Graphs of Polynomaials | Turning Point

Let’s explore turning points in the graphs of polynomial functions! 🎢

A turning point on a polynomial graph is a specific location where the graph changes its direction. Imagine you’re tracing the graph with your finger:

• If your finger was moving upwards (the function was increasing 📈) and now it starts moving downwards (the function is decreasing 📉), that point is a turning point. This is called a local maximum.
• Conversely, if your finger was moving downwards (the function was decreasing 📉) and now it starts moving upwards (the function is increasing 📈), that point is also a turning point. This is called a local minimum.

These “ups and downs” are typical features of polynomial functions. You can visualise them as the peaks and valleys on the curve.

The Link Between Degree and Turning Points 🔗

The number of turning points a polynomial graph can have is directly related to its degree (the highest exponent of the variable).

For a polynomial of degree n, it can have at most (maximum) n - 1 turning points. This means it can have n-1 turning points, or fewer, but never more than n-1.

Let’s look at some examples:

• A linear polynomial (degree n=1) like f(x) = x has no turning points (at most 1-1 = 0).
• A quadratic polynomial (degree n=2) like f(x) = x² has at most 2-1 = 1 turning point, which is its vertex.
• A cubic polynomial (degree n=3) like f(x) = x³ - x can have at most 3-1 = 2 turning points.
• A polynomial of degree 4 can have at most 4-1 = 3 turning points.
• A polynomial of degree 5 can have at most 5-1 = 4 turning points.

This relationship is a useful check when sketching graphs: if your graph of a degree n polynomial shows more than n-1 turning points, you know your graph is incorrect.

Locating Turning Points (A Glimpse into Calculus) 🔍

While we can identify how many turning points a polynomial can have based on its degree, precisely locating these turning points on a graph often requires more advanced mathematical tools, specifically from calculus.

In calculus, you learn that at a turning point, the slope of the function becomes zero. However, there can be points where the slope is zero but it’s not a peak or a valley (e.g., an inflection point where the graph flattens momentarily before continuing in the same general direction). Without calculus, you might only be able to roughly estimate the turning points.

Practice Questions 📝

Question 1: Maximum Turning Points

Determine the maximum number of turning points for each polynomial function:

(a) f(x) = 5x⁷ - 2x⁴ + 9x (b) g(x) = -x² + 6x - 10 (c) h(x) = x¹⁰⁰ + 3x⁵⁰ - 1

Question 2: Inferring Degree

A polynomial graph has the following characteristics:

• It has 3 turning points.
• Its left end goes down, and its right end goes up.

(a) What is the minimum possible degree of this polynomial? (b) Is its leading coefficient positive or negative?

Question 3: Identifying Turning Points from Description

Which of the following describes a turning point of a function? (a) A point where the graph crosses the x-axis. (b) A point where the graph changes from increasing to decreasing. (c) A point where the graph remains constant. (d) A point where the graph changes from a positive slope to a negative slope.

Solutions to Practice Questions ✅

Solution 1:

The maximum number of turning points for a polynomial of degree n is n - 1.

(a) f(x) = 5x⁷ - 2x⁴ + 9x

• Degree: n = 7
• Maximum Turning Points: 7 - 1 = 6 🔄

(b) g(x) = -x² + 6x - 10

• Degree: n = 2
• Maximum Turning Points: 2 - 1 = 1 ⛰️

(c) h(x) = x¹⁰⁰ + 3x⁵⁰ - 1

• Degree: n = 100
• Maximum Turning Points: 100 - 1 = 99 🏔️

Solution 2:

(a) What is the minimum possible degree of this polynomial?

• A polynomial of degree n has at most n - 1 turning points.
• If the polynomial has 3 turning points, then n - 1 ≥ 3, which means n ≥ 4.
• Therefore, the minimum possible degree of this polynomial is 4 (a quartic polynomial).

(b) Is its leading coefficient positive or negative?

• The end behaviour indicates: as x → ∞, f(x) → ∞ (right end up) and as x → -∞, f(x) → -∞ (left end down).
• For a polynomial whose ends point in opposite directions, the degree must be odd.
• However, we determined the minimum degree is 4, which is an even number.
• Let’s re-evaluate the end behaviour description: “left end down, right end up”.
  • For an odd degree polynomial:
    • aₙ > 0: Left end down, right end up (like y=x³). This matches the description.
    • aₙ < 0: Left end up, right end down (like y=-x³).
  • For an even degree polynomial:
    • aₙ > 0: Both ends up (like y=x²).
    • aₙ < 0: Both ends down (like y=-x²).
• Since the description states “left end down, right end up”, this contradicts an even-degree polynomial. There might be an ambiguity in the question or an assumption that the number of turning points directly corresponds to the degree for all polynomials.
• Clarification based on sources: The number of turning points is at most n-1. An odd-degree polynomial always has its ends going in opposite directions. An even-degree polynomial always has its ends going in the same direction.
• Given the end behaviour “left end down, right end up”, the polynomial must be of odd degree.
• If it’s an odd-degree polynomial, and it has 3 turning points, then the minimum degree n must satisfy n - 1 ≥ 3, so n ≥ 4. The smallest odd integer that is ≥ 4 is 5.
• Therefore, the minimum possible degree is 5 for this polynomial to satisfy both conditions.
• For an odd-degree polynomial that goes “left end down, right end up”, the leading coefficient must be positive.

Solution 3:

(a) A point where the graph crosses the x-axis. (This describes an x-intercept or zero, not necessarily a turning point.) (b) A point where the graph changes from increasing to decreasing. (This is the definition of a local maximum, which is a type of turning point.) (c) A point where the graph remains constant. (This would be a horizontal line segment, not a turning point as defined for polynomials.) (d) A point where the graph changes from a positive slope to a negative slope. (This is another way of describing a local maximum, matching definition. This statement is also correct.)

Correct options: (b) and (d).