Question

The graph `y=(x^2+5x+7)/(x+2)` is shown below. Find the value of the product `abcd`?

The dashed vertical line through D (d,0) is a vertical asymptote. The dashed line BC is a slant asymptote. The coordinates of B and C are B(0,b) and C(c,0). The curve crosses the y-axis at A(0,a).

Answer 1

`abcd=63`

Explanation:

The question asks us to find various properties of the curve and combine them.

1) Vertical asymptote, `d`

Find the vertical asymptote by setting the denominator to 0:
`x+2=0rArrx=-2rArrd=-2`.

Check that the numerator is not also 0 at the same point:
`(-2)^2+5*(-2)+7=4-10+7=1!=0`
So this point is not a function hole.

2) Curve y-intercept, `a`

We seek the function value `y(0)`.

`y(0)=7/2rArra=7/2`.

3) Slant intercept

Use polynomial long division to express `y(x)` as a sum of terms.

Divide leading terms: `x` goes into `x^2` `x` times. So subtract off `x(x+2)=x^2+2x`:
`x^2+5x+7-x^2-2x=3x+7`
Repeat: `x` goes into `3x` `3` times. So subtract off `3(x+2)=3x+6`:
`3x+7-3x-6=1`

So `y(x)=(x^2+5x+7)/(x+2)=x+3+1/(x+2)`, an identity which you can confirm by multiplying back out.

We know from this that the slant asymptote line is `y=x+3` - the fractional term goes to zero at the infinities.

4) Slant intercept `x`- and `y`-intercepts, `c` and `b`

For the `x`-intercept, `y=0`, so
`0=x+3rArrx=-3rArrc=-3`

For the `y`-intercept, `x=0`, so
`y=0+3rArry=3rArrb=3`

5) Combine results

From above, `a=7/2`, `b=3`, `c=-3`, and `d=-2`.

So `abcd=7/2*3*(-3)*(-2)=63`.