Consider the following graph of a quadratic function.
Write the equation for the quadratic function in standard form.
The standard form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants.
Looking at the graph, we can see that the vertex is at (-1, -3) and the y-intercept is at (0, -1).
Since the vertex is (-1, -3), we can substitute this point into the equation:
-3 = a(-1)^2 + b(-1) + c
-3 = a + b + c (equation 1)
Since the y-intercept is (0, -1), we can substitute this point into the equation:
-1 = a(0)^2 + b(0) + c
-1 = c (equation 2)
From equation 2, we find that c = -1. Now substitute c = -1 into equation 1:
-3 = a + b - 1
a + b = 2
a = 2 - b (equation 3)
Substitute equation 2 and 3 into the standard form of the quadratic function:
y = a*x^2 + b*x + c
y = (2 - b)*x^2 + b*x - 1
y = 2x^2 - bx + b*x - 1
y = 2x^2 + (b - 1)*x - 1
Therefore, the equation for the quadratic function in standard form is y = 2x^2 + (b - 1)*x - 1.