Find the quadratic function that has a vertex at (−2,1) and goes through the point (−1,−1). f(x)=

Question

Find the quadratic function that has a vertex at (-2,1) and goes through the point (-1,-1) .

f(x)=

Answer 1

We start with the formula for a quadratic function f(x) = ax² + bx + c

The vertex of a quadratic function is at x = -b/(2a). So a vertex at x = -2 gives us -2 = -b/(2a) ⇒ b = 4a.

Our updated quadratic equation is:

f(x) = ax² + 4ax + c

Next, a vertex at f(-2) = 1 gives us 1 = 4a - 8a + c ⇒ c = 1 + 4a

Our updated quadratic equation is now f(x) = ax² + 4ax + 4a + 1

Finally, we apply the condition f(-1) = -1 = a - 4a + 4a + 1 ⇒ a = -2

This gives us f(x) = -2x² - 8x - 7

Answer 2

For a quadratic,

y = A(x-h)²+k where (h,k) is the vertex and A is the amplitude

For the vertex and the point (-1,-1)

y = A(-1-(-2))²+ 1 = -1

A(1)² + 1 = -1

A = -2

y = -2(x+2)² +1 = -2(x²+4x+4)+1 = -2x²-8x-8+1

y = -2x²-8x-7