Final answer:
To rewrite the function g(x) = 2x^2 - 8x - 24 in vertex form, factor out the leading coefficient, complete the square, and then simplify the constant terms to get g(x) = 2(x - 2)^2 - 32.
Step-by-step explanation:
To rewrite the function g(x) = 2x^2 - 8x - 24 in vertex form, we need to complete the square. First, we factor out the coefficient of the x^2 term in the quadratic part of the function:
g(x) = 2(x^2 - 4x) - 24
Next, we find a value that, when added and subtracted inside the parenthesis, completes the square:
g(x) = 2(x^2 - 4x + 4 - 4) - 24
We added and subtracted (4), which is (4/2)^2, to complete the square. Now, we express the complete square and simplify the constant terms:
g(x) = 2((x - 2)^2 - 4) - 24
g(x) = 2(x - 2)^2 - 8 - 24
Finally, we combine the constant terms to get the vertex form:
g(x) = 2(x - 2)^2 - 32
Now, the function is in vertex form g(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola, and it can be read directly from the equation as (2, -32).