A function of the form \( p(x)=a x^{2}+q \) has two \( x \)-intercepts, one of which is 9 . Determine the other \( x \)-intercept and explain how you arrived at your answer. \( p(x)=a x^{2}+q \)

To find the second \( x \)-intercept, we can use the property of the roots of a quadratic function. The function \( p(x) = a x^{2} + q \) has a parabolic shape and is symmetrical around its vertex. If one \( x \)-intercept is 9, you can express the two \( x \)-intercepts in terms of their sum. The formula for the sum of the roots \( r_1 + r_2 \) is given by \( -\frac{b}{a} \) for the general form \( ax^2 + bx + c \). In this case, since \( b = 0 \), the sum of the roots is zero. Therefore, if one intercept is 9, the other must be -9. This gives us the two intercepts as 9 and -9, ensuring that their sum equals zero, depicting the symmetry of the quadratic function.