Answer:
1. y = -(x + 1)(x - 3)
2. y = (x + 3)²
Explanation:
1.
The zeros of a function are the x-values for which its output (y-value) is 0. We are given that the zeros of the function we are solving for are -1 and 3. We can use the zero product property to show the function as a multiplication of two terms whose solutions are -1 and 3:
- x = -1 → 0 = x + 1
- x = 3 → 0 = x - 3
So, we can construct the function:
y = a(x + 1)(x - 3)
where 'a' is the proportional deviation from a standard parabola.
We can solve for 'a' by plugging the x- and y-coordinates for the given point (1, 4) into the equation we just constructed.
4 = a(1 + 1)(1 - 3)
4 = a(2)(-2)
4 = -4a
a = -1
So, the equation of a quadratic function with the zeros -1 and 3 that goes through the point (1, 4) is:
y = -(x + 1)(x - 3)
2.
We can find the equation that models the given graph using the vertex form of a parabola:
y = (x - a)^2 + b
where (a, b) is the vertex of the parabola, and the curve isn't stretched from the standard parabola (which it isn't in this case).
We can identify the vertex as (-3, 0), giving us the variable values:
Plugging these into the vertex form equation:
y = (x - (-3))² + 0
y = (x + 3)²