Summarize all the pertinent information obtained by applying the final version of the graphing strategy Curve-sketching techniques to f,and sketch the graph of f.
f(x)=(x-1)^3(x+3)
well, surely you did some of the steps. What did you find? We can help fill in any missing info.
To obtain the pertinent information and sketch the graph of the function f(x) = (x - 1)^3(x + 3), we can follow a step-by-step process for curve sketching techniques.
Step 1: Determine the domain of the function.
The domain of any polynomial function is all real numbers. Therefore, the domain of f(x) is (-∞, +∞).
Step 2: Find the x-intercepts.
To find the x-intercepts, we set f(x) equal to zero and solve for x.
(x - 1)^3(x + 3) = 0
Setting each factor equal to zero gives us:
x - 1 = 0 or x + 3 = 0
Solving these equations, we find x = 1 and x = -3 as the x-intercepts.
Step 3: Find the y-intercept.
To find the y-intercept, we set x = 0 in the expression f(x) = (x - 1)^3(x + 3).
f(0) = (0 - 1)^3(0 + 3)
f(0) = (-1)^3(3)
f(0) = -3
Therefore, the y-intercept is (0, -3).
Step 4: Determine symmetry.
To determine the symmetry of the graph, we check if the function is even or odd.
f(-x) = (-x - 1)^3(-x + 3)
Simplifying, we find f(-x) = -(x + 1)^3(x - 3)
Since f(x) is not equal to f(-x), the function does not exhibit symmetry about the y-axis or origin.
Step 5: Analyze end behavior.
For large positive and negative values of x, we analyze the behavior of the function.
As x approaches positive infinity, f(x) also approaches positive infinity.
As x approaches negative infinity, f(x) approaches negative infinity.
Step 6: Determine critical points and intervals of increase/decrease.
To find critical points, we take the derivative of f(x) and set it equal to zero.
f'(x) = 3(x - 1)^2(x + 3) + (x - 1)^3
Setting f'(x) equal to zero, we solve for x to find any critical points.
The critical points are x = -3 and x = 1.
Next, we determine the intervals of increase and decrease by plotting these critical points on a number line or by evaluating f'(x) in each interval.
For x < -3, f'(x) < 0 -> Decreasing interval
For -3 < x < 1, f'(x) > 0 -> Increasing interval
For x > 1, f'(x) < 0 -> Decreasing interval
Step 7: Determine concavity and inflection points.
To find the concavity, we take the second derivative of f(x) and evaluate it at critical points.
f''(x) = 6(x - 1)(x + 2)
Evaluating f''(x) at x = -3 gives f''(-3) = 6(-3 - 1)(-3 + 2) = -24
Evaluating f''(x) at x = 1 gives f''(1) = 6(1 - 1)(1 + 2) = 0
Since f''(-3) < 0 and f''(1) = 0, we have an inflection point at x = -3.
Step 8: Sketch the graph of f(x).
Using the information obtained from the previous steps, we can sketch the graph of f(x). Here's a rough sketch:
- As x approaches negative infinity, the graph decreases.
- The graph has a local minimum at x = -3.
- The graph increases in the interval -3 < x < 1.
- The graph has a local maximum at x = 1.
- As x approaches positive infinity, the graph decreases.
The graph passes through the x-intercepts at x = -3 and x = 1, and the y-intercept at (0, -3).
Please note that the sketch may not be perfectly accurate, and it is always better to use appropriate software or tools for an accurate representation of a graph.