関数のグラフと曲線の概略図

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

Three families of curves

Three families every mathematician should recognise: line, parabola, reciprocal.

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 $(-3, -8)$.

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

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.

Solve a graph question completely

• For $y=x^2-2$, calculate $y=2,-1,-2,-1,2$ at $x=-2,-1,0,1,2$ and plot a smooth symmetric curve. Draw $y=2$ with a ruler; crossings give solutions $x=-2,2$ to $x^2-2=2$.
• For $y=6/x$, use positive and negative inputs, but never $x=0$. Points $(2,3),(3,2),(-2,-3),(-3,-2)$ lie on separate branches. Core quadratic sketches require roots and symmetry, not turning-point coordinates.

Extended cubic and exponential graphs

• For $y=x^3-1$, the integer table at $x=-2,-1,0,1,2$ is $-9,-2,-1,0,7$. Draw a smooth increasing S-shape; the root is 1 and the $y$-intercept is $-1$.
• For $y=2^x+1$, the table at $x=-1,0,1,2,3$ is $1.5,2,3,5,9$. The curve approaches horizontal asymptote $y=1$ to the left and rises to the right; it does not touch the asymptote.

Shifted reciprocals and fractional powers (Extended)

• $y=2/x+3$ has vertical asymptote $x=0$ and horizontal asymptote $y=3$. Shift both branches of $2/x$ up 3; do not join them across zero.
• For $y=\sqrt{x}$ use $x\ge0$ and points $(0,0),(1,1),(4,2),(9,3)$. For $y=1/\sqrt{x}$ use $x>0$ and points $(1,1),(4,1/2),(9,1/3)$. For $y=1/x^2$, positive and negative inputs give positive outputs, with symmetry about the $y$-axis.

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