To draw graphs of functions, start with a parent graph. Examples include ( y = x^2 ), ( y = sqrt{x} ), ( y = lvert xrvert ), or ( y = frac1x ). Apply transformations. ( f(x) + k ) shifts up k. ( f(x – h) ) shifts right h. ( a,f(x) ) stretches vertically by a. ( -f(x) ) reflects across the x-axis. Mark intercepts and asymptotes. The range is the set of y-values its graph reaches. Read it from bottom to top.
A graph turns a formula into a picture: you can see where a function is positive, where it rises and falls, where it has a highest or lowest point, and which values it can never take. This guide shows the shapes every student should recognize, the rules for moving them around, and a step-by-step method for graphing any function by hand.
The parent functions and their graphs of functions
| Function | Graph | Domain | Range |
|---|---|---|---|
| ( y = x ) | straight line through the origin, slope 1 | all reals | all reals |
| ( y = x^2 ) | U-shaped parabola, vertex ( (0, 0) ) | all reals | ( [0, infty) ) |
| ( y = x^3 ) | S-shaped curve through the origin | all reals | all reals |
| ( y = sqrt{x} ) | half a sideways parabola starting at ( (0, 0) ) | ( [0, infty) ) | ( [0, infty) ) |
| ( y = lvert xrvert ) | V shape, vertex ( (0, 0) ) | all reals | ( [0, infty) ) |
| ( y = frac1x ) | two branches, asymptotes ( x = 0 ) and ( y = 0 ) | ( x neq 0 ) | ( y neq 0 ) |
| ( y = 2^x ) | rising curve through ( (0, 1) ), asymptote ( y = 0 ) | all reals | ( (0, infty) ) |
| ( y = ln x ) | rising curve through ( (1, 0) ), asymptote ( x = 0 ) | ( (0, infty) ) | all reals |
| ( y = sin x ) | wave between ( -1 ) and 1, period ( 2pi ) | all reals | ( [-1, 1] ) |
Recognizing these shapes is most of the work. More detail on each family: polynomials, exponential functions, logarithm rules (log graphs) and trigonometric functions.
Transformations
| Change to the formula | Effect on the graph |
|---|---|
| ( f(x) + k ) | shift up k (down if ( k < 0 )) |
| ( f(x – h) ) | shift right h (left if ( h < 0 )) |
| ( a,f(x) ), ( a > 1 ) | vertical stretch by a |
| ( a,f(x) ), ( 0 < a < 1 ) | vertical compression |
| ( -f(x) ) | reflect across the x-axis |
| ( f(-x) ) | reflect across the y-axis |
| ( f(bx) ), ( b > 1 ) | horizontal compression by a factor of b |
The horizontal shift feels backwards: ( (x – 3)^2 ) moves the parabola right 3, because the vertex is now where ( x – 3 = 0 ).
Order of operations. For ( y = a,f(x – h) + k ): shift horizontally, stretch or reflect, then shift vertically. The point ( (0, 0) ) of the parent graph moves to ( (h, k) ).
Example 1: a transformed parabola
( y = -2(x – 1)^2 + 3 ): start with ( x^2 ), shift right 1, stretch by 2, flip upside down, shift up 3. The vertex is ( (1, 3) ), the parabola opens downward, and the range is ( (-infty, 3] ).
Example 2: a transformed square root
( y = sqrt{x + 4} – 2 ): shift ( sqrt{x} ) left 4 and down 2. It starts at ( (-4, -2) ). Domain ( [-4, infty) ), range ( [-2, infty) ).
Example 3: a transformed reciprocal
( y = frac{1}{x – 2} + 1 ): asymptotes move to ( x = 2 ) and ( y = 1 ). Domain ( x neq 2 ), range ( y neq 1 ).
How to graph any function by hand
- Domain. Where is the function defined? (See domain and range.)
- Intercepts. y-intercept: set ( x = 0 ). x-intercepts: solve ( f(x) = 0 ).
- Symmetry. If ( f(-x) = f(x) ) the function is even (symmetric about the y-axis, like ( x^2 )). If ( f(-x) = -f(x) ) it’s odd (symmetric about the origin, like ( x^3 )).
- Asymptotes. Vertical: where a denominator is zero. Horizontal: the behavior as ( x to pminfty ) (see limits at infinity).
- End behavior. For polynomials, the leading term decides it.
- A few points. Plot values on each side of every intercept and asymptote.
- Connect smoothly, respecting asymptotes and the domain.
Example 4: graph ( f(x) = x^3 – 3x )
- Domain: all reals.
- Intercepts: ( f(0) = 0 ); ( x(x^2 – 3) = 0 ) gives ( x = 0, pmsqrt3 approx pm1.73 ).
- Symmetry: ( f(-x) = -f(x) ), so it’s odd.
- End behavior: falls on the left, rises on the right.
- Turning points: ( f'(x) = 3x^2 – 3 = 0 ) at ( x = pm1 ), giving a local maximum ( (-1, 2) ) and a local minimum ( (1, -2) ). Finding these with derivatives is covered in critical points.
Example 5: graph ( g(x) = frac{2x}{x – 1} )
Domain ( x neq 1 ) with a vertical asymptote there. Horizontal asymptote ( y = 2 ) (ratio of leading coefficients). Intercept at the origin. Points: ( g(-1) = 1 ), ( g(2) = 4 ), ( g(3) = 3 ). The graph has two branches hugging the asymptotes.
What is the range of a function?
The range is the set of all output values, the y-values the graph actually reaches. To read it from a graph, scan from the bottom of the graph to the top and note the lowest and highest values reached, plus any gaps.
- ( y = x^2 + 1 ): lowest point at ( y = 1 ), no highest point. Range ( [1, infty) ).
- ( y = frac1x ): every y-value except 0. Range ( (-infty, 0) cup (0, infty) ).
- ( y = 3sin x ): between ( -3 ) and 3. Range ( [-3, 3] ).
Algebraically, solve ( y = f(x) ) for x and see which y-values are allowed. The domain and range guide covers this in detail.
Graph it yourself
Type any function to see its graph, and toggle the derivative to see where it rises and falls. Try the examples above, or <code>-2(x – 1)^2 + 3</code>, <code>sqrt(x + 4) – 2</code> and <code>1/(x – 2) + 1</code>.
The full graphing calculator lets you compare several functions.
Reading information from a graph
| Question | Where to look |
|---|---|
| Where is ( f(x) = 0 )? | x-intercepts |
| Where is ( f(x) > 0 )? | where the graph is above the x-axis |
| Where is f increasing? | where the graph rises from left to right |
| Maximum and minimum values | highest and lowest points |
| Domain | horizontal extent of the graph |
| Range | vertical extent of the graph |
| Is it a function? | vertical line test: no vertical line hits it twice |
Common mistakes
- Shifting the wrong way. ( f(x + 2) ) moves left 2, not right.
- Applying transformations in the wrong order. Stretch before the vertical shift.
- Crossing a vertical asymptote. The graph can’t touch it.
- Assuming a horizontal asymptote is never crossed. It can be crossed in the middle of the graph; it only describes the far ends.
- Forgetting domain restrictions from roots and logs.
Practice problems
Describe the transformations, then give the domain and range.
- ( y = (x + 2)^2 – 5 )
- ( y = -lvert xrvert + 4 )
- ( y = sqrt{x – 1} + 3 )
- ( y = 2^x – 1 )
- ( y = frac{1}{x + 3} )
- Is ( f(x) = x^4 – x^2 ) even, odd or neither?
Answers: (1) left 2, down 5; domain all reals, range ( [-5, infty) ). (2) reflect, up 4; domain all reals, range ( (-infty, 4] ). (3) right 1, up 3; domain ( [1, infty) ), range ( [3, infty) ). (4) down 1; domain all reals, range ( (-1, infty) ). (5) left 3; domain ( x neq -3 ), range ( y neq 0 ). (6) even.
FAQ
How do you graph a function?
Find the domain, intercepts, symmetry and asymptotes, plot a few points, and connect them. Or recognize the parent graph and apply the transformations in the formula.
What does f(x − h) + k do to a graph?
It shifts the graph of f right by h and up by k.
What is the range of a function?
The set of all y-values the function produces. On a graph, it’s everything from the lowest to the highest point the graph reaches.
What is a parent function?
The simplest function of a family, such as ( y = x^2 ) for quadratics, from which every other member is made by transformations.
How can you tell if a graph is a function?
Use the vertical line test: if any vertical line crosses the graph more than once, it isn’t a function.
Further reading
- Graph of a function (Wikipedia) — definitions and examples of function graphs.
- Graphing Functions (Paul’s Online Math Notes) — graphing by plotting points, with worked examples.
Calculators for this topic
Keep learning
The Unit Circle: Angles, Coordinates and Trig Values
The unit circle explained: why the coordinates are cos and sin, every special angle in degrees and radians, tangent on the unit circle, and memory tricks.
Trigonometric Functions: Sin, Cos, Tan, Table and Graphs
The six trig functions with SOH-CAH-TOA and the unit circle, a table of exact values (sin 30, cos 180, sec 45), graphs, period and amplitude.