Turning Points: The Story of Changing Direction
The Basics: Peaks and Valleys on Simple Graphs
The easiest place to start is with the graphs you already know. Picture a smooth, curvy path on a map. A turning point is where that path changes from going uphill to going downhill, or from downhill to uphill. On a graph, the "direction" is determined by whether the y-values are getting bigger (increasing) or smaller (decreasing) as you move from left to right.
The most common graph with a clear turning point is a parabola. The graph of $y = x^2$ is a U-shaped curve that opens upward. Its turning point is at the very bottom, at the origin (0, 0). At this point, the graph changes from decreasing (as you move from left toward the center) to increasing (as you move away from the center to the right). This is called a minimum point.
Conversely, the graph of $y = -x^2$ is an upside-down U. It opens downward. Its turning point is at the very top, also at (0, 0). Here, the graph changes from increasing to decreasing. This is called a maximum point.
Moving to Higher Powers: Cubic Functions and Beyond
Not all graphs have just one turning point. As functions become more complex, they can have multiple turning points. A classic example is the cubic function, like $y = x^3 - 3x$.
Let's trace its path. Starting from far left, the graph is decreasing. Then, it turns and starts increasing. That's the first turning point (a local minimum). Later, it turns again and starts decreasing. That's the second turning point (a local maximum). This creates a wave-like pattern. The number of turning points a polynomial function can have is at most one less than its degree. So a cubic (degree 3) can have up to 2 turning points, a quartic (degree 4) can have up to 3, and so on.
The Calculus Connection: Finding Turning Points with Derivatives
For students in later high school grades, a powerful tool for precisely locating turning points is introduced: differentiation. The derivative of a function, often written as $f'(x)$ or $\frac{dy}{dx}$, gives us the slope of the graph at any point.
Think of the slope as the steepness of the hill. When you are at the very peak of a hill (a maximum), you are neither going up nor down—the slope is flat. Similarly, at the very bottom of a valley (a minimum), the slope is also flat. Therefore, at a turning point, the slope (or derivative) is zero.
1. Given a function $y = f(x)$, find its derivative $f'(x)$.
2. Set the derivative equal to zero and solve: $f'(x) = 0$.
3. The solutions (x-values) are the critical points where turning points might occur.
4. Test these x-values in the original function $f(x)$ to find the corresponding y-coordinates and determine if each is a maximum or minimum.
For example, take $f(x) = x^2 - 4x + 3$. Its derivative is $f'(x) = 2x - 4$. Setting $2x - 4 = 0$ gives $x = 2$. Plugging $x = 2$ back into the original function gives $y = (2)^2 - 4(2) + 3 = -1$. So, the turning point is at (2, -1). Since the graph is a U-shaped parabola ($x^2$ has a positive coefficient), this is a minimum point.
Real-World Applications: From Business to Basketball
Turning points are not just abstract math; they help solve real problems. Imagine you run a lemonade stand. Your profit depends on the price you charge. If you charge too little, you sell a lot but make little per cup. If you charge too much, you sell very few. Somewhere in between is the "perfect" price that gives you the maximum profit. On a graph of Profit vs. Price, this perfect price is the x-coordinate of the turning point at the very top (the maximum).
In physics, if you throw a basketball, its height over time follows a parabolic path. The turning point at the top of that parabola is the maximum height the ball reaches before it starts coming back down. Knowing how to find this turning point tells you the peak of the shot.
| Field | What is Graphed? | Turning Point Meaning |
|---|---|---|
| Business | Profit vs. Price | Price for maximum profit (Maximum) |
| Sports | Height of Ball vs. Time | Maximum height reached |
| Medicine | Drug Effectiveness vs. Dose | Optimal dose before side effects increase (Maximum) |
| Environment | Pollution Level vs. Time | Point when cleanup efforts start reducing pollution (Maximum) |
Important Questions
Q1: Does every function have a turning point?
No. Many functions do not change direction at all. A straight line ($y = mx + b$) always has a constant slope and never turns. Functions like $y = x^3$ (without the $-3x$) have a point where the slope is zero, but they don't change direction from increasing to decreasing—they just momentarily flatten out before continuing to increase. This is called an inflection point, not a turning point.
Q2: What's the difference between a 'local' and an 'absolute' maximum/minimum?
A local maximum is the highest point in its immediate neighborhood—it's the top of a hill, but there might be a higher hill elsewhere on the graph. An absolute (or global) maximum is the highest point on the entire graph. All absolute maximums are local maximums, but not all local maximums are absolute. The same logic applies to minimums. On a parabola that opens upward, the single turning point is both the local and absolute minimum.
Q3: Can we find turning points without using calculus?
Yes, for simpler functions. For quadratic functions ($ax^2 + bx + c$), you can use the vertex formula $x = -\frac{b}{2a}$ or complete the square. For graphs that are symmetric, you can sometimes find the turning point by looking at the symmetry. However, for more complicated functions, calculus provides a reliable and universal method that always works.
Conclusion
Footnote
[1] Derivative: A fundamental concept in calculus that measures how a function changes as its input changes. It represents the instantaneous rate of change or the slope of the tangent line to the graph of the function at a point.
[2] Parabola: The U-shaped curve that is the graph of a quadratic function of the form $y = ax^2 + bx + c$.
[3] Vertex: The "tip" or turning point of a parabola. For a quadratic function, it is the point where the graph reaches its maximum or minimum value.
[4] Critical Point: A point on the graph of a function where its derivative is zero or undefined. Turning points occur at critical points where the derivative changes sign.
[5] Inflection Point: A point on a curve where the curvature changes sign, meaning the graph changes from being concave up (like a cup) to concave down (like a cap), or vice versa. The slope may be zero, but it is not a turning point because the direction of increase/decrease does not change.