2. To find the x coordinate of the vertex, use the equation Then substitute the value of x back into the equation of the parabola and solve for y. You are given the equation y=-x 2 + 4x –1. Find the coordinates of the vertex. x b a  2 x  2  ab14, (  x  4 21)() 2 yxx  41 y  481y  3 y  – 2 2421()() The coordinates of the vertex are (2,3) Substitute and solve for y

3. Standard Form Features Let f(x) = ax 2 + bx + c, where a ≠ 0. If |a| > 1, the parabola becomes narrower. If |a| < 1, the parabola becomes wider. The axis of symmetry is x = - b / 2a and passes through the graph’s vertex. The vertex has the coordinates (- b / 2a, f(- b / 2a )). (0, c) is the y-intercept for the graph.

4.  Choose two values of x that are to the right or left of the x-coordinate of the vertex.  Substitute those values in the equation and solve for y.  Graph the points. (Keep in mind the value of a as this will help you determine which way the graph opens.)  Since a parabola is symmetric about the vertical line through the vertex, you can plot mirror image points with the same y-values on the “other side” of the parabola. xy = -x 2 + 4x – 1y 1 y = -(1) 2 + 4(1) – 1 y = -1 +4 – 1 2 y = -(-1) 2 + 4(-1) – 1 y = -1 – 4 – 1 -6 You can check these solutions by comparing the y value with the y-value for the symmetric x-values (1  3, -1  5) (3,2), (5,-6)

5. x y Plot the vertex and the points from your table of values: (2,3), (1,2), (-1,-6). Use the symmetry of parabolas to plot two more points on the “other side” of the graph. The point (1,2) is one unit away from the line of symmetry, so we can also plot the point (3,2). The point (-1,-6) is three units away from the line of symmetry, so we can also plot the point (5,-6). Sketch in the parabola.

6. Korean terms / US terms Standard Form / Vertex Form 기본형 y = a(x-h) 2 + k (h,k) is the vertex General Form / Standard Form 일반형 y=ax 2 +bx + c (0,c) is the y-intercept Forms

7. Example Given the equation f(x) = - 1 / 2 (x - 1) 2 + 3, determine: Given the equation f(x) = - 1 / 2 (x - 1) 2 + 3, determine: Opens up or down? Opens up or down? Vertex: ________ Vertex: ________ Axis of Sym.: _____ Axis of Sym.: _____ Sketch the graph. Sketch the graph. a = - ½ not 1 a < 0 ( - ) means opens down 1 means over 1 up 1 -1 means over 1 down 1 over 2 down 4 over 2 down 4 - ½ means over 1 down ½ - ½ means over 1 down ½ over 2 down 2 over 2 down 2 y x 5 5 -5 (1,3) x=1

8. Standard form to Vertex form For the function f(x) = x 2 - 2x - 3, find: For the function f(x) = x 2 - 2x - 3, find: 1. X-intercepts: _______ 4. Y-intercept: ________ 5. The equation in vertex form: ____________________ 6. Opens up or down?___ 3. Vertex: _______ 2. Axis of Sym.: ______ y x 5 5 -5 (-1,0), (3,0) by solving x 2 - 2x - 3 =0 and from this we can determine the Axis of Symmetry (0,-3) the constant in standard form (0,c) y=(x-1) 2 – 4 a=1 in original equation Up a>0 (1,-4) x=1 substituting 1 for x into the equation will give us the y-value of the vertex

9. Using the Vertex A soccer goalie kicks the ball straight up with an initial velocity of 56 feet per second. The equation h = -16t 2 + 56t describes the height, h, of the ball t seconds after being kicked. A soccer goalie kicks the ball straight up with an initial velocity of 56 feet per second. The equation h = -16t 2 + 56t describes the height, h, of the ball t seconds after being kicked. How long until the ball reached its peak? How long until the ball reached its peak? What is the maximum height? What is the maximum height? How long was the ball in the air? How long was the ball in the air?

10. h = -16t 2 + 56t

11. Practice worksheets