Define a quadratic function $$ y=f(x) $$ that satisfies the given conditions. Vertex $$ (-4,5) $$ and passes through $$ (0,-27) $$.

Question

UpStudy AI · Accepted Answer

1. The vertex form of a quadratic function is given by  
   $$
   y = a(x - h)^2 + k,
   $$  
   where $$(h,k)$$ is the vertex.

2. Given the vertex $$(-4,5)$$, substitute $$h = -4$$ and $$k = 5$$:
   $$
   y = a(x - (-4))^2 + 5 = a(x+4)^2 + 5.
   $$

3. The function passes through the point $$(0,-27)$$. Substitute $$x=0$$ and $$y=-27$$ into the equation:
   $$
   -27 = a(0+4)^2 + 5.
   $$

4. Simplify the equation:
   $$
   -27 = a(4)^2 + 5,
   $$
   $$
   -27 = 16a + 5.
   $$

5. Solve for $$a$$:
   $$
   16a = -27 - 5 = -32,
   $$
   $$
   a = \frac{-32}{16} = -2.
   $$

6. Substitute $$a = -2$$ back into the vertex form of the quadratic:
   $$
   y = -2(x+4)^2 + 5.
   $$

Thus, the quadratic function that satisfies the given conditions is  
$$
y = -2(x+4)^2 + 5.
$$
The quadratic function is $$ y = -2(x + 4)^2 + 5 $$.

Define a quadratic function \( y=f(x) \) that satisfies the given conditions. Vertex \( (-4,5) \) and passes through \( (0,-27) \).