Find the coordinates of the vertex for the parabola d f(x)=-7(x-2)^2-8 3) f(x)=-x^2-4x-9

To find the coordinates of the vertex for each parabola, we can use different approaches depending on the form of the quadratic function.<br /><br />### 2) \( f(x) = -7(x-2)^2 - 8 \)<br /><br />This equation is in the vertex form of a parabola, which is given by:<br /><br />\[ f(x) = a(x-h)^2 + k \]<br /><br />where \((h, k)\) are the coordinates of the vertex. In this case, comparing with the given function:<br /><br />- \(a = -7\)<br />- \(h = 2\)<br />- \(k = -8\)<br /><br />Thus, the vertex of the parabola is at \((h, k) = (2, -8)\).<br /><br />### 3) \( f(x) = -x^2 - 4x - 9 \)<br /><br />This equation is in the standard form of a quadratic function:<br /><br />\[ f(x) = ax^2 + bx + c \]<br /><br />where \(a = -1\), \(b = -4\), and \(c = -9\). The vertex can be found using the formula for the x-coordinate of the vertex:<br /><br />\[ x = -\frac{b}{2a} \]<br /><br />Substituting the values of \(a\) and \(b\):<br /><br />\[ x = -\frac{-4}{2(-1)} = \frac{4}{-2} = -2 \]<br /><br />Now, substitute \(x = -2\) back into the original function to find the y-coordinate:<br /><br />\[ f(-2) = -(-2)^2 - 4(-2) - 9 \]<br />\[ f(-2) = -4 + 8 - 9 \]<br />\[ f(-2) = -5 \]<br /><br />Therefore, the vertex of the parabola is at \((-2, -5)\).<br /><br />In summary:<br />- The vertex of the first parabola is \((2, -8)\).<br />- The vertex of the second parabola is \((-2, -5)\).