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Find the y-intercept and the horizontal asymptote of the following exponential function: 𝑓(𝑥)=1(3)𝑥+6  Group of answer choicesy-intercept: (0,9); horizontal asymptote: y=3y-intercept: (0,7); horizontal asymptote: y=6y-intercept: (0,4); horizontal asymptote: y=6y-intercept: (0,4); horizontal asymptote: y=1y-intercept: (0,6); horizontal asymptote: y=7

Question

Find the y-intercept and the horizontal asymptote of the following exponential function: 𝑓(𝑥)=1(3)𝑥+6  Group of answer choicesy-intercept: (0,9); horizontal asymptote: y=3y-intercept: (0,7); horizontal asymptote: y=6y-intercept: (0,4); horizontal asymptote: y=6y-intercept: (0,4); horizontal asymptote: y=1y-intercept: (0,6); horizontal asymptote: y=7

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Solution

Para encontrar la intersección con el eje y y la asíntota horizontal de la función exponencial f(x)=1(3)x+6 f(x) = 1(3)^x + 6 , sigamos los siguientes pasos:

  1. Intersección con el eje y:

    • La intersección con el eje y se encuentra evaluando la función en x=0 x = 0 .
    • f(0)=1(3)0+6 f(0) = 1(3)^0 + 6
    • f(0)=1(1)+6 f(0) = 1(1) + 6
    • f(0)=1+6 f(0) = 1 + 6
    • f(0)=7 f(0) = 7
    • Por lo tanto, la intersección con el eje y es (0,7) (0, 7) .

  2. Asíntota horizontal:

    • La asíntota horizontal de una función exponencial de la forma f(x)=a(b)x+c f(x) = a(b)^x + c es y=c y = c .
    • En este caso, c=6 c = 6 .
    • Por lo tanto, la asíntota horizontal es y=6 y = 6 .

Entonces, la respuesta correcta es:

  • Intersección con el eje y: (0,7) (0, 7)
  • Asíntota horizontal: y=6 y = 6

La opción correcta es: y-intercept: (0,7); horizontal asymptote: y=6

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