 Another method of writing an equation in vertex form is to complete the square  If you have an equation in the form h = -2.25x x , where ‘h’ is height, how do you find the maximum height?  1 st, factor out -2.25:  h = -2.25(x 2 - 2x – 3)  We need the equation in vertex form, so try finding a perfect square, or make one

 h = -2.25(x 2 - 2x – 3)  When figuring out what to factor out, consider the first 2 terms (i.e. x 2 – 2x)  h = -2.25((x – 1) 2 – 4)  Now expand the expression:  h = -2.25(x – 1) 2 – (-2.25)(4)  h = -2.25(x – 1)  The vertex is (1, 9), therefore, the maximum height is 9 meters.

 Write y = x 2 + 6x + 2 in vertex form, and then graph the relation.  Notice that the equation can’t be factored normally, so let’s try to complete the square  y = x 2 + 6x + 2 – let’s try (x + 3) 2 because the first 2 terms are x 2 + 6x  (x + 3) 2 = x 2 + 6x + 9, but our last term is +2, so we need to subtract 7 to make the expressions match  y = (x + 3) 2 - 7

 Therefore, the vertex is at (-3, -7).  Since a > 0, the parabola opens upwards  The equation of axis of symmetry is x = -3  The y-intercept is 2 (set x = 0)

 Carrie’s diving platform is 6 ft above the water. One of her dives can be modeled by the equation d = x 2 – 7x + 6, where d is her position relative to the surface of the water and x is her horizontal distance from the platform. How deep did Carrie go before coming back up to the surface?

 d = x 2 – 7x + 6  This looks more strange than the others, but it’s the same. We are dealing with x 2 – 7x  Try (x – 3.5) 2 = x 2 – 7x  We need +6, not , so we need to subtract 6.25 from the perfect square to make the 2 expressions equal  d = (x – 3.5) 2 – 6.25  The vertex is (3.5, 6.25), so Carrie dove to a depth of 6.25 ft before turning back.

 A football’s height h after t seconds is:  h = -4.9t t  h = -4.9(t 2 – 2.4t – 0.29)  For (t 2 – 2.4), try (t – 1.2) 2  (t – 1.2) 2 = t 2 – 2.4t  We need not so we need to subtract 1.73 from (t – 1.2) 2 to make the expressions equal  h = -4.9((t – 1.2) 2 – 1.73)  h = -4.9(t – 1.2) 2 – 8.48  Therefore, the football reached a maximum height of 8.48m after 1.2 seconds.

 A quadratic relation in standard form, y = ax 2 + bx + c an be rewritten in its equivalent vertex form, y = a(x – h) 2 + k, by creating a perfect square within the expression and then factoring it  This technique is called completing the square