Question

write an equation for the parabola that has the given vertex and passes through the given point.

vertex
(2, -7)

point
(0, 5)

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 \((2,-7)\), so \(h = 2\), \(k=-7\). Substitute into the formula: \( f(x)=a(x - 2)^2 - 7 \).

Step2: Substitute the point \((0,5)\)

We know the parabola passes through \((0,5)\), so substitute \(x = 0\), \(f(x)=5\) into \( f(x)=a(x - 2)^2 - 7 \):
\( 5=a(0 - 2)^2 - 7 \)
Simplify: \( 5 = 4a - 7 \)

Step3: Solve for \(a\)

Add 7 to both sides: \( 5 + 7=4a \) → \( 12 = 4a \)
Divide by 4: \( a = 3 \)

Step4: Write the equation

Substitute \(a = 3\), \(h = 2\), \(k=-7\) into vertex form. Note that \(x - 2=x+(-2)\), so the equation is \( f(x)=3(x - 2)^2 - 7 \), which matches the form \( f(x)=a(x + \square)^2 + \square \) as \(x - 2=x+(-2)\), \(a = 3\), and the constant term is \(-7\).

Answer:

\( f(x) = \boldsymbol{3}(x + \boldsymbol{(-2)})^2 + \boldsymbol{(-7)} \) (or simplified as \( f(x) = 3(x - 2)^2 - 7 \))