Final answer:
Two quadratic functions sharing x-intercepts (-1,0) and (1,0) are y = x^2 - 1 for the upward opening parabola and y = -x^2 + 1 for the downward opening parabola.
Step-by-step explanation:
To find two quadratic functions, one that opens upward and one that opens downward, with the given x-intercepts (-1,0) and (1,0), we can use the factored form of a quadratic equation, which is y = a(x - r)(x - s), where r and s are the roots of the equation, and a is a nonzero coefficient that determines the direction of the parabola.
In this case, the roots are -1 and 1. For the upward opening parabola, we choose a positive value for a, and for the downward opening parabola, we choose a negative value for a.
An example of a quadratic function that opens upwards could be y = (x + 1)(x - 1).
Expanding this, we get y = x2 - 1.
This parabola opens upwards because the coefficient of x2 (which is 1) is positive.
An example of a quadratic function that opens downwards could be y = -1(x + 1)(x - 1).
Expanding this, we get y = -x2 + 1.
This parabola opens downwards because the coefficient of x2 (which is -1) is negative.