Question

Find the coordinates of the vertex and write the equation of the axis of symmetry. $$y=-x^{2}+4 x+16$$

Short answer

The coordinates of the vertex are (2,12) and the equation of the axis of symmetry is \(x = 2.\)

Step-by-step solution

Identify a, b, and c from the quadratic equation

Our standard form equation \(y=ax^{2}+bx+c\) is given as \(y=-x^{2}+4x+16\), where \(a = -1\), \(b = 4\) and \(c = 16\).

Find the x-coordinate of the vertex

The formula for the x-coordinate of the vertex (h) is \(h = -\frac{b}{2a}\). Substituting in our identified \(a\) and \(b\) values, we calculate \(h = -\frac{4}{2(-1)} = 2\).

Find the y-coordinate of the vertex

Substitute \(h = 2\) into the equation to calculate the y-coordinate of the vertex (k). So, \(k = -2^{2}+4(2)+16 = 12.\)

Write the equation of the axis of symmetry

The equation of the axis of symmetry is in the form \(x = h\). Since \(h = 2\), the equation of the axis of symmetry is \(x = 2\).