Question

Use the method of completing the square to write the equation of the given parabola: (y - k) = a(x - 5)² , where (a > 0 , h, k) is the vertex, and x = h is the axis of symmetry.

a) (y + 2) = 2(x + 5)²
b) (y - 2) = 3(x + 5)²
c) (y + 2) = 3(x + 5)²
d) (y - 2) = 2(x + 5)²

Answer 1

To write the equation of the parabola using the method of completing the square, expand the right side, move terms around, factor out 'a' from the coefficients of x, and simplify the equation. The correct equation is (y - k) = a(x - 5)², and the corresponding option is b) (y - 2) = 3(x + 5)².

Step-by-step explanation:

The given equation is (y - k) = a(x - 5)². To write the equation of the parabola using the method of completing the square, we need to expand the right side and simplify the equation. Let's go through the steps:

1. Start by expanding the right side:

• (y - k) = a(x - 5)² --> (y - k) = a(x² - 10x + 25)

Expand further:

• (y - k) = ax² - 10ax + 25a

Move the term with 'y' to the left side:

• y - k - ax² + 10ax - 25a = 0

Move the constant term (- k - 25a) to the right side:

• y - ax² + 10ax = 25a + k

Factor out 'a' from the coefficients of x:

• y - a(x² - 10x) = 25a + k

Complete the square:

• y - a(x² - 10x + 25) = 25a + k + a(25)
• y - a(x - 5)² = 25a + k + 25a
• y - a(x - 5)² = 50a + k

So, the equation of the given parabola is (y - k) = a(x - 5)², and the correct option is b) (y - 2) = 3(x + 5)².