Graphs and Trig Values

to memorize for quiz

y = x		Clearly, this is the easiest of all, but make sure it looks like it goes through the correct points.
y = x²		A basic parabola. Notice that it passes through (1, 1) and (2, 4), along with the corresponding points on the other side.
y = x³		A basic cubic. It passes through the origin, and, like sine, is odd with symmetry about the origin. It passes through (1, 1), (–1, –1), and the next integer coordinates are at (2, 8) and (–2, –8). It’s steep.
y =		This is the principal square root – the positive one. It’s a function, and passes the vertical line test. It’s also precisely half a parabola. The origin, (1, 1), and (4, 2) are good points to locate.
y =		The reciprocal function. It’s got two symmetric halves. You can plot points on it pretty easily; (1, 1) and (–1, –1) are important ones to locate.
y =		Notice that the curve gets close to the x-axis very quickly, but not to the y-axis. If you consider the y-values for x = ½ and x = 2, you’ll see why.
y = sin x		Sine goes through the origin, and is an odd function (symmetric about the origin). Important things happen every units along the x-axis, and that’s why the horizontal scale is in multiples of . The top, middle, and bottom points should be correctly located.
y = cos x		Cosine and sine are horizontal translations of each other. Cosine “starts” at the top of the curve and slopes down on both sides, while sine “starts” in the middle. Cosine is even, and has symmetry about the y-axis. Once again, important things happen every units along the x-axis, and that’s why the horizontal scale is in multiples of .
y = tan x		Tangent has vertical asymptotes at odd multiples of , because you can calculate it as sine divided by cosine. Whenever cosine is 0, you get a vertical asymptote. Its x-intercepts are at integer multiples of π, and it also passes through .
y = sec x		Secant is the reciprocal of cosine, and you can get values of it by taking reciprocals of cosine values. Notice that the vertical asymptotes are in the same places as tangent’s, and for the same reason. Also, since cosine itself never gets more than one unit away from the x-axis, secant is always at least one unit away.
y =		This is the graph of absolute value. Notice that it does go through the origin. Therefore the statement that “absolute value makes numbers positive” cannot be true.
y =		This is the greatest integer function, a/k/a the floor function. It rounds down to the next lower integer. So every value between 2 and 3 on the x-axis is graphed with a y-coordinate of 2. At x = 2, y is 2, but at x= 3, y is suddenly 3. That’s what the open and closed circles mean.
y =		This one is a semicircle. If the variable a gives you trouble, replace it mentally with 1. Notice that if x is outside the interval [–a, a], there will be a negative under the radical, and no y-coordinate. The domain is finite.
y = e^x		All exponential functions look pretty much like this. Since e⁰ = 1, the y-intercept is 1. Raising a positive number to a power will never give a negative value or 0, so the x-axis is a horizontal asymptote of the graph.
y = ln x		This is the basic logarithmic function; they’re all pretty much like this. They’re also the inverse functions to exponentials. In y = e^x, the y-intercept was 1; here, the x-intercept is 1. See the “opposite” nature? And since the horizontal asymptote of y = e^x was the x-axis, the vertical asymptote of the natural logarithm is the y-axis. Symmetry is a beautiful thing.

You will need to know sine, cosine, and tangent for 0°, 30°, 45°, 60°, 90°, 180°, 270°, and three others with those reference angles in other quadrants. Notice that in the table, I’ve given the angles in radians. That’s because in calculus, you always use radians. If you need a degree answer at the end, you convert it.

	sin	cos	tan
0	0	1



	1	0	, undefined
π	0	–1
	–1	0	, undefined

On the unit circle, sin q is the y-coordinate of the point where the angle q in standard position intersects the circle; cos q is the x-coordinate. That means we have sine positive in quadrants I and II, where y is positive, and cosine positive in quadrants I and IV, where x is positive. Tangent is the ratio of sine to cosine, so it is positive in quadrant I (sine and cosine are both positive) and in quadrant III (sine and cosine are both negative).

To find the sine, cosine, or tangent for an angle with the reference angles above but not in the first quadrant, first determine if the result will be positive or negative according to the rules above. Then decide if the reference angle (the angle to the closest side of the x-axis) is , , or. Use that ratio from the table with the sign you picked. For instance, to find cos , first recognize that this is in the second quadrant (close to π), so the cosine value will be negative. Secondly, the reference angle must be , since that’s how far is from π. In the table, cos = , so using the sign we can conclude that .