Graphs Of Functions
Graph shapes quick reference
Full interactive graphs guide: This note gives you the quick-reference summary for all graph shapes. For in-depth worked examples, transformations and graph-reading exercises, open the full guide:
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Linear graphs — \( y = mx + c \)
Shape: Straight line
Key features:
Key features:
- \( m \) = gradient (slope). Positive \( m \): rises left to right. Negative \( m \): falls left to right. \( m = 0 \): horizontal line.
- \( c \) = y-intercept (where the line crosses the y-axis).
- x-intercept: set \( y = 0 \), solve for \( x = -c/m \).
Quadratic graphs — \( y = ax^2 + bx + c \)
Shape: Parabola (U-shape if \( a > 0 \); ∩-shape if \( a < 0 \))
Key features:
Key features:
- y-intercept: \( (0, c) \)
- x-intercepts (roots): solve \( ax^2 + bx + c = 0 \) by factorisation or the quadratic formula.
- Line of symmetry: \( x = -\dfrac{b}{2a} \)
- Vertex (turning point): substitute \( x = -\dfrac{b}{2a} \) into the equation to find the y-coordinate.
- Discriminant: \( b^2 - 4ac > 0 \): two roots; \( = 0 \): one repeated root (touches x-axis); \( < 0 \): no real roots.
Cubic graphs — \( y = ax^3 \) and beyond
Shape: S-shaped curve
Key features:
Key features:
- \( y = x^3 \): passes through origin (0,0), always increasing; rotational symmetry about origin.
- \( a > 0 \): bottom-left to top-right. \( a < 0 \): top-left to bottom-right.
- A cubic can have up to 3 roots (crosses x-axis up to 3 times) and up to 2 turning points.
- \( y = ax^3 + bx^2 + cx + d \): y-intercept at \( (0, d) \); shape depends on sign of \( a \).
Reciprocal graphs — \( y = \dfrac{k}{x} \) and \( y = \dfrac{k}{x^2} \)
\( y = k/x \) — Shape: Hyperbola (two curves, one in each of the opposite quadrants)
- Asymptotes: \( x = 0 \) (y-axis) and \( y = 0 \) (x-axis) — the curve never touches either axis.
- \( k > 0 \): curves in Q1 (top-right) and Q3 (bottom-left). \( k < 0 \): curves in Q2 and Q4.
- No x-intercepts, no y-intercepts.
- Asymptotes: same as \( y = k/x \) — both axes.
- \( k > 0 \): both branches above x-axis (Q1 and Q2). \( k < 0 \): both below.
The most common sketching error on reciprocal graphs is drawing the curve touching the asymptote. The curve approaches the axis but never reaches it. Mark the asymptote as a dashed line and label it.
Exponential graphs — \( y = ab^x \)
Shape: Rapidly increasing or decreasing curve
- y-intercept always at \( (0, a) \) — since \( b^0 = 1 \), so \( y = a \cdot 1 = a \).
- \( b > 1 \): exponential growth — curve rises steeply to the right.
- \( 0 < b < 1 \): exponential decay — curve falls, approaching the x-axis as \( x \to \infty \).
- Asymptote: \( y = 0 \) (the x-axis) — the curve never crosses it (stays positive for \( a > 0 \)).
- No x-intercepts (when \( a > 0 \) and \( b > 0 \)).
Other standard curves
| Equation | Shape | Key features |
|---|---|---|
| \( y = x^{1/2} = \sqrt{x} \) | Half-parabola, rising | Starts at (0,0); defined only for \( x \geq 0 \); always positive |
| \( y = |x| \) | V-shape | Vertex at origin; left arm: \( y = -x \); right arm: \( y = x \) |
| \( y = x^0 = 1 \) | Horizontal line | Constant function; gradient = 0 |
Quick summary table
| Type | Equation | Shape | Asymptotes? | y-intercept |
|---|---|---|---|---|
| Linear | \( mx + c \) | Straight line | None | \( (0, c) \) |
| Quadratic | \( ax^2 + bx + c \) | Parabola (U or ∩) | None | \( (0, c) \) |
| Cubic | \( ax^3 + \ldots \) | S-curve | None | \( (0, d) \) |
| Reciprocal | \( k/x \) | Hyperbola | \( x=0,\; y=0 \) | None |
| Reciprocal² | \( k/x^2 \) | Two branches, same side | \( x=0,\; y=0 \) | None |
| Exponential | \( ab^x \) | J-curve (growth/decay) | \( y = 0 \) | \( (0, a) \) |
When asked to "sketch" — always: (1) draw the correct shape, (2) label the y-intercept as a coordinate, (3) mark any asymptotes as dashed lines with their equations, (4) if roots exist, mark them on the x-axis. These four steps cover most of the available marks.