SOLUTION: This question is from the CEMC math competition from last year. The parabola y = -x^2 has vertex labelled P and intersects the x-axis at the points labelled A and B. The parabo

Algebra ->  Functions -> SOLUTION: This question is from the CEMC math competition from last year. The parabola y = -x^2 has vertex labelled P and intersects the x-axis at the points labelled A and B. The parabo      Log On


   

Question 1180174: This question is from the CEMC math competition from last year.
The parabola y = -x^2 has vertex labelled P and intersects the x-axis at the points labelled A and B. The parabola translated from it's original position so that its vertex moves along the line y = x+4 to the point Q. In this new position, the parabola intersects the x-axis at the points labelled B and C. What are the coordinates of C?
Please view diagram here:
https://docs.google.com/document/d/1Cl3HrcoR6jibNzxfDTIilS3imSplLY8nQUU0OpU9vLA/edit?usp=sharing
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!




For the original parabola y=-x%5E2%2B+4, the vertex is P(0,4) and the x-intercepts are A(-2,0) and B(2,0).
Let the vertex of the translated parabola be Q(q, p).
Since the new parabola is a translation of the original, the equation of this new parabola is
y=-%28x-q%29%5E2%2Bp
Since Q lies on the line y=x%2B+4, we have p=q%2B+4 and the equation of the newparabola is
y=-%28x-q%29%5E2%2Bq%2B+4
Since B( 2,0) lies on the new parabola, we can substitute (2,0) into this equation:
0+=-%282-q%29%5E2%2Bq%2B+4
0+=-%28q%5E2-4q%2B+4%29+%2Bq%2B+4
0+=-q%5E2%2B+4q-4+%2Bq%2B+4
0+=-q%5E2%2B+5q
0+=-q%28q-5%29
Therefore, q=+0 or q=+5.
The value q=+0 corresponds to point P(0,4) in theoriginal parabola. Therefore, q=+5 and the vertex of the new parabola is Q(5,9) and the equation of this parabola is
y=-%28x-5%29%5E2%2B+9
Since C is an x-intercept of this parabola, to determine C set y=+0 in the equation for the parabola and solve for x:
0+=-%28x-5%29%5E2%2B+9
%28x-5%29%5E2=+9
x-5 = ± 3
x=+8 or x=2,
x=2 corresponds to point B, and the value x=+8 corresponds topointC.
Therefore, the coordinates of C are (8,0).