"Which h and k Values Transform a Function into Vertex Form?"

When it comes to representing a quadratic function in vertex form, the values of 'h' and 'k' play a pivotal role. These parameters determine the horizontal and vertical shifts of the parabola, respectively. Let's delve into understanding these values and how to use them to write a function in vertex form.

Understanding Quadratic Functions in Vertex Form

Quadratic functions can be written in three different forms: standard, intercept, and vertex. The vertex form is particularly useful as it provides immediate information about the vertex of the parabola. The general form of a quadratic function in vertex form is:

f(x) = a(x-h)² + k

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Identifying 'h' and 'k' in the Vertex Form

'h' is the x-coordinate of the vertex of the parabola. It represents the horizontal shift from the standard form of the quadratic function.
'k' is the y-coordinate of the vertex. It represents the vertical shift from the standard form and also indicates the y-value of the vertex.

Determining 'h' and 'k' Values

To write a quadratic function in vertex form, you need to know the values of 'a', 'h', and 'k'. If you're given the standard form of the function (f(x) = ax² + bx + c), you can find 'h' and 'k' using the following steps:

Finding 'h'

The x-coordinate of the vertex ('h') can be found using the formula:

h = -b / (2a)

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Finding 'k'

To find 'k', substitute 'h' back into the standard form of the function:

k = f(h) = a(h)² + b(h) + c

Examples

Let's consider an example to illustrate the process. Suppose we have the following quadratic function in standard form:

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f(x) = 2x² - 8x + 6

Finding 'h'

Using the formula for 'h', we have:

h = -(-8) / (2*2) = 2

Finding 'k'

Now, substitute 'h' back into the standard form to find 'k':

k = f(2) = 2(2)² - 8(2) + 6 = 2

So, the function in vertex form is:

f(x) = 2(x-2)² + 2

Using 'h' and 'k' to Graph the Function

Once you have the values of 'h' and 'k', you can graph the function by plotting the vertex ('h', 'k') and then using the coefficient 'a' to determine the direction and steepness of the parabola.

In conclusion, understanding and using the values of 'h' and 'k' in the vertex form of a quadratic function allows for easier graphing, analysis, and comparison of functions. By following the steps outlined above, you can accurately write any quadratic function in vertex form.

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