What is the graph of a function?
A function gives exactly one output y for each input x. Its graph is the set of all points (x, y) with y = f(x).
To draw a graph by hand:
- Make a table with a few x values (include negatives, 0 and fractions if needed).
- Work out y for each x.
- Plot the points.
- Join them with a smooth curve (not a ruler), unless the function is a straight line or has a corner.
A point (a, b) is on the graph if f(a) = b. For example, (3, 9) is on y = x² because 3² = 9.
Properties we read from a graph
- Domain: all x values allowed. Range: all y values the graph reaches.
- Zeros: where the graph meets the x-axis (y = 0).
- Increasing / decreasing: going left to right, does the graph go up or down?
- Even function: f(−x) = f(x); the graph is symmetric about the y-axis.
- Odd function: f(−x) = −f(x); the graph is symmetric about the origin (turn it half a circle and it looks the same).
- Greatest / least value: the highest or lowest point, if there is one.
The five basic graphs
| Function | Shape | Domain | Range | Symmetry | Rises / falls |
|---|---|---|---|---|---|
| y = x² | parabola (U) | all x | y ≥ 0 | even | falls for x < 0, rises for x > 0; least value 0 at x = 0 |
| y = x³ | cubic (S) | all x | all y | odd | always rises |
| y = √x | half-parabola | x ≥ 0 | y ≥ 0 | none | always rises (slower and slower) |
| y = |x| | V | all x | y ≥ 0 | even | falls for x < 0, rises for x > 0; corner at (0, 0) |
| y = k/x (k > 0) | hyperbola | x ≠ 0 | y ≠ 0 | odd | falls on each branch; branches in 1st and 3rd quarters |
For y = k/x with k < 0 the branches move to the 2nd and 4th quarters and rise on each branch. The axes are asymptotes: lines the graph gets closer and closer to but never touches. y = k/x describes inverse proportion: x·y = k stays the same.
Multiplying by a number a, y = a·f(x), stretches the graph up and down (|a| > 1 steeper, |a| < 1 flatter). A negative a flips it upside down.
Solving equations and systems with graphs
To solve f(x) = g(x): draw y = f(x) and y = g(x) on the same grid. The x-values of the crossing points are the solutions. The number of crossing points tells you the number of solutions.
Worked example (shown in step 5 of the 3D): solve x² = x + 2.
- Draw the parabola y = x² and the line y = x + 2.
- They cross at (−1, 1) and (2, 4).
- Check: (−1)² = 1 and −1 + 2 = 1 ✓; 2² = 4 and 2 + 2 = 4 ✓.
- Answer: x = −1 or x = 2.
For a system of two equations in x and y, draw both graphs: each crossing point (x, y) is one solution of the system. Graph answers can be rough, so always check by substituting.
Solving inequalities with graphs
f(x) < g(x) is true where the graph of f is below the graph of g. f(x) > g(x) is true where it is above.
From the example: the parabola is below the line between the crossing points, so x² < x + 2 for −1 < x < 2, and x² > x + 2 for x < −1 or x > 2.
Simple case: √x > 2 where the curve is above the line y = 2, which is x > 4.
Try it: predict, then check
On paper: make a table for y = 4/x with x = 1, 2, 4, 8, and the negatives. Plot and join each branch. Predict: what happens as x gets very large? Very close to 0?
In the 3D: at the last step choose y = x³, set a = −1 and guess the picture before you let go of the slider. Then try y = |x| with a = 0.5.
Key formulas and definitions
- Point (a, b) is on y = f(x) ⇔ f(a) = b
- Even: f(−x) = f(x); Odd: f(−x) = −f(x)
- y = x²: domain all x, range y ≥ 0
- y = √x: domain x ≥ 0, range y ≥ 0
- y = |x| = x if x ≥ 0, −x if x < 0
- y = k/x: x ≠ 0, y ≠ 0, x·y = k
- f(x) = g(x) ⇔ x-coordinates of crossing points
Worked examples
1. Is the point (−3, 9) on the graph of y = x²? Is (−2, −8) on y = x³?
(−3)² = 9, so yes. (−2)³ = −8, so yes.
2. Make a table for y = √x with x = 0, 1, 4, 9 and describe the graph.
y = 0, 1, 2, 3. The points are (0,0), (1,1), (4,2), (9,3). The curve starts at the origin and rises, but more and more slowly. There is nothing to the left of x = 0.
3. For y = 6/x find y when x = 2, 3, −1, and say which quarters the graph is in.
y = 3, 2, −6. k = 6 > 0, so the branches are in the 1st quarter (x, y > 0) and the 3rd quarter (x, y < 0).
4. Find the range of y = |x| − 2 and its zeros.
|x| ≥ 0, so y ≥ −2: range y ≥ −2. Zeros: |x| − 2 = 0 → |x| = 2 → x = 2 or x = −2.
5. Solve x² = x + 2 graphically and check.
Draw y = x² and y = x + 2. They cross at (−1, 1) and (2, 4). Check: 1 = −1 + 2 ✓, 4 = 2 + 2 ✓. So x = −1 or x = 2.
6. How many solutions does x³ = x have? Use graphs.
Draw y = x³ and y = x. They cross at (−1, −1), (0, 0) and (1, 1). So 3 solutions: x = −1, 0, 1. Check: (−1)³ = −1, 0³ = 0, 1³ = 1 ✓.
7. Solve the inequality 4/x > x for x > 0 using graphs.
For x > 0, y = 4/x and y = x cross where 4/x = x → x² = 4 → x = 2. To the left of 2 the hyperbola is above the line. So 4/x > x for 0 < x < 2.
Common mistakes
- Joining the two branches of y = k/x across x = 0. The graph has a gap there; x = 0 is not allowed.
- Drawing √x to the left of the y-axis. Negative x has no real square root.
- Giving the crossing point (2, 4) as the answer to x² = x + 2. The solution is the x-value: x = 2.
- Drawing y = |x| as a smooth U. It is a V with a sharp corner at the origin.