Graphs Of Functions
Quadratic and cubic graphs
Quadratic graphs — full treatment
Shape depends on the sign of \( a \)
| If \( a > 0 \) | If \( a < 0 \) |
|---|---|
| U-shape (opens upward); minimum turning point | ∩-shape (opens downward); maximum turning point |
Key features to find and label
- y-intercept: Set \( x = 0 \). The y-intercept is always \( (0, c) \).
- x-intercepts (roots): Set \( y = 0 \) and solve \( ax^2 + bx + c = 0 \). Methods: factorisation, completing the square, or the quadratic formula \( x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- Line of symmetry: \( x = -\dfrac{b}{2a} \). This vertical line passes through the turning point.
- Turning point (vertex): Substitute \( x = -\dfrac{b}{2a} \) back to find \( y \). Label as coordinate pair.
Worked example — sketch \( y = x^2 - 4x + 3 \)
1. y-intercept: \( x = 0 \Rightarrow y = 3 \). Point: \( (0, 3) \).
2. Roots (set \( y = 0 \)): \( x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3) = 0 \Rightarrow x = 1 \) or \( x = 3 \). Points: \( (1, 0) \) and \( (3, 0) \).
3. Line of symmetry: \( x = -\dfrac{-4}{2(1)} = 2 \)
4. Turning point: \( y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 \). Minimum at \( (2, -1) \).
Sketch: U-shaped curve; y-intercept at 3; roots at x = 1 and x = 3; minimum at (2, −1).
The discriminant — how many roots?
| Discriminant | Number of roots | Graph behaviour |
|---|---|---|
| \( b^2 - 4ac > 0 \) | Two distinct real roots | Crosses x-axis at two points |
| \( b^2 - 4ac = 0 \) | One repeated root | Touches x-axis at exactly one point (vertex on x-axis) |
| \( b^2 - 4ac < 0 \) | No real roots | Does not cross x-axis at all |
Vertex form — completing the square
Writing \( y = a(x - h)^2 + k \) immediately reveals the vertex at \( (h, k) \) and the line of symmetry \( x = h \).
Worked example — completing the square for \( y = 2x^2 - 8x + 5 \)
\( y = 2(x^2 - 4x) + 5 \)
\( = 2\left[(x - 2)^2 - 4\right] + 5 \)
\( = 2(x-2)^2 - 8 + 5 \)
\( = 2(x-2)^2 - 3 \)
Vertex: \( (2, -3) \) — a minimum since \( a = 2 > 0 \).
Line of symmetry: \( x = 2 \)
Cubic graphs
Shape and features of cubics
- \( a > 0 \): the curve goes from bottom-left to top-right (starts low, ends high).
- \( a < 0 \): the curve goes from top-left to bottom-right.
- y-intercept: set \( x = 0 \) — gives \( (0, d) \).
- x-intercepts: solve \( ax^3 + bx^2 + cx + d = 0 \). A cubic has at most 3 real roots and at least 1 (since every real cubic must cross the x-axis at least once).
- Turning points: a cubic has at most 2 local turning points. Find by differentiating and setting \( dy/dx = 0 \) (if required at this level).
- \( y = x^3 \) specifically: passes through origin; one root at \( x = 0 \); no turning points; point of inflection at origin.
Worked example — sketch \( y = x^3 - 3x \)
y-intercept: \( x = 0 \Rightarrow y = 0 \). Point: \( (0, 0) \).
Roots: \( x^3 - 3x = 0 \Rightarrow x(x^2 - 3) = 0 \Rightarrow x = 0,\; x = \sqrt{3} \approx 1.73,\; x = -\sqrt{3} \approx -1.73 \).
Turning points (via differentiation): \( \dfrac{dy}{dx} = 3x^2 - 3 = 0 \Rightarrow x = \pm 1 \).
At \( x = 1 \): \( y = 1 - 3 = -2 \) → local minimum \( (1, -2) \).
At \( x = -1 \): \( y = -1 + 3 = 2 \) → local maximum \( (-1, 2) \).
Shape: S-curve, \( a = 1 > 0 \), crossing x-axis at \( -\sqrt{3},\; 0,\; \sqrt{3} \).