Graphing Functions
A graph is just a picture of a function's input-output pairs. Every input produces an output , and together they make a point on the coordinate plane. Plot enough of those points and the shape of the function appears.
The method is the same no matter what the function looks like: build a table of values, plot each row as a point, then connect the points. Lines need only two or three points; curves need a few more so you can see the bend.
The three-step method
Step 1: make a table. Pick a handful of -values — small numbers around zero like work well — and compute the output for each one by substituting into the rule.
Step 2: plot the points. Each row of the table is a point : the input is the horizontal coordinate and the output is the vertical coordinate.
Step 3: connect the points. If the function is defined for every in between, draw a smooth line or curve through the points and add arrows on the ends to show it keeps going.
The shape tells you the family
A rule like , where is only multiplied and shifted, always graphs as a straight line. A rule with in it, like , graphs as a U-shaped curve called a parabola. If your plotted points refuse to line up straight, that is not a mistake — the function is telling you it curves.
This is why curves need more points than lines. Two points pin down a line completely, but a parabola needs points on both sides of its turning point before the U-shape shows up.
Worked examples
Example 1: graph a line
Graph .
Answer: A straight line through rising for every step right
Example 2: check whether a point is on the graph
Is on the graph of ?
Answer: Yes — , so is on the graph
Example 3: graph a curve
Graph .
Answer: A parabola with its lowest point at
Try one yourself
Common questions
How many points do I need to plot?
For a line, two points are enough and a third is a good check. For a curve like , plot at least five points, including inputs on both sides of zero, so the shape of the bend is visible.
Which x-values should I pick for the table?
Small numbers centered on zero, like . They are quick to compute and they usually fit on the grid. If the outputs get too big for your graph, swap in smaller inputs.
Do I always connect the points?
Connect them when the function accepts every input in between, which is the usual case for rules like . If the function only makes sense for separate inputs — like the number of tickets bought — leave the points unconnected.
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