Question

Consider the following function: y=3x-x²/15-8x+x² Complete each step to graph the function. Show all work at each step for credit. 1. Find the domain of the function. 2. Simplify the function if possible. 3. Find the x and y intercepts. 4. Find any odd or even symmetry. 5. Find all vertical and horizontal asymptotes. 6. Find the intervals on increase \& decrease along with any local max or min points using a sign chart. 7. Find the intervals of concavity along with any inflection points using a sign chart. 4. Sketch the function showing all details.

Answer 1

To graph the function, start by determining its domain, which is all real numbers, then simplify the expression. Identify the x and y intercepts, symmetry, and asymptotes. Next, use calculus to find intervals of increase, decrease, and concavity and sketch the graph with these details.

Step-by-step explanation:

Graphing a Function Step-by-Step

To graph the function y = 3x - x² / 15 - 8x + x², certain steps must be followed:

1. Find the domain of the function: Since there are no denominators or even roots, the domain is all real numbers.
2. Simplify the function if possible: The simplified form is y = -x² / 15 + 5x.
3. Find the x and y intercepts: Set y to 0 to find x-intercepts and x to 0 to find the y-intercept.
4. Determine any odd or even symmetry: This is a quadratic function, so it has even symmetry (symmetrical about the y-axis).
5. Identify all vertical and horizontal asymptotes: There are none for this function as it is a polynomial.
6. Find intervals of increase & decrease, and any local max or min points using a sign chart: Calculate the derivative and analyze the signs to find these intervals.
7. Determine intervals of concavity and any inflection points with a sign chart: Calculate the second derivative and use it to determine concavity.
8. Sketch the function showing all details: Plot intercepts, max and min points, and concavity to sketch the graph.