Aradio tower is located on a coordinate system measured in miles. the range of a signal in a particular direction is modeled by a quadratic function where the boundary of the signal starts at the vertex at (4, 2). it passes through the point (5, 4). a linear road connects points (–3, 7) and (8, 2). which system of equations can be used to determine whether the road intersects the boundary of the tower’s signal?

We will determine a quadratic function in vertex form:
y = a ( x - 4 )² + 2
4 = a ( 5 - 4 )² + 2
4 = a + 2
a = 2
y = 2 ( x - 4 )² + 2
Then we will determine a linear function in standard form, that passes through the points ( -3, 7 ) and ( 8, 2 ).
7 = -3 m + b
2 = 8 m + b   / ·(-1) 

7 = - 3 m + b
+
- 2 = - 8 m - b

5 = - 11 m
m = -5/11,              2 = -5/11 · 8 + b,        b= 2 + 40/11,     b = 62/11
y = - 5/11 x + 62/11  /  ·11
 11 y = - 5 x + 62
 5 x + 11 y = 62
Finally the system of equations is:
y = 2 ( x - 4 )² + 2
5 x + 11 y = 62
    

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