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Question

As long as the coefficients a and b are nonzero, the parabolic graph y = ax^2 + bx has two x-intercepts. Without using the quadratic formula, show what they are. Use them to find the axis of symmetry for this parabola.

Answer 1

X intercepts when y=0

so ax² + bx = 0. So x(ax+b) = 0

so x = 0

ax + b = 0. So x = -b/a

so two x intercepts are 0, -b/a

on a parabola the axis of symmetry is the vertical line through the vertex

the x coordinate of the line is the mean of the two x intercepts

so x = (0 + -b/a) / 2 = -b/2a

Answer 2

The X intercepts occur when Y = 0. Let Y = 0 and solve the equation for X.

X ( aX + b) = 0

there are two roots, X = 0 and X = - b / a

The axis of symmetry in this case lies half-way between X = 0 and X = -b/a on the X axis or at X = - b / 2a.