relationship between a quadratic function's equation and its vertex.