Polynomial Functions from a Graph 🕵️‍♀️

Understanding the graphs of polynomial functions involves a twofold mission: first, being able to identify whether a given graph represents a polynomial function, and second, understanding the key characteristics that shape the graph of a polynomial.

Identifying Polynomial Functions from a Graph 🕵️‍♀️

When presented with a graph, you can determine if it’s a polynomial function by checking for two main properties:

• Smoothness ✨: Polynomial functions always display smooth curves, meaning they do not have any sharp corners or edges. If you try to draw a polynomial graph, you should be able to join the points effortlessly without experiencing any “abrupt jerk”. If a graph has a corner or an edge, it is unlikely to be a polynomial function. For example, graphs of linear and quadratic functions (which are types of polynomials) are drawn smoothly without jerks.
• Continuity 〰️: Polynomial functions are continuous curves, meaning they have no breaks. You should be able to draw the entire graph without lifting your pen. If a graph requires you to lift your pen to continue drawing (indicating a break or discontinuity), then it is not a polynomial function.

For instance, a graph that is smooth but has a sharp corner, like one shown in the sources, would not qualify as a polynomial function. Similarly, a graph that is smooth in sections but has a break where you’d need to lift your pen, like the one illustrated with a discontinuity at x=0, would also be disqualified. Conversely, a graph that is visibly smooth and continuous, resembling a line or a curve with gentle turns, would qualify as a polynomial function.

Characterising Graphs of Polynomial Functions 📊

Once a function is identified as a polynomial, its graph displays specific characteristics:

• Zeros/X-intercepts 🎯: These are the values of ‘x’ for which the polynomial function f(x) equals zero. On a graph, these correspond to the points where the function intersects or touches the x-axis, also known as x-intercepts. Factoring the polynomial’s equation is a crucial technique to find its zeros.

• Multiplicities of Zeros 🔢: The behavior of a polynomial graph at its x-intercepts is determined by the “multiplicity” of the corresponding factor – how often that factor appears in the polynomial’s factored form.

  • If a zero has an even multiplicity (e.g., the factor is squared, like (x-a)²), the graph will touch the x-axis and bounce off it at that intercept, rather than crossing it. As the even power increases (e.g., from x² to x⁴ or x⁶), the graph appears flatter as it approaches and leaves the zero.
  • If a zero has an odd multiplicity (e.g., the factor is cubed, like (x-a)³), the graph will cross or intersect the x-axis at that intercept. If the graph appears almost linear at the intercept, it indicates a single (first) order multiplicity. Higher odd powers (e.g., x⁵, x⁷) will make the graph appear flatter while approaching and leaving the zero.
  • The sum of the multiplicities of all real zeros should be less than or equal to the degree of the polynomial. This is because some polynomials may have complex roots that do not appear as x-intercepts on a real coordinate plane.

• End Behavior 🧭: This describes how the graph behaves as ‘x’ approaches positive or negative infinity (i.e., the far left and far right ends of the graph). The end behavior of a polynomial is determined solely by its leading term (the term with the highest degree and its coefficient).

  • Even Degree Exponent (n) and Positive Leading Coefficient (aₙ > 0): As ‘x’ goes to either positive or negative infinity, f(x) will go up to positive infinity (↗️↖️). This is similar to a quadratic function opening upwards.
  • Even Degree Exponent (n) and Negative Leading Coefficient (aₙ < 0): As ‘x’ goes to either positive or negative infinity, f(x) will go down to negative infinity (↘️↙️). This is similar to a quadratic function opening downwards.
  • Odd Degree Exponent (n) and Positive Leading Coefficient (aₙ > 0): As ‘x’ goes to positive infinity, f(x) goes up to positive infinity (↗️); as ‘x’ goes to negative infinity, f(x) goes down to negative infinity (↙️). This resembles the graph of x³.
  • Odd Degree Exponent (n) and Negative Leading Coefficient (aₙ < 0): As ‘x’ goes to positive infinity, f(x) goes down to negative infinity (↘️); as ‘x’ goes to negative infinity, f(x) goes up to positive infinity (↖️). This is the opposite of the previous odd degree case.

• Turning Points 🎢: These are points on the graph where the function changes its behavior from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum).

  • A polynomial of degree ’n’ can have at most n-1 turning points. For example, a quadratic function (degree 2) has at most one turning point, and a cubic polynomial (degree 3) can have at most two turning points.
  • The exact location of turning points typically requires calculus to identify precisely. However, this property helps verify if a sketched graph is reasonable.

Graphing and Polynomial Creation Strategy ✍️

To sketch a polynomial graph from its equation:

1. Identify x-intercepts (zeros) and their multiplicities.
2. Determine the y-intercept (by setting x=0).
3. Analyse the end behavior using the leading term.
4. Consider the maximum number of turning points (n-1).
5. Sketch the graph by connecting these points smoothly, respecting the behavior at intercepts and the end behavior.

To derive an algebraic equation from a given polynomial graph:

1. Identify all x-intercepts from the graph, as these correspond to the factors of the polynomial.
2. Determine the multiplicity of each factor by observing the graph’s behavior at each x-intercept (bounces off for even multiplicity, crosses for odd multiplicity).
3. Infer the least possible degree of the polynomial based on the sum of multiplicities and the end behavior (odd/even degree).
4. Set up a preliminary equation using the factors and their multiplicities, with an unknown “stretch factor” (often denoted as ‘a’).
5. Use another known point from the graph, such as the y-intercept, to solve for the stretch factor ‘a’.
6. Substitute the value of ‘a’ back into the equation to get the full algebraic expression.

By understanding these properties, you can effectively identify and characterize polynomial functions and their graphs.