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Objective Develop the formulas for the axis of symmetry and the vertex of a parabola, both geometrically and algebraically.

You should know by know how changing the coefficients in a quadratic function y = ax 2 + bx + c affects the shape of its graph. You should also know that the axis of symmetry is the vertical line and that this formula can be used to find the coordinates of the vertex of a parabola. The goal of this lesson is that you learn why this is true.

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

Remind students how a parabola changes shape as the coefficients of y = ax 2 + bx + c change. For example, the graph of y = ( x + d ) 2 has an axis of symmetry x = -d . This comes from the formula for the axis of symmetry. Why is this true? Notice that the value of ( x + d ) 2 is always positive or zero (since the square of a number can never be negative). That means that the smallest value of ( x + d ) 2 is zero, which occurs when x + d = 0, or x = -d . So the minimum point on the graph of y = ( x + d ) 2 occurs when x = -d and y = 0. Since the axis of symmetry goes through the vertex (which in this case is a minimum point), it therefore occurs when x = -d. • Notice that when a is a positive number, then the function y = a ( x + d ) 2 is always greater than or equal to zero. The function is equal to zero when a( x + d ) 2 = 0, which occurs when ( x + d ) = 0, or x = -d . The axis of symmetry for these functions is the vertical line through the minimum point (vertex), x = -d .

• When a is a negative number, a similar argument applies. Namely, since the value of ( x + d ) 2 is always greater than or equal to zero, then assuming a is a negative number, y = a ( x + d ) 2 is always less than or equal to zero. (Ask your students why.) This expression therefore has its maximum value at zero, which occurs when a( x + d ) 2 = 0, or x = -d . Thus, the axis of symmetry, the vertical line through the maximum point (vertex), has the equation x = -d .

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

For any nonzero number a , the graph of y = a ( x + d ) 2 has a vertex at ( - d , 0), and the axis of symmetry is the vertical line x = -d. If a > 0, the vertex is a minimum point, and if a < 0, the vertex is a maximum point.  