Question

write an equation for the parabola that has the given vertex and passes through the given point.
vertex
(4, 3)
point
(6, -17)
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 \((4,3)\), so \(h = 4\), \(k = 3\). Substitute into the formula: \( f(x)=a(x - 4)^2 + 3 \).

Step2: Find the value of \(a\)

The parabola passes through the point \((6, -17)\). Substitute \(x = 6\) and \(f(x)= -17\) into the equation:
\[
-17=a(6 - 4)^2 + 3
\]
Simplify the right - hand side: \(6 - 4 = 2\), so \((6 - 4)^2=4\). The equation becomes \(-17 = 4a+3\).
Subtract 3 from both sides: \(-17 - 3=4a\), i.e., \(-20 = 4a\).
Divide both sides by 4: \(a=\frac{-20}{4}=-5\).

Step3: Write the final equation

Substitute \(a = - 5\), \(h = 4\) (note that in the form \(f(x)=a(x + \square)^2+\square\), \(x - 4=x+(-4)\)) and \(k = 3\) into the vertex form. So the equation is \(f(x)=-5(x - 4)^2 + 3=-5(x+(-4))^2 + 3\).

Answer:

\(f(x)=\boldsymbol{-5}(x+\boldsymbol{-4})^{2}+\boldsymbol{3}\)