Question

For the function g(x) = -3x^2 + 9x + 5, write the function in vertex form.

a) g(x) = -3(x - 3)^2 + 23
b) g(x) = -3(x + 3)^2 + 23
c) g(x) = -3(x - 3)^2 - 23
d) g(x) = -3(x + 3)^2 - 23

Answer 1

To write the function g(x) = -3x^2 + 9x + 5 in vertex form, we can follow these steps: 1) Factor out the common factor -3. 2) Complete the square to get the quadratic expression in the form (x - h)^2. 3) Simplify and rewrite the function in vertex form. The function g(x) = -3(x - 3/2)^2 + 47/4 is the vertex form of the given function.

Step-by-step explanation:

To write the function g(x) = -3x^2 + 9x + 5 in vertex form, we need to complete the square. The vertex form of a quadratic function is g(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Step 1: Factor out the common factor -3 from the first two terms: g(x) = -3(x^2 - 3x) + 5.

Step 2: Complete the square by adding and subtracting (b/2a)^2, which in this case is (3/2)^2 = 9/4: g(x) = -3(x^2 - 3x + 9/4 - 9/4) + 5.

Step 3: Factor the quadratic expression inside the parentheses and simplify: g(x) = -3((x - 3/2)^2 - 9/4) + 5.

Step 4: Distribute -3 to the terms inside the parentheses: g(x) = -3(x - 3/2)^2 + 27/4 + 5.

Step 5: Simplify: g(x) = -3(x - 3/2)^2 + 27/4 + 20/4. g(x) = -3(x - 3/2)^2 + 47/4.

Therefore, the function g(x) = -3(x - 3/2)^2 + 47/4 is in vertex form.