1. Use calculus and algebraic methods to do a complete analysis (i.e., intercepts, critical points, intervals of increase and decrease, points of inflection, intervals of concavity, local maximum or minimum points) for each of the following functions and then sketch the graph of the function. You can use graphing software (Geometer’s Sketchpad or Winplot) to verify your analysis and graph. Keep in mind that your analysis is the most important aspect of this question, not the graph.
ƒ(x) = −x2 + 4x − 3
g(x) = x3 − 6x2 + 9x
h(x) = x4 − 6x2
p(x) = x4 − 4x3
2. In this question, you should determine the asymptotes and then continue with rest of the analysis.
q(x)= -2/(x+3)^2
After all the help you have received here, you should be able to most of these questions yourself. You should also make the effort to type a ^ before exponents. Show your work and someone will help you.
Factor the equations to find the x-axis interecepts. plug in x = 0 to get the y axis intercepts. Take the first derivatives and set it equal to zero to find the critical points. Take the second derivatives at critical points to determine concavity intervals. You will have to do your own graphs.
Do you have a graphing calculator in your possession. Just curious.
Whom are you asking?
To analyze each function and sketch its graph, we need to follow these steps:
1. Find and analyze intercepts:
- To find the x-intercepts, set f(x) = 0 and solve for x.
- To find the y-intercept, evaluate f(0).
2. Find critical points:
- Calculate the derivative, f'(x), of the function.
- Set f'(x) = 0 and solve for x to find critical points.
- Determine the behavior of the function on each side of the critical points.
3. Determine intervals of increase and decrease:
- Use the sign of the derivative, f'(x), to determine where the function is increasing or decreasing.
4. Find points of inflection:
- Calculate the second derivative, f''(x), of the function.
- Set f''(x) = 0 and solve for x to find possible points of inflection.
- Determine the concavity of the function on each side of the points of inflection.
5. Determine intervals of concavity:
- Use the sign of the second derivative, f''(x), to determine where the function is concave up or concave down.
6. Find local maximum or minimum points:
- Use the first derivative test (sign change of f'(x)) to find local maximum or minimum points.
Now let's apply these steps to each function:
1. f(x) = -x^2 + 4x - 3
- Intercept analysis:
- x-intercepts: Set f(x) = 0: -x^2 + 4x - 3 = 0. Solve to find x-intercepts if any.
- y-intercept: Evaluate f(0) = -0^2 + 4(0) - 3.
- Critical points analysis:
- Find f'(x): Take the derivative of f(x).
- Solve f'(x) = 0 for x to find any critical points.
- Look at the behavior of f(x) on each side of the critical points.
- Intervals of increase and decrease:
- Determine the sign of f'(x) on different intervals to determine where f(x) is increasing or decreasing.
- Points of inflection analysis:
- Find f''(x): Calculate the second derivative of f(x).
- Solve f''(x) = 0 for x to find possible points of inflection.
- Note the behavior of f(x) on each side of the points of inflection.
- Intervals of concavity:
- Determine the sign of f''(x) to determine where f(x) is concave up or concave down.
- Local maximum or minimum points:
- Use the first derivative test (sign change of f'(x)) to find any local maximum or minimum points.
- Sketch the graph using the information collected.
Repeat the above steps for the remaining functions: g(x), h(x), p(x), and q(x). Use graphing software to verify your analysis and graph if needed.