The Logic of the “Double Bend”#
A cubic function ($f(x) = ax^3 + bx^2 + cx + d$) can have up to two turning points and three x-intercepts. Calculus gives us the tools to find all of them precisely.
1. The 7-Step Sketching Method#
Follow these steps in order to sketch any cubic graph:
Step 1: Shape ($a$ value)#
- If $a > 0$: graph goes from bottom-left to top-right (↗)
- If $a < 0$: graph goes from top-left to bottom-right (↘)
Step 2: y-intercept#
Set $x = 0$: $y = d$ (the constant term). Plot the point $(0; d)$.
Step 3: x-intercepts#
Set $f(x) = 0$ and solve. Use the Factor Theorem to find the roots:
- Find one root by trial.
- Divide to get a quadratic.
- Solve the quadratic for the remaining roots.
Step 4: Stationary (Turning) Points#
Find $f'(x)$ and set $f'(x) = 0$. Solve for $x$. Substitute back into $f(x)$ (NOT $f'(x)$!) to get the $y$-coordinates.
Step 5: Nature of Turning Points#
Use the second derivative test:
- $f''(x) > 0$ at the point → Local Minimum ($\smile$)
- $f''(x) < 0$ at the point → Local Maximum ($\frown$)
Step 6: Point of Inflection#
Set $f''(x) = 0$ and solve for $x$. Substitute into $f(x)$ for the $y$-coordinate. This is where concavity changes from $\smile$ to $\frown$ (or vice versa).
Step 7: Plot and Connect#
Plot all points found above, and draw a smooth curve through them following the shape determined by $a$.
2. Full Worked Example#
Sketch $f(x) = x^3 - 3x^2 - 9x + 27$
Step 1: Shape#
$a = 1 > 0$ → graph goes from bottom-left to top-right (↗).
Step 2: y-intercept#
$f(0) = 27$. Point: $(0; 27)$.
Step 3: x-intercepts#
$f(x) = 0$: Try $x = 3$: $f(3) = 27 - 27 - 27 + 27 = 0$ ✓
Divide by $(x - 3)$: $x^3 - 3x^2 - 9x + 27 = (x-3)(x^2 - 9) = (x-3)(x-3)(x+3) = (x-3)^2(x+3)$
Roots: $x = 3$ (repeated — graph touches the axis) and $x = -3$ (graph crosses).
Step 4: Turning Points#
$f'(x) = 3x^2 - 6x - 9 = 3(x^2 - 2x - 3) = 3(x-3)(x+1)$
$f'(x) = 0$: $x = 3$ or $x = -1$
$f(3) = 0$ → Turning point at $(3; 0)$ $f(-1) = -1 - 3 + 9 + 27 = 32$ → Turning point at $(-1; 32)$
Step 5: Nature#
$f''(x) = 6x - 6$
At $x = 3$: $f''(3) = 18 - 6 = 12 > 0$ → Local Minimum ✓ At $x = -1$: $f''(-1) = -6 - 6 = -12 < 0$ → Local Maximum ✓
Step 6: Point of Inflection#
$f''(x) = 0$: $6x - 6 = 0$ → $x = 1$ $f(1) = 1 - 3 - 9 + 27 = 16$. Inflection point at $(1; 16)$.
Summary of Key Points:#
| Feature | Value |
|---|---|
| y-intercept | $(0; 27)$ |
| x-intercepts | $(-3; 0)$ and $(3; 0)$ |
| Local Maximum | $(-1; 32)$ |
| Local Minimum | $(3; 0)$ |
| Point of Inflection | $(1; 16)$ |
3. Reading Information from a Given Graph#
Exams also give you the graph and ask you to determine the equation. The strategy:
Given turning points#
If you know the turning points, use the fact that $f'(x) = 0$ at those $x$-values:
- Find $f'(x) = 3ax^2 + 2bx + c$.
- The turning point x-values are the roots of $f'(x) = 0$.
- Use additional information (y-intercept, a specific point) to find the remaining constants.
Given x-intercepts#
If you know all three x-intercepts ($p$, $q$, $r$):
$$ f(x) = a(x - p)(x - q)(x - r) $$Use one more point to find $a$.
4. Intervals of Increase and Decrease#
- $f$ is increasing where $f'(x) > 0$.
- $f$ is decreasing where $f'(x) < 0$.
For the example above: $f'(x) = 3(x-3)(x+1)$
| Interval | Sign of $f'(x)$ | $f$ is… |
|---|---|---|
| $x < -1$ | $(-)(-) = +$ | Increasing |
| $-1 < x < 3$ | $(-)(+) = -$ | Decreasing |
| $x > 3$ | $(+)(+) = +$ | Increasing |
5. Concavity#
- $f$ is concave up ($\smile$) where $f''(x) > 0$.
- $f$ is concave down ($\frown$) where $f''(x) < 0$.
For the example: $f''(x) = 6x - 6$
| Interval | Sign of $f''(x)$ | Concavity |
|---|---|---|
| $x < 1$ | Negative | Concave down ($\frown$) |
| $x > 1$ | Positive | Concave up ($\smile$) |
The inflection point at $x = 1$ is where concavity changes.
🚨 Common Mistakes#
- Plugging into $f'(x)$ instead of $f(x)$: After finding the turning point $x$-values from $f'(x) = 0$, you must substitute into the original $f(x)$ to get $y$. Substituting into $f'(x)$ always gives 0!
- Confusing x-intercepts with turning points: A cubic can have up to 3 x-intercepts ($f(x) = 0$) and up to 2 turning points ($f'(x) = 0$). These are completely different questions.
- Forgetting the repeated root: If $(x-3)^2$ is a factor, the graph touches the x-axis at $x = 3$ but doesn’t cross. A single factor means the graph crosses.
- Wrong shape direction: If $a < 0$, the graph starts high and ends low. The local max is on the RIGHT and the local min is on the LEFT — the reverse of $a > 0$.
💡 Pro Tip: The “Symmetry” Logic#
The Point of Inflection is ALWAYS the exact midpoint between the two turning points. If your turning points are at $x = -1$ and $x = 3$, the inflection is at $x = \frac{-1 + 3}{2} = 1$. Use this as a quick check!
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