Use the guidelines of this section to sketch the curve.y =x2 - 4yyXX-55-55-5-5y55-5-5-5-5 Guidelines for Sketching a CurveThe following checklist is intended as a guide to sketching a curve y = f(x) by hand. Notevery item is relevant to every function. (For instance, a given curve might not have anasymptote or possess symmetry.) But the guidelines provide all the information you needto make a sketch that displays the most important aspects of the function.A. Domain It's often useful to start by determining the domain D of f, that is, the set ofvalues of x for which f(x) is defined.B. Intercepts The y-intercept is f(0) and this tells us where the curve intersects the y-axis.To find the x-intercepts,tion is difficult to solve.)C. Symmetryset y = 0 and solve for x. (You can omit this step if the equa-(i) If f(-x) = f(x) for all x in D, that is, the equation of the curve is unchangedwhen x is replaced by -x, then f is an even function and the curve is symmetric aboutthe y-axis. This means that our work is cut in half. If we know what the curve looks likefor x 0, then we need only reflect about the y-axis to obtain the complete curve [seeFigure 1(a)]. Here are some examples: y = x², y = x*, y = |x|, and y = cos x.(a) Even function: reflectional symmetry(ii) If f(-x) = -f(x) for all x in D, then f is an odd function and the curve is sym-metric about the origin. Again we can obtain the complete curve if we know what it lookslike for x 0. [Rotate 180° about the origin; see Figure 1(b).] Some simple examplesof odd functions are y = x, y = x°, y = x', and y = sin x.( iii) If f(x + p) = f(x) for all x in D, where p is a positive constant, then f is calleda periodic function and the smallest such number p is called the period. For instance,y = sin x has period 27 and y = tan x has period T. If we know what the graph lookslike in an interval of length p, then we can use translation to sketch the entire graph (seeFigure 2).(b) Odd function: rotational symmetryFIGURE 1FIGURE 2Periodic function:translational symmetryа -рa +pa + 2paD. Asymptotes(i) Horizontal Asymptotes. Recall from Section 2.6 that if either lim, f(x) = Lor limx--« f(x) = L, then the line y = L is a horizontal asymptote of the curve y = f(x).If it turns out that lim, f(x) = 0 (or -0), then we do not have an asymptote to theright, but that is still useful information for sketching the curve.(ii) Vertical Asymptotes. Recall from Section 2.2 that the line x = a is a verticalasymptote if at least one of the following statements is true:1lim f(x) = olim f(x) = 0Xa+Xa-lim f(x) = -00lim f(x) = -0(For rational functions you can locate the vertical asymptotes by equating the denomi-nator to 0 after canceling any common factors. But for other functions this method doesnot apply.) Furthermore, in sketching the curve it is very useful to know exactly whichof the statements in [1 is true. If f(a) is not defined but a is an endpoint of the domainof f, then you should compute lim,a-f(x) or lim,a+ f(x), whether or not this limit isinfinite.(iii) Slant Asymptotes. These are discussed at the end of this section.E. Intervals of Increase or Decrease Use the I/D Test. Compute f'(x) and find the intervalson which f(x) is positive (f is increasing) and the intervals on which f'(x) is negative(f is decreasing).F. Local Maximum and Minimum Values Find the critical numbers of f [the numbers c wheref'(c) = 0 or f'(c) does not exist]. Then use the First Derivative Test. If f' changes frompositive to negative at a critical number c, then f (c) is a local maximum. If f' changesfrom negative to positive at c, then f(c) is a local minimum. Although it is usually prefer-able to use the First Derivative Test, you can use the Second Derivative Test if f'(c) = 0and f"(c) + 0. Then f "(c) 0 implies that f(c) is a local minimum, whereas f"(c) 0 and concave downward where f"(x)

Question

Use the guidelines of this section to sketch the curve.y =x2 - 4yyXX-55-55-5-5y55-5-5-5-5 Guidelines for Sketching a CurveThe following checklist is intended as a guide to sketching a curve y = f(x) by hand. Notevery item is relevant to every function. (For instance, a given curve might not have anasymptote or possess symmetry.) But the guidelines provide all the information you needto make a sketch that displays the most important aspects of the function.A. Domain It's often useful to start by determining the domain D of f, that is, the set ofvalues of x for which f(x) is defined.B. Intercepts The y-intercept is f(0) and this tells us where the curve intersects the y-axis.To find the x-intercepts,tion is difficult to solve.)C. Symmetryset y = 0 and solve for x. (You can omit this step if the equa-(i) If f(-x) = f(x) for all x in D, that is, the equation of the curve is unchangedwhen x is replaced by -x, then f is an even function and the curve is symmetric aboutthe y-axis. This means that our work is cut in half. If we know what the curve looks likefor x> 0, then we need only reflect about the y-axis to obtain the complete curve [seeFigure 1(a)]. Here are some examples: y = x², y = x*, y = |x|, and y = cos x.(a) Even function: reflectional symmetry(ii) If f(-x) = -f(x) for all x in D, then f is an odd function and the curve is sym-metric about the origin. Again we can obtain the complete curve if we know what it lookslike for x > 0. [Rotate 180° about the origin; see Figure 1(b).] Some simple examplesof odd functions are y = x, y = x°, y = x', and y = sin x.( iii) If f(x + p) = f(x) for all x in D, where p is a positive constant, then f is calleda periodic function and the smallest such number p is called the period. For instance,y = sin x has period 27 and y = tan x has period T. If we know what the graph lookslike in an interval of length p, then we can use translation to sketch the entire graph (seeFigure 2).(b) Odd function: rotational symmetryFIGURE 1FIGURE 2Periodic function:translational symmetryа -рa +pa + 2paD. Asymptotes(i) Horizontal Asymptotes. Recall from Section 2.6 that if either lim, f(x) = Lor limx--« f(x) = L, then the line y = L is a horizontal asymptote of the curve y = f(x).If it turns out that lim, f(x) = 0 (or -0), then we do not have an asymptote to theright, but that is still useful information for sketching the curve.(ii) Vertical Asymptotes. Recall from Section 2.2 that the line x = a is a verticalasymptote if at least one of the following statements is true:1lim f(x) = olim f(x) = 0Xa+Xa-lim f(x) = -00lim f(x) = -0(For rational functions you can locate the vertical asymptotes by equating the denomi-nator to 0 after canceling any common factors. But for other functions this method doesnot apply.) Furthermore, in sketching the curve it is very useful to know exactly whichof the statements in [1 is true. If f(a) is not defined but a is an endpoint of the domainof f, then you should compute lim,a-f(x) or lim,a+ f(x), whether or not this limit isinfinite.(iii) Slant Asymptotes. These are discussed at the end of this section.E. Intervals of Increase or Decrease Use the I/D Test. Compute f'(x) and find the intervalson which f(x) is positive (f is increasing) and the intervals on which f'(x) is negative(f is decreasing).F. Local Maximum and Minimum Values Find the critical numbers of f [the numbers c wheref'(c) = 0 or f'(c) does not exist]. Then use the First Derivative Test. If f' changes frompositive to negative at a critical number c, then f (c) is a local maximum. If f' changesfrom negative to positive at c, then f(c) is a local minimum. Although it is usually prefer-able to use the First Derivative Test, you can use the Second Derivative Test if f'(c) = 0and f"(c) + 0. Then f "(c) > 0 implies that f(c) is a local minimum, whereas f"(c) <0implies that f(c) is a local maximum.G. Concavity and Points of Inflection Compute f"(x) and use the Concavity Test. The curveis concave upward where f"(x) > 0 and concave downward where f"(x) < 0. Inflec-tion points occur where the direction of concavity changes.H. Sketch the Curve Using the information in items A-G, draw the graph. Sketch theasymptotes as dashed lines. Plot the intercepts, maximum and minimum points, andinflection points. Then make the curve pass through these points, rising and fallingaccording to E, with concavity according to G, and approaching the asymptotes. Ifadditional accuracy is desired near any point, you can compute the value of the derivativethere. The tangent indicates the direction in which the curve proceeds.

Accepted Answer

Given that: y=xx2-4

Jse the guidelines of this section to sketch the curve. y = x2 - 4 y y 5 X 5