Question

Which representation of the function g(x) = 2x² - 20x + 42 is in vertex form? Also, identify the vertex.

a) g(x) = 2(x - 5)² + 8 at vertex (5, 8)
b) g(x) = 2(x + 5)² - 8 at vertex (-5, -8)
c) g(x) = 2(x + 5)² + 8 at vertex (-5, 8)
d) g(x) = 2(x - 5)² - 8 at vertex (5, -8)

Answer 1

The quadratic function g(x) = 2x² - 20x + 42 can be rewritten in vertex form by completing the square, giving us the function 2(x - 5)² + 8 with its vertex at (5, 8), corresponding to option (a).

Step-by-step explanation:

The function g(x) = 2x² - 20x + 42 can be rewritten in vertex form, which is g(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

To convert the given quadratic function into vertex form, we complete the square:

• Factor out the coefficient of x² from the first two terms: 2(x² - 10x).
• Complete the square inside the parentheses: x² - 10x + 25 is a perfect square trinomial ((x - 5)²).
• Add 25 inside the parentheses and subtract 2 × 25 outside to balance the equation: 2((x - 5)² - 25) + 42.
• Simplify to get the vertex form: 2(x - 5)² + 8.

Therefore, the function in vertex form is 2(x - 5)² + 8 and the vertex is at (5, 8), matching the option (a).