Question

Help?! No clue about functions...

Answer 1

F(x) = x² - 8x + 21

This is a parabola with minimum, or opening upward (since the coefficient of  x² is positive).

1st let's calculate the axis of symmetry: x = -b/2a = - (- 8)/2===> x= + 4

2nd plug (x= - 4) into the equation, y = 5.

Hence the coordinates of the vertex are (4,5)

Answer 2

Method 1: Using Calculus

The first method in going about this is to use Calculus, a branch of mathematics interested in the rate of change of a certain graph. In order to use Calculus, we need to understand what we are required to do when finding minimum values.

We first need to take the derivative of the function, because this becomes our tangent or slope at any arbitrary point on the graph.

$f'(x) = 2x - 8$

We now need the derivative to be equal to zero, as this is will give us a horizontal tangent, which means we are able to find the turning point of a graph.

$f'(x) = 0; 2x - 8 = 0$
$2x = 8; x = 4$

So, at x = 4, there will be a minimum, since the second derivative is clearly greater than zero. By substituting this point back to f(x), we can find a value for y, and this is our minimum point of the curve.

$f(4) = 4^{2} - 8(4) + 21$
$f(4) = 16 - 32 + 21 = 5$

Thus, we know the minimum point of this curve, f(x), is at: (4, 5)

Method 2: Vertex form

For any given quadratic, or a polynomial with degree 2, 4, 6, etc., we can easily find the minimum point, as that simply is our vertex. By converting this into vertex form, we can find the coordinates that will satisfy the vertex of this function, which becomes our minimum point.

Start by completing the square:
$f(x) = y= x^{2} - 8x + 16 - 16 + 21$
$y = x^{2} - 8x + 16 + 5$
$x^{2} - 8x + 16 = y - 5$
$(x - 4)^{2} = (y - 5)$

Thus, the vertex becomes: (4, 5) which is our lowest point (ie our minimum point)