(Graphing Calculator)Sketch the graph of the function

f(x)= 2x^3-7x^2-27x-18/x^2+5

Please insert adequate parentheses in the expression.

Well,

the numerator is (2x+3)(x+1)(x-6)

so the graph crosses the x-axis at -3/2, -1, and 6

As x grows large, the curve approaches the line y=2x, but is always slightly below it, since

2x^3/(x^2+5) < 2x^3/x^2

The curve crosses the line y=2x at about x = -4.75

To sketch the graph of the function f(x) = (2x^3 - 7x^2 - 27x - 18) / (x^2 + 5), we can follow certain steps:

1. Determine the vertical asymptotes: The vertical asymptotes occur where the denominator (x^2 + 5) equals zero. However, in this case, the denominator is always positive, so there are no vertical asymptotes.

2. Find the x-intercepts: Set the numerator (2x^3 - 7x^2 - 27x - 18) equal to zero and solve for x to find the x-intercepts.

2x^3 - 7x^2 - 27x - 18 = 0

You can solve this equation by factoring or using a numerical method like the Newton-Raphson method or graphical methods. Once you find the x-intercepts, let's say x1, x2, etc., they become key points on the graph.

3. Determine the horizontal asymptotes: To find the horizontal asymptotes, compare the degree of the numerator (3) and the degree of the denominator (2).
- If the degree of the numerator is greater, there is no horizontal asymptote.
- If the degrees are the same, there is a horizontal asymptote at the ratio of the leading coefficients (2/1 = 2 in this case).
- If the degree of the denominator is greater, there is a horizontal asymptote at y = 0.

In this case, the degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote.

4. Determine the behavior of the graph: You can analyze the behavior of the graph by looking at the leading coefficients and their signs. In this function, the leading coefficient is positive, so the graph will approach positive infinity as x approaches both positive and negative infinity.

5. Sketch the graph using the information above: Plot the x-intercepts found in step 2 and use them as reference points. Then, draw the curve connecting these points, taking into account the behavior of the graph mentioned in step 4.

If you have a graphing calculator, you can simply input the function f(x) = (2x^3 - 7x^2 - 27x - 18) / (x^2 + 5) and it will plot the graph for you.