Graphs of Polynomials Behavior at X-intercepts

Let’s explore how the graph of a polynomial behaves when it touches or crosses the x-axis, which is where its “zeros” or “x-intercepts” are found! 🧐

Understanding Zeros and Multiplicities 🎯

First, what are zeros of a polynomial? They are simply the values of ‘x’ for which the polynomial function ‘f(x)’ equals zero. Graphically, these are the points where the graph crosses or touches the x-axis.

When a polynomial is written in its factored form, such as f(x) = (x - a)ᵐ, the number ‘a’ is a zero of the polynomial. The exponent ’m’ in this factor is called the multiplicity of that zero. The multiplicity is essentially how often that factor is appearing. The behaviour of the graph at each x-intercept is critically determined by this multiplicity.

How Graphs Behave at X-Intercepts (Based on Multiplicity) 📈

The graph of a polynomial function exhibits distinct behaviours at its x-intercepts depending on whether the multiplicity of that zero is even or odd.

• Even Multiplicity (e.g., 2, 4, 6…) ↩️↪️

  • If a zero has an even multiplicity, the graph will touch the x-axis and bounce off. It will not cross the x-axis at this point.
  • This behaviour is similar to a quadratic function (like y = x²) where the parabola touches the x-axis at its vertex and turns around.
  • For higher even powers (e.g., multiplicity 4, 6), the graph will appear flatter as it approaches the x-intercept and as it leaves. Think of it as a “wider bounce”.

• Odd Multiplicity (e.g., 1, 3, 5…) ➡️⬅️

  • If a zero has an odd multiplicity, the graph will cross or intersect the x-axis.
  • If the multiplicity is 1 (a single order zero), the graph will appear almost linear as it crosses the x-axis. It’s like a straight line passing through the point.
  • For higher odd powers (e.g., multiplicity 3, 5), the graph will still cross the x-axis, but it will appear flatter as it crosses. This is often described as an “S-shape” or a “twist” as it passes through the intercept, similar to the graph of y = x³. The graph will also appear flatter while approaching the zero and leaving from it.

In summary, the key takeaway is that even multiplicities lead to a “touch and bounce” behaviour, while odd multiplicities lead to a “cross-through” behaviour, with higher multiplicities causing a “flattening” effect at the intercept.

Practice Questions 📝

Question 1: Identifying Behaviour from Multiplicity

For each given zero and its multiplicity, describe how the graph of the polynomial will behave at that x-intercept. (a) Zero at x = 4 with multiplicity 1 (b) Zero at x = -1 with multiplicity 2 (c) Zero at x = 0 with multiplicity 3 (d) Zero at x = 2 with multiplicity 4

Question 2: Matching Factors to Graph Behaviour

A polynomial P(x) has the following factored form: P(x) = (x + 3)² (x - 1) (x - 5)³. Describe the behaviour of the graph of P(x) at each of its x-intercepts.

Question 3: Sketching Based on Multiplicity (Conceptual)

Imagine a polynomial with zeros at x = -2 (multiplicity 1), x = 0 (multiplicity 2), and x = 3 (multiplicity 3). (a) Which x-intercept(s) will the graph cross? (b) Which x-intercept(s) will the graph touch and bounce off? (c) At which x-intercept will the graph appear to flatten as it crosses?

Solutions to Practice Questions ✅

Solution 1:

(a) Zero at x = 4 with multiplicity 1.

• Since the multiplicity is odd (1), the graph will cross the x-axis at x = 4. It will appear almost linear at this intercept.

(b) Zero at x = -1 with multiplicity 2.

• Since the multiplicity is even (2), the graph will touch the x-axis and bounce off at x = -1.

(c) Zero at x = 0 with multiplicity 3.

• Since the multiplicity is odd (3), the graph will cross the x-axis at x = 0. Due to the higher odd multiplicity (3), the graph will appear flatter as it crosses the x-axis compared to a linear crossing.

(d) Zero at x = 2 with multiplicity 4.

• Since the multiplicity is even (4), the graph will touch the x-axis and bounce off at x = 2. Due to the higher even multiplicity (4), the graph will appear flatter as it approaches and leaves the x-axis at this point.

Solution 2:

The polynomial is P(x) = (x + 3)² (x - 1) (x - 5)³. Let’s identify the zeros and their multiplicities:

• Zero at x = -3: The factor is (x + 3)². The multiplicity is 2 (even).
  • Behaviour: At x = -3, the graph will touch the x-axis and bounce off.
• Zero at x = 1: The factor is (x - 1). The multiplicity is 1 (odd).
  • Behaviour: At x = 1, the graph will cross the x-axis, appearing almost linear at the intercept.
• Zero at x = 5: The factor is (x - 5)³. The multiplicity is 3 (odd).
  • Behaviour: At x = 5, the graph will cross the x-axis. Due to the higher odd multiplicity (3), it will appear flatter as it crosses the x-axis.

Solution 3:

A polynomial with zeros at x = -2 (multiplicity 1), x = 0 (multiplicity 2), and x = 3 (multiplicity 3).

(a) Which x-intercept(s) will the graph cross?

• The graph will cross at zeros with odd multiplicities.
• Therefore, it will cross at x = -2 (multiplicity 1) and x = 3 (multiplicity 3).

(b) Which x-intercept(s) will the graph touch and bounce off?

• The graph will touch and bounce off at zeros with even multiplicities.
• Therefore, it will touch and bounce off at x = 0 (multiplicity 2).

(c) At which x-intercept will the graph appear to flatten as it crosses?

• Flattening at crossing occurs with higher odd multiplicities.
• Therefore, the graph will appear to flatten as it crosses at x = 3 (multiplicity 3).

This understanding of multiplicity is a powerful tool for sketching polynomial graphs and interpreting them visually! 🎨