Graphs of functions and sketching curves
| English | Chinese | Pinyin |
|---|---|---|
| function | 函数 | hán shù |
| parabola | 抛物线 | pāo wù xiàn |
| reciprocal | 倒数 | dào shǔ |
| roots | 根 | gēn |
| intersection | 交集 | jiāo jí |
| asymptotes | 渐近线 | jiàn jìn xiàn |
| turning point | 转折点 | zhuǎn zhé diǎn |
The shape of a function 函数
- Every function has a signature shape, like a fingerprint. A linear function is a straight line; a quadratic is a parabola 抛物线; a reciprocal 倒数 has two branches that never touch the axes.
- Recognising the shape lets you sketch a graph in seconds — without plotting dozens of points.
Plotting and reading graphs
- Make a table of values: pick $x$-values, calculate $y$-values, plot the coordinates, join with a smooth curve.
- Roots 根 are where the graph crosses the $x$-axis (the solutions of $y = 0$).
- The intersection 交集 of two graphs solves the two equations simultaneously.
A suspension bridge cable hangs in a parabola, a quadratic curve
Sketching curves
y = ax³ + bx² + cx + d
Sketch a curve from its turning points and where it crosses the axes.
The roots of a graph are where it:
Roots are the x-values where y = 0 — where the curve meets the x-axis.
Three families of curves
Three families every mathematician should recognise: line, parabola, reciprocal.
| Function | Shape | Key features |
|---|---|---|
| $y = mx + c$ | straight line | gradient $m$, intercept $c$ |
| $y = ax^2 + bx + c$ | parabola | U if $a > 0$, ∩ if $a < 0$ |
| $y = \dfrac{a}{x}$ | reciprocal | two branches, asymptotes 渐近线 at axes |
Parabola direction. $y = x^2 - 4$ opens upward (U-shape) because the coefficient of $x^2$ is positive. $y = -x^2 + 4$ opens downward (∩-shape).
The graph of y = ax² + bx + c (with a > 0) is a:
A positive a gives a U-shaped parabola; a negative a gives an ∩ shape.
The graph of y = −x² + 4 opens upward.
The coefficient of x² is negative (−1), so the parabola opens downward (∩-shape).
The graph of y = a/x has two branches that never touch the axes. These lines are called ______.
Asymptotes are lines the curve approaches but never touches. For y = a/x, the axes are asymptotes.
Turning points 转折点 (Extended)
- A parabola has one turning point (vertex) — its minimum or maximum.
- Completing the square reveals it directly:
- $y = (x + p)^2 + q$ has its turning point at $(-p,\; q)$.
- Example: $y = (x + 3)^2 - 8$ → turning point $(-3, -8)$.
Signs flip. In $(x + 3)^2 - 8$, the turning point is $(-3, -8)$ — the $x$-coordinate has the opposite sign to what's inside the bracket.
Completing the square, $y=(x+3)^2-8$, shows the turning point $(-3,-8)$ and the line of symmetry $x=-3$
The five basic graph shapes; knowing each shape lets you sketch quickly from the equation
The curve y = (x + 3)² − 8 has its turning point at (−3, q). What is q?
Completed-square form (x+p)² + q has turning point (−p, q), so q = −8.
Using graphs to solve equations
- Where a line and a curve intersect, both equations are satisfied.
- Solving $x^2 - 4 = 2x + 1$ graphically: plot $y = x^2 - 4$ and $y = 2x + 1$, read off the intersection $x$-values.
The line y = x + 2 intersects the curve y = x² at two points. One has x = 2. What is the other x-value?
x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 or x = −1.
You've got it
- roots $=$ where the curve crosses the $x$-axis; intersection solves two equations
- a quadratic is a parabola (U if $a > 0$, ∩ if $a < 0$)
- the turning point of $(x+p)^2 + q$ is $(-p,\; q)$
- reciprocal curves have asymptotes they never touch