Welche quadratische Funktion wird unter Verwendung eines Directrix von y = -2 und eines Fokus von (1, 6) erzeugt? ein. f (x) = (1 / 8) (x-1) ^ 2 - 2 b. f (x) = - (1 / 8) (x + 1) ^ 2 - 2 c. f (x) = - (1 / 16) (x + 1) ^ 2 - 2 d. f (x) = (1 / 16) (x-1) ^ 2 + 2

Antworten:

$f (x) = \frac{1}{16} {(x - 1)}^{2} + 2 \to (d)$ $f(x)=1/16(x-1)^2+2to(d)$

Erläuterung:

$from any point (x, y) on the parabola the focus and$ $"from any point "(x,y)" on the parabola the focus and"$
$directrix are equidistant$ $"directrix are equidistant"$

$using the distance formula$ $"using the "color(blue)"distance formula"$

$\sqrt{{(x - 1)}^{2} + {(y - 6)}^{2}} = | y + 2 |$ $sqrt((x-1)^2+(y-6)^2)=|y+2|$

$squaring both sides$ $color(blue)"squaring both sides"$

${(x - 1)}^{2} + {(y - 6)}^{2} = {(y + 2)}^{2}$ $(x-1)^2+(y-6)^2=(y+2)^2$

${(y - 6)}^{2} - {(y + 2)}^{2} = - {(x - 1)}^{2}$ $(y-6)^2-(y+2)^2=-(x-1)^2$

$cancel(y^2)-12y+36cancel(-y^2)-4y-4=-(x-1)^2$

$-16y+32=-(x-1)^2$

$-16y=-(x-1)^2-32$

$rArry=f(x)=1/16(x-1)^2+2to(d)$