Graphs of Rational Functions

Vertical Asymptotes and Holes

100

Find the vertical asymptote and hole of the function, if it exists:

f(x) = 2/(x-3)

.

vertical asymptote: x = 3

Hole: none/ no hay

200

Find the vertical asymptote and hole of the function, if it exists:

f(x) = (x + 5)/(2x + 1)

 x = -1/2 

vertical asymptote

Hole: none/ no hay

300

Find the vertical asymptote and hole of the function, if it exists:

f(x) = x^2/(3x - 6)

 vertical asymptote: 

x=2

Hole: none/ no hay

400

Find the vertical asymptote and hole of the function, if it exists:

f(x) = (x^2 - 4)/(x^2 +x - 2)

vertical asymptote: x = 1

Hole: 

(-2, 1.33)

500

Find the vertical asymptote and hole of the function, if it exists:

f(x) = (2x + 6)/ (x^2 +5x+6)

vertical asymptote:  x=-2 

Hole: 

(-3,-2)

Horizontal Asymptotes

100

What is the horizontal asymptote of the function

f(x) = (x - 3)/(x^2 + 4)

y = 0

200

What is the horizontal asymptote of the function

f(x) = - 1/(x -3)

y=0

300

What is the horizontal asymptote of the function

f(x) = (2x - 1)/(x -3)

y = 2

400

What is the vertical asymptote of the function

f(x) = (x^3 + 2x^2 + 3)/(2x^3 + 1)

y=1/2

500

What is the horizontal asymptote of the function:

f(x) = (x^3 + 3x^2 - 2)/(2x - 3)

Does not exist

No existe

Projectile Motion

100

The height 𝒉 (in meters) of a thrown pop-up baseball can be modeled by the quadratic function: La altura 𝒉 (en metros) de una pelota de béisbol tirada se puede modelar por la función cuadrática:

h(x)= −4.9x^2+30x+1.5

where 𝒙 (in seconds) is the time passed since being thrown. donde 𝒙 (en segundos) es tiempo que ha pasado desde que se tiró la pelota.

What is the initial height? ¿Qué es la altura inicial?

1.5 meters/metros

200

The height 𝒉 (in meters) of a thrown pop-up baseball can be modeled by the quadratic function: La altura 𝒉 (en metros) de una pelota de béisbol tirada se puede modelar por la función cuadrática:

h(x)= −4.9x^2+30x+1.5

where 𝒙 (in seconds) is the time passed since being thrown. donde 𝒙 (en segundos) es tiempo que ha pasado desde que se tiró la pelota.

What is the vertex of the parabola? ¿Qué es el vértice?

(3.061, 47.42)

300

The height 𝒉 (in meters) of a thrown pop-up baseball can be modeled by the quadratic function: La altura 𝒉 (en metros) de una pelota de béisbol tirada se puede modelar por la función cuadrática:

h(x)= −4.9x^2+30x+1.5

where 𝒙 (in seconds) is the time passed since being thrown. donde 𝒙 (en segundos) es tiempo que ha pasado desde que se tiró la pelota.

What is the maximum height of the baseball? ¿Qué es la altura máxima de la pelota?

47.42 meters

y-value of vertex

400

The height 𝒉 (in meters) of a thrown pop-up baseball can be modeled by the quadratic function: La altura 𝒉 (en metros) de una pelota de béisbol tirada se puede modelar por la función cuadrática:

h(x)= −4.9x^2+30x+1.5

where 𝒙 (in seconds) is the time passed since being thrown. donde 𝒙 (en segundos) es tiempo que ha pasado desde que se tiró la pelota.

At what time does the baseball reach its maximum height? ¿A qué tiempo alcanza la pelota su altura máxima?

3.061 seconds

x-value of vertex

500

The height 𝒉 (in meters) of a thrown pop-up baseball can be modeled by the quadratic function: La altura 𝒉 (en metros) de una pelota de béisbol tirada se puede modelar por la función cuadrática:

h(x)= −4.9x^2+30x+1.5

where 𝒙 (in seconds) is the time passed since being thrown. donde 𝒙 (en segundos) es tiempo que ha pasado desde que se tiró la pelota.

At what time does the baseball hit the ground? ¿A qué tiempo choca la pelota con el suelo?

6.172 seconds

positive x-intercept

Domain and Range

100

What is the domain of the function:

f(x) = 2/(x-3)

(-infty, 3)cup(3,infty)

200

What is the domain of

f(x)=(x+5)/(2x+1)

(-infty, -1/2)cup(-1/2,infty)

300

What is the domain of

f(x) = (x^2 - 4)/(x^2 +x - 2)

(-infty, -2)cup(-2,1)cup(1,infty)

400

What is the range of

f(x) = (x+5)/(2x+1)

(-infty,1/2)cup(1/2,infty)

500

What is the range of

f(x) = (x^2 - 4)/(x^2 +x - 2)

(-infty, 1)cup(1, 1.33)cup(1.33, infty)

Intercepts

100

What is the y-intercept of the function

f(x) = (x + 5)/(2x + 1)

(0, 5)

200

What is the y-intercept of the function

f(x) = (2x + 6)/ (x^2 +5x+6)

(0,1)

300

What is the x-intercept of the function:

f(x) = (x + 5)/(2x + 1)

(-5, 0)

400

What is the x-intercept of the function

f(x) = 3/(x + 5)

There is no x-intercept.

No hay intercepto del eje x.

500

This is the x-intercept of the function

f(x) = (x^2 - 4)/(x^2 +x - 2)

(2,0)