Graphs
Plotting points works, and it is slow. The faster route is to know a handful of parent shapes by heart and read the rest of the rule as instructions for moving one of them about. Once you can look at an equation and say where its graph sits without drawing it, sketching stops being arithmetic and becomes recognition.
Calculus — graphing and functions
Graphing calculator- y-interceptset x = 0
There is at most one, because a function gives one output per input.
- x-interceptsset y = 0 and solve
There can be any number of these. They are the roots, and factoring is usually how you find them.
- Even — symmetric about the y-axisf(−x) = f(x)
The left half mirrors the right. x2, x4 and cos x are even, and so is anything built only from even powers.
- Odd — symmetric about the originf(−x) = −f(x)
Rotating the graph half a turn about the origin leaves it unchanged. x, x3 and sin x are odd.
- Most things are neitherf(−x) ≠ f(x) and f(−x) ≠ −f(x)
Mixing even and odd powers usually destroys both symmetries. x2 − 2x is neither, and that is the ordinary case rather than a failure.
- Symmetry about the x-axispossible for a curve, never for a function
It would put two y values above one x, which the vertical line test forbids. A sideways parabola has it; no function does.
- Liney = x
Straight, through the origin, at 45°. Odd.
- Parabolay = x2
A U with its vertex at the origin, opening upwards. Even.
- Cubicy = x3
Flattens at the origin, then rises steeply both ways — down on the left, up on the right. Odd.
- Square rooty = √x
Half a parabola on its side. Starts at the origin and only exists for x ≥ 0.
- Absolute valuey = |x|
A V with its corner at the origin. Even. The corner is what makes it interesting later.
- Reciprocaly = 1x
Two branches that never touch either axis. Odd, and the first graph here with asymptotes.
- Up and downf(x) + c moves up by c
Outside, and it behaves: plus goes up, minus goes down.
- Left and rightf(x + c) moves LEFT by c
Inside, and it is backwards. f(x − 3) moves right. The reason: to get the old output you now need an x that is 3 bigger, so every point shifts right.
- Flip vertically−f(x)
Outside, so it reflects in the x-axis — the whole graph turns upside down.
- Flip horizontallyf(−x)
Inside, so it reflects in the y-axis — left and right swap.
- Taller and shortera · f(x)
Outside. a > 1 stretches away from the x-axis; 0 < a < 1 squashes towards it.
- Narrower and widerf(bx)
Inside, so backwards again. b > 1 squeezes the graph horizontally; b < 1 stretches it out.
Order matters when both a stretch and a shift are involved. Write the rule as a · f(x − h) + k and read it as: start from the parent, stretch by a, then move by h across and k up.
- Vertical asymptotesolve for y , then set the denominator to 0
Where the bottom vanishes the output runs off to infinity, so each such x = a is a vertical asymptote. The curve can never cross it \u2014 there is no output there at all.
- Horizontal asymptotesolve for x , then set that denominator to 0
The same move with the variables swapped. Each y = b it rules out is a horizontal asymptote, describing where the curve settles far from the origin.
- The numerator has to survivedenominator = 0 while numerator ≠ 0
If both vanish at the same value the factor cancels, and what you have is a hole in the curve rather than an asymptote.
- Each rearrangement pays twicesolve for y → domain, solve for x → range
Solving for y hands you the domain and the vertical asymptotes together; solving for x hands you the range and the horizontal ones. Do each rearrangement once and read both answers off it.
- 1
Intercepts. Set y = 0 for the x-intercepts and x = 0 for the y-intercept. If the resulting equation has no solution then that intercept does not exist, which is itself worth knowing.
- 2
Symmetry. Replace y with −y, then x with −x, then both. Every substitution that leaves the equation unchanged halves the drawing you have left to do.
- 3
Solve for y. This hands you the domain and every vertical asymptote at once.
- 4
Solve for x. And this hands you the range and every horizontal asymptote.
- 5
Dash the asymptotes, then join up. Draw the asymptotes as dashed lines first and plot the intercepts. Then test one large value on each branch to see which side of the asymptote that branch runs along.
- f(x + 3) moves the graph right by 3f(x + 3) moves the graph LEFT by 3
Check it on a point. For f(x) = x2, the vertex of f(x+3) = (x+3)2 is where x + 3 = 0, that is x = −3. Left.
- −f(x) and f(−x) are the same−f(x) flips top to bottom; f(−x) flips left to right
Try f(x) = √x: −√x is the same curve hanging below the axis, while √−x is the mirror image on the left of the axis. Different graphs, and even different domains.
- f(−x) = −f(x) for every functionOnly for odd functions
It is a test, not an identity — it is the question being asked, not a rule that can be applied. Most functions fail it, and failing is a valid answer.
- a curve can never cross an asymptotenever a vertical one; a horizontal one it may
A vertical asymptote sits at a missing input, so nothing can be plotted there at all. A horizontal asymptote only describes where the curve settles far out, and near the origin it is free to cross it and come back.
- denominator = 0 always gives an asymptotecancel first, then look
If the top vanishes at the same value, the factor cancels and leaves a hole rather than an asymptote. x2 − 9x − 3 has no asymptote at x = 3 — only a single missing point on an otherwise ordinary line.
- A graph symmetric about the x-axis is a functionIt cannot be.
Symmetry about the x-axis puts a matching point below every point above, so one input has two outputs. The vertical line test fails at once.
Type an answer and it gets marked. What you get right is kept on this device.
- 1.Intercepts of y = x2 − 4x + 3
- 2.Symmetry of f(x) = 2x4 + 5
- 3.Symmetry of f(x) = x3 + x2
- 4.Describe y = (x − 4)2 + 1
- 5.Describe y = 1x + 2
- 6.Asymptotes of y = x + 1x − 2
- 7.Asymptotes of y = 3xx2 − 4
- 8.Symmetry of x2 + 4y2 = 16
- 9.Describe y = −|x − 1| + 3