Graphing Functions: From a Rule to a Picture

Graphing turns a rule into something you can see. It is the fastest way to answer questions about a function that would otherwise take real algebra — where it is zero, how high it goes, whether it ever levels off.

What a Graph Actually Is

The graph of is the set of all points — every input paired with the output it produces.

So each point carries two facts: how far across (the input) and how far up (the output). Reading a graph is nothing more than reading those two numbers back off. See Function Notation for that skill in detail.

Because is a function, the graph automatically passes the vertical line test. That is a guarantee, not a coincidence.

The Table Method

This works for absolutely anything, and it is what to fall back on when the shape is unfamiliar.

1. Choose inputs. Small integers around zero unless the function suggests otherwise.
2. Evaluate the function at each.
3. Plot the resulting points.
4. Join them with a smooth curve, extending as far as the domain allows.

For :

Choose inputs deliberately. If the function has a square root, start where the inside is zero. If it has a denominator, avoid the value that breaks it and take points either side. If it is a parabola, centre your inputs on the vertex rather than on zero — a table from to tells you almost nothing about .

Join smoothly, not with straight segments, unless the function really is piecewise linear. And never join across a break in the domain.

Shapes Worth Knowing on Sight

Most functions you meet are one of a few basic shapes, moved around. Recognising the family lets you sketch from two or three points instead of eight.

Every other member of a family is the parent shifted, stretched or reflected. is the standard parabola moved 3 right and 1 up. A minus sign in front flips it vertically. Getting the family right first means the sketch only needs adjusting, not rebuilding.

Features Worth Marking

A good sketch is not a hundred plotted points — it is a handful of correct ones, in the right places.

• -intercept — evaluate . One substitution, always available unless 0 is outside the domain.
• -intercepts — solve . Also called the zeros or roots. See x- and y-Intercepts.
• Turning points — the vertex of a parabola, or any peak or trough.
• Asymptotes — lines the curve approaches without touching, usually where the domain has a gap.
• Endpoints — where a restricted domain stops, drawn filled or open.

Mark those and the shape between them is nearly forced.

Increasing, Decreasing, and Turning

Read a graph left to right, the way you read text.

• Increasing where the curve rises as you move right.
• Decreasing where it falls.
• Constant where it is flat.

A turning point is where the behaviour switches. decreases on and increases on , turning at the origin.

These intervals are always stated in terms of , not — a common slip. The parabola above is decreasing on the interval , not on ””.

Graphing a Piecewise Function

A piecewise function uses different rules on different stretches of the domain. Graph each rule only where it applies. Piecewise Functions goes through evaluating, the domain and the boundary conventions in detail.

The boundary is where care is needed:

• Filled dot — that endpoint belongs to the graph ( or ).
• Open circle — it does not ( or ).
• Never join across the break. The pieces are separate.

Exactly one dot should be filled at any boundary value in the domain — two filled dots would mean two outputs for one input, breaking the function rule.

Using Domain and Range as a Check

Before you commit to a sketch, compare it against what you already know.

If you worked out the domain is , your curve should start at and go right, with nothing to the left. If the range is , the lowest point should sit at . A sketch that contradicts either one has an error in it.

This works in both directions: the graph is also the most reliable way to find the range in the first place.

Worked Example A: A Line

Graph .

The -intercept is . Slope 2 means up 2, right 1, giving . Two points is enough for a line; confirms it.

Worked Example B: A Parabola

Graph .

• -intercept: .
• -intercepts: , so and .
• Vertex: halfway between the roots at , where .

Plot , , , and the mirror point , then join. Range: .

Worked Example C: A Square Root

Graph .

The domain starts where the inside is zero, at . Table:

Perfect squares plus one make the arithmetic clean. The curve starts at and rises, flattening as it goes. Range: .

Worked Example D: An Absolute Value

Graph .

The V has its point where the inside is zero, at , giving . One unit either side gives and ; two units gives and . Draw two straight arms from the point.

Worked Example E: A Reciprocal

Graph .

Domain excludes 0, so there is a vertical asymptote there and a horizontal one at . Take points either side:

Two separate branches. Do not join them across the asymptote.

Common Mistakes to Avoid

• Too few points on a curve. Two points only ever determine a line.
• Joining across a break. A gap in the domain means separate pieces.
• Plotting points as . The input goes across.
• Centring the table on zero regardless. Put the points where the action is.
• Straight segments between points. Most functions curve.
• Filling both dots at a piecewise boundary. That would give one input two outputs.
• Extending a restricted domain. has nothing to the left of the origin.
• Ignoring a negative sign in front. opens downward.

Where Graphing Leads Next

• Transformations — shifting, stretching and reflecting a parent shape instead of tabulating.
• Solving equations graphically, by reading where two curves cross.
• Systems and inequalities, where solutions are regions rather than points.
• Domain and range, both easiest to read off a picture.
• Calculus, where the derivative formalises increasing, decreasing and turning points.
• Data and modelling, where a fitted curve is judged largely by eye.

Practice Problems

Work each one before opening the answer.

Problem 1. Give three points on the graph of .

Problem 2. What is the -intercept of ?

Problem 3. Find the -intercepts of .

Problem 4. Where is the vertex of ?

Problem 5. Where does the graph of begin?

Problem 6. Where is the point of the V for ?

Problem 7. Which parent function does belong to, and what has changed?

Problem 8. On what interval is decreasing?

Problem 9. Complete a table for at .

Problem 10. For a piecewise function with for and for , what happens at ?

Problem 11. Why does the graph of come in two pieces?

Problem 12. A graph runs from a filled dot at up to a filled dot at . State its domain and range.

Quick Reference

Graphing rests on function notation for evaluating and on functions for the guarantee that the picture passes the vertical line test. Read the extent horizontally as the domain and vertically as the range; the general table technique is in Graphing Equations. More Algebra lessons are available.