Variations et Tableau de Variation des Fonctions

TL;DR

You'll learn how a function's "variation" describes whether it's going up or down. We'll use a "tableau de variation" (variation table) to neatly summarize these changes over its domain. This table helps you quickly understand a function's behavior and find its highest or lowest points.

1. The Mental Model

Think of a function as a path you're walking. Variation tells you if the path is climbing (increasing), descending (decreasing), or flat (constant). A variation table is like a simplified map showing all the ups and downs of that path.

2. The Core Material

When we talk about the variation of a function, we're simply describing how its output (y-value) changes as its input (x-value) increases.

What is a Function Increasing?

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A function $f(x)$ is increasing on an interval if, for any two numbers $a$ and $b$ in that interval, if $a < b$, then $f(a) < f(b)$.
* In plain terms: As you move from left to right on the x-axis, the graph goes up.

What is a Function Decreasing?

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A function $f(x)$ is decreasing on an interval if, for any two numbers $a$ and $b$ in that interval, if $a < b$, then $f(a) > f(b)$.
* In plain terms: As you move from left to right on the x-axis, the graph goes down.

What is a Function Constant?

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A function $f(x)$ is constant on an interval if, for any two numbers $a$ and $b$ in that interval, $f(a) = f(b)$.
* In plain terms: As you move from left to right on the x-axis, the graph stays flat.

How to Build a Tableau de Variation (Variation Table)

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A variation table is a powerful tool to summarize a function's behavior. It has two main rows:

1. x: This row lists key x-values in ascending order, including:
   • The boundaries of the function's domain.
   • Any x-values where the function changes direction (from increasing to decreasing, or vice-versa).
   • Any x-values where the function is undefined (vertical asymptotes, etc.).
2. f(x) (or y): This row shows the behavior of the function over the intervals defined by the x-values above. We use arrows to indicate:
   • An upward arrow (⬆️) for increasing.
   • A downward arrow (⬇️) for decreasing.
   • A horizontal line (—) for constant.
   • At the x-values listed, you write the corresponding y-values, or symbols for limits ($\infty$, $-\infty$).

graph TD
    A["Identify Domain of Function"] --> B["Find Points Where Slope Changes (e.g., Derivative = 0 or undefined)"]
    B --> C["Order Critical X-values (and Domain Boundaries)"]
    C --> D["Choose Test Values in Each Interval"]
    D --> E["Determine Function's Behavior (Increasing/Decreasing/Constant) in Each Interval"]
    E --> F["Fill the Tableau de Variation: X-values & Corresponding Y-values/Arrows"]
    F --> G["Identify Local Maxima/Minima from the Table"]

Finding Critical Points (where direction changes)

For many functions you'll encounter, especially polynomials, the points where the function changes direction are often where its derivative is zero.

• If $f'(x) > 0$, the function is increasing.
• If $f'(x) < 0$, the function is decreasing.
• If $f'(x) = 0$, it's a potential local maximum or minimum (or a saddle point).

You'll learn more about derivatives soon, but for now, you can often find these points by observing the graph or using algebra for simpler functions.

3. Worked Example

Let's analyze the function $f(x) = x^2 - 4x + 3$ on the interval $[-1, 5]$.

1. Domain: The problem specifies $[-1, 5]$.
2. Find where direction changes: This is a parabola opening upwards. Its vertex is where it changes direction. The x-coordinate of the vertex of $ax^2 + bx + c$ is $-b/(2a)$.
   For $f(x) = x^2 - 4x + 3$, $a=1, b=-4$. So, $x = -(-4)/(2*1) = 4/2 = 2$.
   The function changes direction at $x=2$.

3. Calculate y-values at key points:

   • At $x = -1$: $f(-1) = (-1)^2 - 4(-1) + 3 = 1 + 4 + 3 = 8$
   • At $x = 2$: $f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$
   • At $x = 5$: $f(5) = (5)^2 - 4(5) + 3 = 25 - 20 + 3 = 8$

4. Construct the Tableau de Variation:

*   From $x=-1$ to $x=2$, the function goes from $y=8$ down to $y=-1$. It's decreasing.
*   From $x=2$ to $x=5$, the function goes from $y=-1$ up to $y=8$. It's increasing.
*   At $x=2$, $f(2)=-1$ is a local minimum.

4. Key Takeaways

• Variation describes whether a function's value is increasing, decreasing, or constant as its input increases.
• A tableau de variation is a table summarizing these changes, using arrows and key function values.
• You need to identify the function's domain and any points where its direction of variation changes.
• Points where the function changes from increasing to decreasing are local maxima.
• Points where the function changes from decreasing to increasing are local minima.

• The values in the "x" row of the table should always be in ascending order.

• Common Mistakes to Avoid:

  • Not including the domain boundaries in your "x" row.
  • Confusing the x-values with the y-values in the table.
  • Forgetting to calculate the y-values (or limits) corresponding to each x-value in the table.
  • Incorrectly determining if the function is increasing or decreasing between critical points.

5. Now Try It

Consider the function $g(x) = -x^3 + 3x^2$ on the interval $[-1, 3]$.

1. Identify the domain.
2. Find the x-values where the function changes direction. (Hint: For this type of function, you can often find these by finding points where the graph looks like it "flattens out" before changing direction. You'll formally learn about derivatives soon to make this exact.) For now, I'll tell you the points of interest are $x=0$ and $x=2$.
3. Calculate $g(x)$ at $x=-1$, $x=0$, $x=2$, and $x=3$.
4. Construct the complete tableau de variation for $g(x)$.

Success looks like: A clear, two-row table with the correct x-values, corresponding y-values, and arrows indicating whether the function is increasing or decreasing between those points.