How do I work out this problem?

Question

Complete the square of the given function, then graph the function using the technique of shifting: 𝑓(𝑥) = −3𝑥2 − 6𝑥 + 1

a. Write in Vertex form:

b. Vertex is located at:_____________ with a min/max. (circle one)

c. Is the function shifted horizontally, if so how?

d. Is the function reflected? If so on which axis?

e. Is the function shifted vertically, if so how?

Answer 1

Completing the square, we know that we will add and subtract a(b/2a)^2 = b^2/4a

Alternatively, as 6/3 = 2, we can think about x^2 + 2x + 1 being a square, so -3x^2 - 6x - 3 will be -3 * a square.

Then, f(x) = -3x^2 - 6x - 3 + 3 + 1 = -3(x+1)^2 + 4

This is vertex form; the vertex will be at (-1, 4); the maximum will be at 4 (there is a negative with the squared term).

The function is shifted 1 to the left (x+a) means a shift of a to the left; if a is negative, of course, this means the shift is right).

The function is reflected about the x-axis; there is a negative coefficient with the squared x term

It is shifted vertically up 4 (the + 4 constant term in the vertex form of the quadratic..