Question

write an equation for the parabola that has the given vertex and passes through the given point.
vertex
(-1, 5)
point
(2, -13)
f(x) = ?(x + )² +

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given vertex \((-1, 5)\), so \( h = -1 \), \( k = 5 \). Substituting into the formula, we get \( f(x) = a(x - (-1))^2 + 5 = a(x + 1)^2 + 5 \).

Step2: Substitute the point into the equation

The parabola passes through \((2, -13)\). Substitute \( x = 2 \), \( f(x) = -13 \) into \( f(x) = a(x + 1)^2 + 5 \):
\[
-13 = a(2 + 1)^2 + 5
\]

Step3: Solve for \( a \)

Simplify the equation:
\[
-13 = 9a + 5
\]
Subtract 5 from both sides:
\[
-18 = 9a
\]
Divide both sides by 9:
\[
a = -2
\]

Step4: Write the final equation

Substitute \( a = -2 \), \( h = -1 \), \( k = 5 \) into the vertex form: \( f(x) = -2(x + 1)^2 + 5 \).

Answer:

\( f(x) = -2(x + 1)^2 + 5 \) (So the values are -2, 1, 5 respectively for the boxes)