Graphing Polynomial Functions
Overview
Graphing polynomial functions allows us to understand their behavior by plotting them on a coordinate plane. The general form of a polynomial function is:
\( f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \)
Where \( a_n \) is the leading coefficient, and \( n \) is the degree of the polynomial. The graph of a polynomial function is smooth and continuous. Key characteristics of polynomial graphs include:
- Degree of the Polynomial: The degree determines the overall shape of the graph. For example, even-degree polynomials have symmetric graphs, while odd-degree polynomials have graphs that go in opposite directions at the ends.
- End Behavior: The end behavior refers to the direction in which the graph moves as \( x \) approaches infinity or negative infinity. This is determined by the degree and the leading coefficient of the polynomial.
- Roots/Zeroes: The x-intercepts (or roots) of the polynomial are the values of \( x \) for which \( f(x) = 0 \). These correspond to the factors of the polynomial.
- Turning Points: The graph of a polynomial function can change direction at turning points. The maximum number of turning points is \( n-1 \), where \( n \) is the degree of the polynomial.
Example 1: Graphing a Quadratic Polynomial
Consider the polynomial function \( f(x) = x^2 - 4x + 3 \). We can find the roots by factoring:
\( f(x) = x^2 - 4x + 3 = (x - 1)(x - 3) \)
The roots are \( x = 1 \) and \( x = 3 \). The graph is a parabola opening upwards because the leading coefficient is positive. The vertex of the parabola is located at the point halfway between the roots, at \( x = 2 \). The graph looks like this:
Example 2: Graphing a Cubic Polynomial
Consider the polynomial function \( f(x) = x^3 - 3x^2 - 4x + 12 \). We can find the roots by factoring:
\( f(x) = x^3 - 3x^2 - 4x + 12 = (x - 2)(x^2 - x - 6) \)
Next, factor the quadratic part:
\( x^2 - x - 6 = (x - 3)(x + 2) \)
The roots are \( x = 2 \), \( x = 3 \), and \( x = -2 \). The graph is an S-shaped curve with these x-intercepts.
Practice Questions
- Question 1: Graph the polynomial \( f(x) = x^2 - 6x + 8 \).
Solution
Factor the quadratic polynomial:
\( f(x) = x^2 - 6x + 8 = (x - 2)(x - 4) \)
The roots are \( x = 2 \) and \( x = 4 \). Since the leading coefficient is positive, the graph is a parabola opening upwards. The vertex is at \( x = 3 \), halfway between the roots.
The graph will be a U-shaped curve with the roots at \( x = 2 \) and \( x = 4 \).
- Question 2: Graph the polynomial \( f(x) = x^3 - 5x^2 + 4x \).
Solution
Factor the polynomial:
\( f(x) = x^3 - 5x^2 + 4x = x(x^2 - 5x + 4) \)
Factor the quadratic part:
\( x^2 - 5x + 4 = (x - 1)(x - 4) \)
The roots are \( x = 0 \), \( x = 1 \), and \( x = 4 \). The graph will pass through the x-axis at these points and will be an S-shaped cubic curve.
- Question 3: Graph the polynomial \( f(x) = x^3 + 3x^2 - 4x - 12 \).
Solution
Factor the polynomial:
\( f(x) = x^3 + 3x^2 - 4x - 12 = (x + 3)(x^2 - 4) \)
Factor the quadratic part:
\( x^2 - 4 = (x - 2)(x + 2) \)
The roots are \( x = -3 \), \( x = 2 \), and \( x = -2 \). The graph is a cubic curve with these x-intercepts, passing through at \( x = -3 \), \( x = 2 \), and \( x = -2 \).
- Question 4: Graph the polynomial \( f(x) = x^2 + 4x + 3 \).
Solution
Factor the quadratic polynomial:
\( f(x) = x^2 + 4x + 3 = (x + 1)(x + 3) \)
The roots are \( x = -1 \) and \( x = -3 \). Since the leading coefficient is positive, the graph is a parabola opening upwards. The vertex is at \( x = -2 \), halfway between the roots.
- Question 5: Graph the polynomial \( f(x) = x^4 - 4x^2 \).
Solution
Factor the polynomial:
\( f(x) = x^4 - 4x^2 = x^2(x^2 - 4) = x^2(x - 2)(x + 2) \)
The roots are \( x = 0 \), \( x = 2 \), and \( x = -2 \). The graph will touch the x-axis at \( x = 0 \) and pass through at \( x = 2 \) and \( x = -2 \).