Asked by Jayla Watson on Mar 10, 2024
Verified
Identify the vertex and focus of the parabola (x−10) 2+36(y+6) =0( x - 10 ) ^ { 2 } + 36 ( y + 6 ) = 0(x−10) 2+36(y+6) =0 .
A) vertex: (10,−6) ( 10 , - 6 ) (10,−6) focus: (−154,−6) ( - 154 , - 6 ) (−154,−6)
B) vertex: (10,−6) ( 10 , - 6 ) (10,−6) focus: (10,−15) ( 10 , - 15 ) (10,−15)
C) vertex: (10,−6) ( 10 , - 6 ) (10,−6) focus: (154,−6) ( 154 , - 6 ) (154,−6)
D) vertex: (−10,6) ( - 10,6 ) (−10,6) focus: (−10,−15) ( - 10 , - 15 ) (−10,−15)
E) vertex: (−10,6) ( - 10,6 ) (−10,6) focus: (−154,6) ( - 154,6 ) (−154,6)
Vertex
A point where two or more curves, lines, or edges meet. In the context of geometry, it is often used to refer to the corner point of a polygon or the apex of a cone or pyramid.
Parabola
The graph of a quadratic function, a U-shaped curve that can open upwards or downwards, determined by the function's coefficients.
Focus
A point used in the definitions of conic sections, such as ellipses and hyperbolas, that helps to define their shapes.
- Determine the specific features of parabolas, including their vertex and focus.
Verified Answer
JC
Jacob CaudillMar 10, 2024
Final Answer :
B
Explanation :
The given equation can be rewritten in the form (x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)(x−h)2=4p(y−k) to identify the vertex (h,k)(h, k)(h,k) and the focus (h,k+p)(h, k + p)(h,k+p) . Here, h=10h = 10h=10 , k=−6k = -6k=−6 , and 4p=364p = 364p=36 , so p=9p = 9p=9 . Thus, the vertex is (10,−6)(10, -6)(10,−6) , and the focus, found by adding ppp to the yyy -coordinate of the vertex, is (10,−6+9)=(10,−15)(10, -6 + 9) = (10, -15)(10,−6+9)=(10,−15) .
Learning Objectives
- Determine the specific features of parabolas, including their vertex and focus.