How to graph a function, step by step
To graph a function, pick x-values, work out y for each one, and plot the (x, y) points. That is the whole method, and everything below is just that sentence done carefully. A function is a rule that turns each input into one output: for y = x² the rule is "square the input", so an input of 2 gives an output of 4, and the point (2, 4) sits on the graph. The x-coordinate of a point is the input, the y-coordinate is the output, and the graph is the picture of every input-output pair the rule can make.
The worked example: y = x²
The worked example. Step 1 — pick x-values: -2, -1, 0, 1, 2 — five values, spread across zero so you see both sides of the curve. Step 2 — work out y for each one: (-2)² is 4, (-1)² is 1, 0² is 0, 1² is 1, 2² is 4. Step 3 — write the pairs: (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4). Step 4 — plot them on the grid, one dot each. Step 5 — join them with a smooth curve: a U-shape, the parabola, with its lowest point at (0, 0). The two sides mirror each other, because squaring -2 and squaring 2 give the same answer — and that symmetry is the check that the points are right.
Picking x-values that actually help
Five values are enough for most graphs, but they have to be the right five. Always include zero: it gives the y-intercept, the point where the graph crosses the y-axis, and for y = x² that point is (0, 0). Always include a negative and a positive value: without both, you cannot see whether the graph is symmetric, and a parabola drawn from positive x-values alone looks like half a curve. And spread them out: -2, -1, 0, 1, 2 shows the shape of y = x² far better than 0, 1, 2, 3, 4, which only shows the rising arm. If the function has a feature — a peak, a gap, a flattening — put an x-value on each side of it.
Reading intercepts and slope
Once the points are plotted, the graph starts answering questions. The y-intercept is where the graph crosses the y-axis — the point with x = 0. For y = x² it is (0, 0); for y = 2x + 1 it is (0, 1), because 2 × 0 + 1 is 1. The x-intercepts are where the graph crosses the x-axis — the points with y = 0. And the slope is how steep the graph is: for a straight line like y = 2x + 1 the slope is 2, the number in front of x, so the line rises 2 units for every 1 unit it runs. For a curve the steepness changes along the graph — the parabola is flat at (0, 0) and steeper at (2, 4) than at (1, 1) — which is why you plot several points instead of two.
A second example: y = 2x + 1
Straight lines need only two points, but three prove you did not slip. Take x = 0, 1, 2. Step 1 — work out y: 2 × 0 + 1 is 1, 2 × 1 + 1 is 3, 2 × 2 + 1 is 5. Step 2 — write the pairs: (0, 1), (1, 3), (2, 5). Step 3 — plot them: they sit on a straight line, rising 2 for every 1 across — the slope is 2 and the y-intercept is 1. Step 4 — draw the line through them with a ruler. If the three points do not sit on one straight line, one of the calculations is wrong: go back and redo the arithmetic before you draw anything.
Checking your graph
Whatever method you used, check it with a point you did not plot. For y = x², pick x = 5 — a value outside your table — and work it out: 5² is 25, so the point (5, 25) should sit on the curve where your smooth line passes. If the curve you drew misses (5, 25), the curve is wrong, not the point. For y = 2x + 1, pick x = 3 the same way: 2 × 3 + 1 is 7, so (3, 7) should sit on the line. A second check is the shape itself: a parabola is symmetric, a straight line is straight, and a function that is undefined somewhere — 1/x at x = 0, or a square root at a negative x — has a gap, not a point. Run your numbers through the graphing-calculator and compare them with the worked examples above.
Common mistakes
The first is plotting only positive x-values: half the graph goes missing, and a parabola looks like a rising curve instead of a U. The second is joining the points with straight segments instead of a smooth curve — the points (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) joined by straight lines make a pointed V, not a parabola. The third is reading the slope off a curve as if it were a line: the slope of y = x² at (1, 1) is not the same as at (2, 4), because the curve gets steeper. And the fourth is forgetting the domain: some functions have no value at some x — 1/x has none at x = 0, and a square root has none for negative x — so the graph has a gap there, and drawing a point across the gap is wrong.
A quick check you can do in your head
You can sanity-check any graph without a calculator. First, test the easy inputs: x = 0 gives the y-intercept, and for y = x² that is (0, 0). Second, test a negative and a positive input and ask whether the outputs should match: for y = x² they do, because squaring forgets the sign, so the graph must be symmetric. Third, test one more input outside your table — x = 5 gives (5, 25) for y = x² and x = 3 gives (3, 7) for y = 2x + 1 — and if the point does not sit on the curve you drew, redraw the curve. If a calculator says the graph of y = x² passes anywhere but through (2, 4), the numbers are wrong.