What is the y-coordinate of the vertex of a parabola with the following equation y = x^2 - 8x + 18?

May 13, 2018

Vertex = (4,2)

Explanation:

To find the vertex of a quadratic equation you can either use use the vertex formula or put the quadratic in vertex form:

Method 1: Vertex formula

a is the coefficient of the first term in the quadratic, b is the coefficient of the second term and c is the coefficient of the third term in the quadratic.

$V e r t e x = \left(- \frac{b}{2 a} , f \left(x\right)\right)$

In this case a = 1 and b = -8, so substituting these values into the formula above gives:

$V e r t e x = \left(- \frac{- 8}{2 \cdot 1} , f \left(- \frac{- 8}{2 \cdot 1}\right)\right)$

which becomes:

$V e r t e x = \left(4 , {4}^{2} - 8 \cdot 4 + 18\right)$

which simplifies to:

$V e r t e x = \left(4 , 2\right)$

Method 2: Vertex form

vertex form looks like this: ${\left(x - h\right)}^{2} + k$

To convert from quadratic form to vertex form substitute the variables in the next equation with the coefficients of the quadratic ${\left(x + \frac{b}{2}\right)}^{2} + c - {\left(\frac{b}{2}\right)}^{2}$

In this case b = -8 and c = 18

Substituting these variables we get

${\left(x - \frac{8}{2}\right)}^{2} + 18 - {\left(- \frac{8}{2}\right)}^{2}$

Which becomes:

${\left(x - 4\right)}^{2} + 18 - {4}^{2}$

which simplifies to:

${\left(x - 4\right)}^{2} + 2$

This is called the vertex form because the vertex can be easily found in this form.

$V e r t e x = \left(h , k\right)$

$V e r t e x = \left(4 , 2\right)$

Note: This method can be quicker than the first method but only works when the coefficient of a is 1.