Please find attached the graph of the function, y = 9·x⁵ - 45·x⁴, created with MS Excel
What is graphing of a function?
Graphing of a function is the visual representation of the function on the coordinate plane.
1. Identifying the domain and any symmetries
The domain of the function y = 9·y⁵ - 45·x⁴ = 9·x⁴·(x - 5), is the set of all real numbers, and the function is not symmetric with respect to the origin or the y-axis
2. Finding the derivatives y' and y'';
The first derivative of the function y' = 45·x⁴ - 180·x³ = 45·x³·(x - 4)
The second derivative, y'' = 180·x³ - 540·x² = 180·x²·(x - 3)
3. Finding the critical points and the behaviour of the function at each one;
The critical points of the function are the x-values where y' equivalent to 0 or is undefined
Where y' = 0, we get;
y' = 45·x⁴ - 180·x³ = 45·x³·(x - 4) = 0
x = 0, and x = 4
The value of the second derivative at x = 0 is; y''(0) = 180×0³ - 540×0² = 0, therefore, the function has an inflection point at x = 0
The value of the second derivative at x = 4 is; y''(4) = 180×4³ - 540×4² = 2880, a positive value, which indicates that the function has a local minimum at the point x = 4
4. Find where the curve is increasing and where it is decreasing; The curve is increasing in the region, where; y' > 0, and it is decreasing where y' < 0. y' = 45·x³·(x - 4), therefore, y' > 0, where x < 0, or x > 4, and the value of y' < 0, where 0 < x < 4, therefore;
The curve is increasing when x < 0 or x > 4, and decreasing where 0 < x < 4
5. Finding the points of inflection of the curve;
The points of inflection are the points where y'' = 0, therefore;
y'' = 180·x²·(x - 3) = 0
x = 0, and x = 3
y(3) = 9×3⁴×(3 - 5) = 1458
The points of inflection are; (0, 0), and (3, -1458)
6. Determine the concavity;
The curve, y , is concave up when the second derivative, y'' > 0, and the curve is concave down when y'' < 0
y'' = 180·x²·(x - 3), therefore, y'' > 0, when x > 3, and y'' < 0 when the x-value < 3, therefor;
The curve is concave up when x > 3, and concave down when x < 3
7. Identify the asymptotes;
The function does not have a vertical or horizontal asymptote
8. Plot key points such as intercepts, critical points, inflection points
The y-intercept is; (0, 0)
The critical points are (0, 0), and (4, 2304)
Inflection point are; (0, 0), (3, -1458)
9. Finding the coordinates of the absolute extreme point;
The domain of all real numbers, indicates that the function does not any absolute maximum or minimum point value
The graph of the function, y = 9·x⁵ - 45·x⁵ = 9·x⁴·(x - 5), showing the features of the graph, created MS Excel is attached.
Learn more on the graph of a function here: https://brainly.com/question/29011558
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