Write the equation of the quadratic function given vertex V and point P. which lies on the function. Write the equation in vertex form f(x)=a(x-h)^2+k V(-9,9) P(4,-498)

To write the equation of the quadratic function in vertex form given the vertex \( V(-9, 9) \) and a point \( P(4, -498) \) that lies on the function, we start with the vertex form of a quadratic function:<br /><br />\[ f(x) = a(x - h)^2 + k \]<br /><br />Here, \( (h, k) \) is the vertex of the parabola. Given the vertex \( V(-9, 9) \), we have:<br /><br />\[ f(x) = a(x + 9)^2 + 9 \]<br /><br />Next, we use the point \( P(4, -498) \) to find the value of \( a \). We substitute \( x = 4 \) and \( f(x) = -498 \) into the equation:<br /><br />\[ -498 = a(4 + 9)^2 + 9 \]<br /><br />Simplify inside the parentheses:<br /><br />\[ -498 = a(13)^2 + 9 \]<br /><br />\[ -498 = 169a + 9 \]<br /><br />Now, solve for \( a \):<br /><br />\[ -498 - 9 = 169a \]<br /><br />\[ -507 = 169a \]<br /><br />\[ a = \frac{-507}{169} \]<br /><br />\[ a = -3 \]<br /><br />Now that we have the value of \( a \), we can write the equation of the quadratic function in vertex form:<br /><br />\[ f(x) = -3(x + 9)^2 + 9 \]<br /><br />So, the equation of the quadratic function in vertex form is:<br /><br />\[ f(x) = -3(x + 9)^2 + 9 \]