Answer:
1) it's a circle, (4x + 12)² + (2y - 2)² = 64
2) it's a circle, (x + 4)² + (y - 3)² = 40
3) it's a parabola, (x + 3)² = -4(y - 1)
Step-by-step explanation:
<u>Equation of a circle: (x - h)² + (y - k)² = r²</u>
factorize the equation:
1) => 16x² + 96x + <u>144 </u>+ 4y² - 8y + <u>4 </u>+ 84 = <u>144 </u>+ <u>4</u>
=> (4x + 12)² + (2y - 2)² = -84 + 144 + 4
=> (4x + 12)² + (2y - 2)² = 64 <em>(equation of the circle)</em>
Graph the circle using the radius and the center.(& compass for drawing circles)
=>(4x - (-12))² + (2y - 2)² = 8²
(x - h )² + (y - k)² = r²
<em>Center:</em> (h, k) => (-12, 2)
<em>Radius:</em> √r² => √64 = 8
2) => x² + 8x + <u>16</u> + y² - 6y + <u>9</u> - 15 = <u>16</u> + <u>9</u>
=> (x + 4)² + (y - 3)² = 15 + 16 + 9
=> (x + 4)² + (y - 3)² = 40 <em>(equation of the circle)</em>
Graph the circle using the radius and the center.(& compass for drawing circles)
=> (x - (-4) )² + (y - 3)² = 40
(x - h )² + (y - k)² = r²
<em>Center:</em> (h, k) => (-4, 3)
<em>Radius:</em> √r² => √40 ≈ 6.32
<u>Equation of a parabola facing down: (x - h)² = 4a(y - k)</u>
3) => x + 6x + 9 = -4y - 5 + 9
=> (x + 3)² = -4y + 4
=> (x + 3)² = -4(y - 1) <em>(equation of the parabola)</em>
Graph the parabola using the direction, vertex, focus, and axis of symmetry.
Direction: Opens Down
=> (x - (-3))² = 4 × -1(y - 1)
(x - h )² = 4 × a(y - k)
Vertex: (h, k) => (−3, 1)
Focus: (h, k + a) => (-3, 1 + (-1)) => (−3, 0)
Axis of Symmetry: x = h => x = −3
Directrix: y = k - a => y = 1 - (-1) => y = 2
<u> x | y </u>
−5 | 0
−4 | 3/4
−3 | 1
−2 | 3/4
−1 | 0
<u><em>(graphs in the pictures below)</em></u>