Question

write an equation for the parabola that has the given vertex and passes through the given point.
vertex
(3, -8)
point
(6, 19)
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 \((3,-8)\), so \(h = 3\), \(k=-8\). Then the equation becomes \( f(x)=a(x - 3)^2-8 \), which can be rewritten as \( f(x)=a(x+(- 3))^2+(-8) \).

Step2: Substitute the point \((6,19)\) into the equation

Substitute \(x = 6\) and \(f(x)=19\) into \( f(x)=a(x - 3)^2-8 \):
\( 19=a(6 - 3)^2-8 \)

Step3: Solve for \(a\)

Simplify the equation:
\( 19=a(3)^2-8 \)
\( 19 = 9a-8 \)
Add 8 to both sides:
\( 19 + 8=9a \)
\( 27 = 9a \)
Divide both sides by 9:
\( a = 3 \)

Step4: Write the equation

Substitute \(a = 3\), \(h = 3\) (so \(x+(-3)\) in the form \(x + [\ ]\)), and \(k=-8\) into the vertex form \(f(x)=a(x - h)^2 + k\) (or the given form \(f(x)=a(x + [\ ])^2+[\ ]\)):
\( f(x)=3(x+(-3))^2+(-8) \)

Answer:

\( f(x) = 3(x + (-3))^2 + (-8) \) (or simplified as \( f(x) = 3(x - 3)^2 - 8 \))