Question 526824
Find the vertex, Line of symmetry (LOS), Maximum (or minimum).
{{{f(x) = x^2-12x-5}}}
1) The x-coordinate of the vertex is given by:
{{{x = -b/2a}}} The a and b come from the general form of the quadratic equation: {{{f(x) = ax^2+bx+c}}}, so a = 1 and b = -12.
{{{x = -(-12)/2(1)}}}
{{{x = 6}}} The y-coordinate is found by substituting this value of into the given quadratic equation and solving for y, thus: (Change f(x) to y).
{{{y = x^2-12x-5}}} Substitute x = 6.
{{{y = (6)^2-12(6)-5}}} Evaluate.
{{{y = 36-72-6}}}
{{{y = -41}}}
The vertex is at (6, -41)
The equation of the line of symmetry is simply:
{{{x = 6}}}
This equation, when graphed, will show a parabola that opens upwards and therefore its vertex will be at its minimum point.
You can tell which way the parabola opens by inspecting the coefficient of the {{{x^2}}} term (a):
If a is positive, the parabola opens upwards.
If a is negative, the parabola opens downwards. Here's the graph.
{{{graph(400,400,-5,15,-45,5,x^2-12x-5)}}}