Answer:

Sure! Let's transform each quadratic function into the vertex form f(x) = a(x - h)² + k and determine the values of a, h, and k.

1. f(x) = 4x² - 16x + 21

To find the vertex form, we need to complete the square. The formula for completing the square is (x - h)² = x² - 2hx + h².

f(x) = 4(x² - 4x) + 21

= 4(x² - 4x + 4 - 4) + 21

= 4((x - 2)² - 4) + 21

= 4(x - 2)² - 16 + 21

= 4(x - 2)² + 5

From this, we can see that a = 4, h = 2, and k = 5.

Therefore, the vertex form of f(x) = 4x² - 16x + 21 is f(x) = 4(x - 2)² + 5.

2. f(x) = -x² - 6x - 9

Following the same process, we complete the square:

f(x) = -(x² + 6x) - 9

= -(x² + 6x + 9 - 9) - 9

= -(x + 3)² + 9 - 9

= -(x + 3)²

Here, we can see that a = -1, h = -3, and k = 0.

Therefore, the vertex form of f(x) = -x² - 6x - 9 is f(x) = -(x + 3)².

3. y = -½ x² - x + 1

Let's complete the square:

y = -½(x² + 2x) + 1

= -½(x² + 2x + 1 - 1) + 1

= -½((x + 1)² - 1) + 1

= -½(x + 1)² + ½ + 1

= -½(x + 1)² + 1½

In this case, a = -½, h = -1, and k = 1½.

Therefore, the vertex form of y = -½ x² - x + 1 is y = -½(x + 1)² + 1½.

These are the transformations of the given quadratic functions into the vertex form f(x) = a(x - h)² + k, along with the values of a, h, and k.