Write an equation where the y intercept=(0,1), and the x intercept is (2,0). And the vertical asymptotes are x=-1 and x=4

Answer 1

To write the equation of a rational function, the location of any vertical asymptotes will determine the linear factors that we place in the denominator (because x-values that make the denominator = 0 will create vertical asymptotes in the graph). So, we know that the denominator in the equation will be (x + 1)(x - 4).

Similarly, the location of any x-intercepts will determine the linear factors that we place in the numerator. Thus, our numerator will contain (x - 2).

Lastly, we can use an a-value as the leading coefficient, and solve for a to ensure that the y-intercept is (0,1):

f(x) = a(x - 2) / [(x + 1)(x - 4)]

f(0) = a (-2) / [(1)(-4)] = 1

1/2 · a = 1

a = 2

f(x) = 2(x - 2) / [(x + 1)(x - 4)]

You should graph this on a TI or Desmos to confirm it has the graphical features as given.

Answer 2

y intercept =1, the point (0,1)

x intercept =2, the point (2,0)

connecting line's slope = (1-0)/(0-2) = -1/2=-.5

line connecting them is y =-x/2 +1= -.5(x-2)

asymptotes x=-1 and 4

y = -.5(x-2)/(x+1) has the given intercepts and x=-1 asymptote

= -.5(x-2)/(x+1)(x-4) doesn't, adding the x-4 in denominator needs multiply numerator by -4

then 2(x-2)/(x+1)(x-4) has the given intercepts and both asymptotes

check results with a graphing calculator such as desmos

Answer 3

Write an equation where the y intercept=(0,1), and the x intercept is (2,0). And the vertical asymptotes are x=-1 and x=4

Rational functions with vertical asymptotes generally follow the form

y = p(x)/q(x). With the given information we can solve for p(x) and q(x).

The x intercept is telling us when x is 2 the equation evaluates to 0. This happens when the numerator, p(x), is 0.

p(x) = (x - 2) * m

(x-2) when x = 2 will always evaluate to 0 regardless of the value of m.

Vertical asymptotes are when the y-value of the function approaches infinity. This happens when you are dividing by 0, when q(x) = 0.

q(x) = (x + 1) * (x - 4) * n

(x + 1) when x = -1 will always evaluate to 0 regardless of the value of (x - 4) or n.

(x - 4) when x = 4 will always evaluate to 0 regardless of the value of (x + 1) or n.

So now we have our numerator and divisor

y = m(x - 2) / n(x + 1)(x - 4)

If you notice we have m and n floating around, since we don't have know what the value of these constants are we can simplify them into 1 variable.

k = m/n

We end up with

y = k(x - 2) / (x + 1)(x - 4)

Finally, we can use the last piece of information and plug the y intercept into the equation to solve for k. (0,1)

1 = k(0 - 2) / (0 + 1)(0 - 4)

1 = k (-2)/(-4)

-4 * 1= -2k

2 = k

With that we can complete out equation

y = 2(x - 2) / (x + 1)(x - 4)

Answer 4

https://www.desmos.com/calculator/7qzl9vgznk