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## Graphs of y = a ( x + d ) 2 + k

Remember that

• when a positive number k is added to a quadratic function, y = f ( x ) + k , then the parabola shifts up k units, and

• when k is subtracted from a quadratic function, y = f ( x ) - k , then the parabola shifts down k units.

This is because if the graph of y = f ( x ) has a minimum point (the parabola opens up), then the minimum value of y = f ( x) + k is k more than the minimum value of y = f ( x ). Moreover, the minimum values of y = f ( x ) + k and y = f ( x ) occur at the same x value. So, if the graph of y = f ( x ) has a minimum point at ( x , y ), then the graph of y = f ( x ) + k has a minimum point at ( x , y + k ). Since the x value of the minimum point is unchanged, the equation of the axis of symmetry is also unchanged.

In the same way, if the quadratic function y = f ( x ) has a maximum point at ( x , y ) (the parabola opens down), then the maximum point of the graph of y = f ( x ) + k is ( x , y + k ). We can now apply this discussion to the functions of the form f ( x ) = a( x + d ) 2 that we have just been studying. The graphs of quadratic functions of the form y = f ( x ) + k or y = a ( x + d ) 2 + k have the following properties, where a is nonzero, and d and k are any numbers.

1. The axis of symmetry is the line x = -d .

2. The vertex is at ( - d , k ). If a is positive, the parabola opens up and the vertex is a minimum point. If a is negative, the parabola opens down and the vertex is a maximum point. Now notice that if we write y = a( x + d ) 2 + k in standard form y = ax 2 + bx + c , we can see how to get the formula for the axis of symmetry. Write this calculation on the chalkboard. Comparing the coefficient of x , we have Since the graph of y = a( x + d ) 2 + k has an axis of symmetry x = -d , this is equal to the equation . End this lesson by providing the following summary. Explain that any quadratic function y = ax2 + bx + c can be written in the form y = a( x + d )2 + k by a process called completing the square. Therefore, these justifications for the formulas for the axis of symmetry and the vertex work for any quadratic function.

 Function Effect on Graph Axis of Symmetry Vertex y = ax 2 Graph widens ( a < 1) or narrows ( a > 1). y = a (0) 2 = 0 Vertex is at (0, 0). Graphs opens up ( a > 0) or down ( a < 0) y = x 2 + c Graph moves up. y = (0) 2 + c = c Vertex is at (0, c ). y = x 2 - c Graph moves down. y = (0) 2 - c = -c Vertex is at (0, - c ). y = ( x + d ) 2 Graph moves left. y = ( - d + d ) 2 + 0 Vertex is at ( - d , 0). y = ( x - d ) 2 Graph moves right. y = ( d - d ) 2 + 0 Vertex is at ( d , 0). y = a( x + d ) 2 + k Width and direction of graph depend on a. x = -d y = a ( - d + d ) 2 + k= 0 + k or k Vertex is at (-d , k ). y = a( x - d ) 2 + k Width and direction of graph depend on a. x = d y = a [ d + ( - d )] 2 + k = 0 + k or k Vertex is at ( d , k ). 