Graphing the Cubic Function y x^3 - 7x^2 36: A Comprehensive Guide

Graphing a cubic function involves understanding its behavior, critical points, and intercepts. In this article, we will guide you through the process of drawing the graph of the function y x^3 - 7x^2 36. Follow these steps to ensure a comprehensive understanding and accurate graphical representation.

1. Identify Key Features of the Graph

1.1 Find the Derivative

To determine the critical points and the behavior of the function, we start by finding its first derivative:

(y frac{d}{dx}(x^3 - 7x^2 36) 3x^2 - 14x)

1.2 Find Critical Points

Set the derivative equal to zero to find the critical points:

3x^2 - 14x 0)

Factor out (x):

(x(3x - 14) 0)

Solve for (x):

(x 0) and (x frac{14}{3} approx 4.67))

1.3 Determine the Nature of Critical Points

Use the second derivative to determine if these points are minima or maxima:

(y'' frac{d^2}{dx^2}(x^3 - 7x^2 36) 6x - 14)

Evaluate the second derivative at the critical points:

(y'(0) 6(0) - 14 -14 quad text{(local maximum)})

(y'left(frac{14}{3}right) 6left(frac{14}{3}right) - 14 28 - 14 14 quad text{(local minimum)})

2. Calculate Function Values at Critical Points

Evaluate (y) at the critical points:

At (x 0):

(y(0) 0^3 - 7(0)^2 36 36)

At (x frac{14}{3}):

(yleft(frac{14}{3}right) left(frac{14}{3}right)^3 - 7left(frac{14}{3}right)^2 36)

Calculate step-by-step:

( frac{2744}{27} - 7 cdot frac{196}{9} 36)

( frac{2744}{27} - frac{1372}{27} frac{972}{27} frac{2744 - 1372 972}{27} frac{2344}{27} approx 86.59)

3. Find Intercepts

3.1 Y-Intercept

Set (x 0):

(y(0) 36 quad text{Y-intercept at (0, 36)})

3.2 X-Intercepts

Set (y 0) and solve for (x):

This may require numerical methods or graphing to find approximate roots.

4. Sketch the Graph

Plot the critical points: 0, 36 is a local maximum and left(frac{14}{3}, frac{2344}{27}right) is a local minimum.

Determine end behavior: As (x) approaches positive infinity, (y) approaches positive infinity, and as (x) approaches negative infinity, (y) approaches negative infinity.

Mark the intercepts: Plot the y-intercept and any x-intercepts found.

Sketch the graph by starting from the left going downwards as (x) approaches negative infinity. Pass through the intercepts and reach the local maximum at (0, 36). Drop to the local minimum at left(frac{14}{3}, frac{2344}{27}right) and finally rise again toward positive infinity.

5. Draw the Curve

Follow the steps mentioned to accurately sketch the graph. Consider using graphing software for a more precise result.

Heres a rough idea of what the graph looks like:

50

This sketch gives you an idea of how the function behaves. You can use graphing software for a more precise graph.