Graphs of functions and sketching curves · Graphiques de fonctions et tracé de courbes
| English | Français |
|---|---|
| function/ˈfʌŋkʃn/ | fonction |
| parabola/pəˈræbələ/ | parabole |
| reciprocal/rɪˈsɪprəkl/ | réciproque |
| roots/ruːts/ | racines |
| intersection/ˌɪntəˈsekʃn/ | intersection |
| asymptotes/ˈæsɪmptəʊts/ | asymptotes |
| turning point/ˈtɜːnɪŋ pɔɪnt/ | point de retournement |
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 · Racines 根 are where the graph crosses the $x$-axis (the solutions of $y = 0$).
- The · Le intersection 交集 of two graphs solves the two equations simultaneously.
A suspension bridge cable hangs in a parabola, a quadratic curve
Sketching curves · Tracer des courbes
y = ax³ + bx² + cx + d
Sketch a curve from its turning points and where it crosses the axes. · Tracer une courbe à partir de ses points de retournement et de ses intersections avec les axes.
The roots of a graph are where it: · Les racines d'un graphique sont là où il :
Roots are the x-values where y = 0 — where the curve meets the x-axis. · Les racines sont les valeurs de x où y = 0 — là où la courbe rencontre l'axe des x.
Three families of curves
Three families every mathematician should recognise: line, parabola, reciprocal.
| Function | Shape · Forme (Shape) | Key features |
|---|---|---|
| $y = mx + c$ | straight line · ligne droite | gradient $m$, intercept $c$ |
| $y = ax^2 + bx + c$ | parabola · parabole | U if $a > 0$, ∩ if $a < 0$ |
| $y = \dfrac{a}{x}$ | reciprocal · réciproque | 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: · Le graphique de y = ax² + bx + c (avec a > 0) est un :
A positive a gives a U-shaped parabola; a negative a gives an ∩ shape. · Un a positif donne une parabole en forme de U ; un a négatif donne une forme ∩.
The graph of y = −x² + 4 opens upward. · Le graphique de y = −x² + 4 s'ouvre vers le haut.
The coefficient of x² is negative (−1), so the parabola opens downward (∩-shape). · Le coefficient de x² est négatif (−1), donc la parabole s'ouvre vers le bas (forme ∩).
The graph of y = a/x has two branches that never touch the axes. These lines are called ______. · Le graphique de y = a/x a deux branches qui ne touchent jamais les axes. Ces lignes sont appelées ______.
Asymptotes are lines the curve approaches but never touches. For y = a/x, the axes are asymptotes. · Les asymptotes sont des lignes que la courbe approche mais ne touche jamais. Pour y = a/x, les axes sont des asymptotes.
Turning points · Points de retournement 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? · La courbe y = (x + 3)² − 8 a son point de retournement en (−3, q). Quelle est q ?
Completed-square form (x+p)² + q has turning point (−p, q), so q = −8. · La forme complétée (x+p)² + q a un point de retournement en (−p, q), donc 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 · et $y = 2x + 1$, 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? · La droite y = x + 2 coupe la courbe y = x² en deux points. L'un a x = 2. Quelle est l'autre valeur de x ?
x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 or x = −1. · x² = x + 2 → x² − x − 2 = 0 → (x−2)(x+1) = 0 → x = 2 ou x = −1.
Solve a graph question completely
- For · Pour $y=x^2-2$, 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 · à $x^2-2=2$.
- For · Pour $y=6/x$, 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.
What is the positive solution read where y = x² − 2 meets y = 2? · Quelle est la solution positive lue où y = x² − 2 rencontre y = 2 ?
The crossing on the right is (2,2), so x = 2. · L'intersection à droite est (2,2), donc x = 2.
Extended cubic and exponential graphs
- For · Pour $y=x^3-1$, 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 · Pour $y=2^x+1$, 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.
For y = 2ˣ + 1, find y at x = 3. · Pour y = 2ˣ + 1, trouvez y en x = 3.
2³ + 1 = 9.
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 · Pour $y=\sqrt{x}$ use · utiliser $x\ge0$ and points $(0,0),(1,1),(4,2),(9,3)$. For $y=1/\sqrt{x}$ use · utiliser $x>0$ 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.
For y = 2/x + 3, give the value of y at the horizontal asymptote. · Pour y = 2/x + 3, donnez la valeur de y sur l'asymptote horizontale.
The 2/x term tends to 0 as |x| grows, so the horizontal asymptote is y = 3. · Le terme 2/x tend vers 0 lorsque |x| augmente, donc l'asymptote horizontale est y = 3.
You've got it
- roots · racines $=$ where the curve crosses the $x$-axis; intersection solves two equations
- a quadratic is a parabola · parabole (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