Understanding Different Types of Function Graphs
Graphs visually represent the relationship between input values (x) and output values (y) of a function. Each function type has a characteristic graph shape and features.
Linear functions have the form y = mx + c, producing straight lines with constant slope and no turning points.
Quadratic functions, y = ax^2 + bx + c, produce parabolas that curve upwards or downwards and have exactly one turning point (vertex). The turning point is either a maximum or minimum depending on the sign of a (positive for minimum, negative for maximum). The parabola is symmetric about a vertical line through the vertex called the axis of symmetry, x = h, where h is the x-coordinate of the turning point.
- Linear: straight line, no turning points.
- Quadratic: parabola, one turning point (vertex), axis of symmetry.
- Cubic: curve with up to two turning points, can change direction twice.
- Reciprocal: hyperbola with vertical and horizontal asymptotes.
- Exponential: curve showing growth or decay with horizontal asymptote.
Key Features: Turning Points, Roots, and Symmetry
Turning points are points where the graph changes direction from increasing to decreasing or vice versa. Quadratic functions have one turning point; cubic functions can have up to two.
Roots (or zeros) are x-values where the graph crosses the x-axis (y=0). Finding roots involves solving the equation f(x) = 0.
Symmetry helps identify graph shape: even functions are symmetric about the y-axis, odd functions about the origin. For example, y = x^2 is even and symmetric about the y-axis, while y = x^3 is odd and symmetric about the origin.
- Turning points: locations where slope changes sign.
- Roots: solutions to f(x) = 0, where graph crosses x-axis.
- Symmetry: even functions mirror on y-axis; odd functions rotate about origin.
Asymptotes in Reciprocal and Exponential Functions
Reciprocal functions have the form y = k/x or variations, where the denominator can be zero, causing vertical asymptotes (lines the graph approaches but never touches). For example, y = 1/x has a vertical asymptote at x = 0.
Horizontal asymptotes describe the behaviour of the graph as x approaches infinity or negative infinity. Reciprocal functions often have horizontal asymptotes at y = 0.
Exponential functions, such as y = a^x (a > 0, a ≠ 1), show rapid growth or decay and have a horizontal asymptote, usually y = 0, which the graph approaches but never crosses.
- Vertical asymptotes: values of x where function is undefined (denominator zero).
- Horizontal asymptotes: y-values approached as x → ±∞.
- Reciprocal functions: vertical and horizontal asymptotes.
- Exponential functions: horizontal asymptote, usually y = 0.
Finding Turning Points of Quadratic Functions by Completing the Square
To find the turning point of a quadratic function y = ax^2 + bx + c, rewrite it in vertex form y = a(x - h)^2 + k, where (h, k) is the turning point.
Steps: 1) Factor out a from the x^2 and x terms if a ≠ 1. 2) Complete the square inside the bracket. 3) Simplify to identify h and k.
Example: Find the turning point of y = 2x^2 - 8x + 3.
- Start with y = 2x^2 - 8x + 3.
- Factor out 2: y = 2(x^2 - 4x) + 3.
- Complete the square inside parentheses: x^2 - 4x + 4 - 4 = (x - 2)^2 - 4.
- Rewrite: y = 2[(x - 2)^2 - 4] + 3 = 2(x - 2)^2 - 8 + 3 = 2(x - 2)^2 - 5.
- Turning point is at (2, -5).
Sketching and Analysing Graphs Using Key Features
When sketching graphs, identify roots, turning points, asymptotes, and symmetry first. Plot these points and draw the curve accordingly.
For example, to sketch y = 1/x, note vertical asymptote at x=0, horizontal asymptote at y=0, and that the graph lies in quadrants I and III.
For y = x^3, note roots at x=0, turning points (found by calculus or graph shape knowledge), and symmetry about the origin.
- Identify roots by solving f(x) = 0.
- Find turning points (quadratic: complete the square; cubic: more advanced methods).
- Locate asymptotes for reciprocal and exponential functions.
- Use symmetry to reduce plotting effort.
- Draw smooth curves through key points respecting asymptotes and turning points.
At-a-glance summary
Comparison table
| Function Type | General Form | Turning Points | Roots (x-intercepts) Example(s) |
|---|---|---|---|
| Linear | y = mx + c | None | One root if m ≠ 0; e.g. y = 2x - 4 has root x=2 |
| Quadratic | y = ax^2 + bx + c | One turning point (vertex) | Up to two roots; e.g. y = x^2 - 4 has roots x=±2 |
| Cubic | y = ax^3 + bx^2 + cx + d | Up to two turning points | Up to three roots; e.g. y = x^3 - x has roots x=0, ±1 |
| Reciprocal | y = k/x or variations | None | No roots; undefined at x=0 (vertical asymptote) |
| Exponential | y = a^x (a>0, a≠1) | None | No roots; approaches y=0 asymptotically |
Active recall
Quick check
1. What is the turning point of y = x^2 - 6x + 8?
Complete the square: y = (x^2 - 6x + 9) - 9 + 8 = (x - 3)^2 - 1. Turning point is at (3, -1).
2. Where are the vertical and horizontal asymptotes of y = 1/(x - 2)?
Vertical asymptote at x = 2 (denominator zero). Horizontal asymptote at y = 0 (as x → ±∞).
3. Does y = x^3 have any turning points?
Yes, y = x^3 has one turning point at the origin (0,0), where the graph changes concavity but not direction.
4. Is the graph of y = x^2 symmetric about the y-axis or origin?
Symmetric about the y-axis (even function).
5. Find the roots of y = x^2 - 9.
Set y=0: x^2 - 9 = 0 → x^2 = 9 → x = ±3. Roots at x = 3 and x = -3.