Use the guidelines highlighted in Section 4.4 p. 271 to explicitly highlight all of the important features and create the complete graph of f (x) = x* + 8x3 – 270x2 + 1 (on the next page). Graphing Guidelines for y = f(x) 1. Identify the domain or interval of interest. On what interval(s) should the function be graphed? It may be the domain of the function or some subset of the domain. 2. Exploit symmetry. Take advantage of symmetry. For example, is the function even (f(-x) = f(x)), odd (f(-x) = -f(x)), or neither? 3. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and intervals of increase and decrease. Computing derivatives-particularly second derivatives–may not be practical, so some functions may need to be graphed without complete derivative information. 4. Find critical points and possible inflection points. Determine points at which f'(x) = 0 or f' is undefined. Determine points at which f"(x) = 0 or f" is undefined. 5. Find intervals on which the function is increasing/decreasing and concave up/down. The first derivative deter- mines the intervals of increase and decrease. The second derivative determines the intervals on which the function is concave up or concave down. 6. Identify extreme values and inflection points. Use either the First or Second Derivative Test to classify the critical points. Both x- and y-coordinates of maxima, minima, and inflection points are needed for graphing. 7. Locate all asymptotes and determine end behavior. Vertical asymptotes often occur at zeros of denominators. Hori- zontal asymptotes require examining limits as x→ ± ∞; these limits determine end behavior. Slant asymptotes occur with rational functions in which the degree of the numerator is one more than the degree of the denominator. 8. Find the intercepts. The y-intercept of the graph is found by setting x = 0. The x-intercepts are found by solving f(x) = 0; they are the real zeros (or roots) of f. 9. Choose an appropriate graphing window and plot a graph. Use the results of the previous steps to graph the function. If you use graphing software, check for consistency with your analytical work. Is your graph complete-that is, does it show all the essential details of the function?

Question

could you help with steps 4-6 in the second picture please. they are the guide lines my book wants me to use. I'm trying to study ahead of the class. my teacher won't teach or assign us these typs of problems for some reason  

Use the guidelines highlighted in Section 4.4 p. 271 to explicitly highlight all of the
important features and create the complete graph of f (x) = x* + 8x3 – 270x2 + 1
(on the next page).
Expand
Transcribed Image Text

Use the guidelines highlighted in Section 4.4 p. 271 to explicitly highlight all of the important features and create the complete graph of f (x) = x* + 8x3 – 270x2 + 1 (on the next page).

Graphing Guidelines for y =
f(x)
1. Identify the domain or interval of interest. On what interval(s) should the function be graphed? It may be the domain
of the function or some subset of the domain.
2. Exploit symmetry. Take advantage of symmetry. For example, is the function even (f(-x) = f(x)), odd (f(-x) = -f(x)),
or neither?
3. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and
intervals of increase and decrease. Computing derivatives-particularly second derivatives–may not be practical, so
some functions may need to be graphed without complete derivative information.
4. Find critical points and possible inflection points. Determine points at which f'(x) = 0 or f' is undefined.
Determine points at which f"(x) = 0 or f" is undefined.
5. Find intervals on which the function is increasing/decreasing and concave up/down. The first derivative deter-
mines the intervals of increase and decrease. The second derivative determines the intervals on which the function is
concave up or concave down.
6. Identify extreme values and inflection points. Use either the First or Second Derivative Test to classify the critical
points. Both x- and y-coordinates of maxima, minima, and inflection points are needed for graphing.
7. Locate all asymptotes and determine end behavior. Vertical asymptotes often occur at zeros of denominators. Hori-
zontal asymptotes require examining limits as x→ ± ∞; these limits determine end behavior. Slant asymptotes occur
with rational functions in which the degree of the numerator is one more than the degree of the denominator.
8. Find the intercepts. The y-intercept of the graph is found by setting x = 0. The x-intercepts are found by solving
f(x) = 0; they are the real zeros (or roots) of f.
9. Choose an appropriate graphing window and plot a graph. Use the results of the previous steps to graph the function.
If you use graphing software, check for consistency with your analytical work. Is your graph complete-that is, does it
show all the essential details of the function?
Expand
Transcribed Image Text

Graphing Guidelines for y = f(x) 1. Identify the domain or interval of interest. On what interval(s) should the function be graphed? It may be the domain of the function or some subset of the domain. 2. Exploit symmetry. Take advantage of symmetry. For example, is the function even (f(-x) = f(x)), odd (f(-x) = -f(x)), or neither? 3. Find the first and second derivatives. They are needed to determine extreme values, concavity, inflection points, and intervals of increase and decrease. Computing derivatives-particularly second derivatives–may not be practical, so some functions may need to be graphed without complete derivative information. 4. Find critical points and possible inflection points. Determine points at which f'(x) = 0 or f' is undefined. Determine points at which f"(x) = 0 or f" is undefined. 5. Find intervals on which the function is increasing/decreasing and concave up/down. The first derivative deter- mines the intervals of increase and decrease. The second derivative determines the intervals on which the function is concave up or concave down. 6. Identify extreme values and inflection points. Use either the First or Second Derivative Test to classify the critical points. Both x- and y-coordinates of maxima, minima, and inflection points are needed for graphing. 7. Locate all asymptotes and determine end behavior. Vertical asymptotes often occur at zeros of denominators. Hori- zontal asymptotes require examining limits as x→ ± ∞; these limits determine end behavior. Slant asymptotes occur with rational functions in which the degree of the numerator is one more than the degree of the denominator. 8. Find the intercepts. The y-intercept of the graph is found by setting x = 0. The x-intercepts are found by solving f(x) = 0; they are the real zeros (or roots) of f. 9. Choose an appropriate graphing window and plot a graph. Use the results of the previous steps to graph the function. If you use graphing software, check for consistency with your analytical work. Is your graph complete-that is, does it show all the essential details of the function?

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