Graph each polynomial function. ƒ(x)=(x-2)²(x+3)

Verified step by step guidance

1. Identify the polynomial function given: \(f(x) = (x-2)^2 (x+3)\). Notice it is a product of factors, which will help determine zeros and their multiplicities.
2. Find the zeros of the function by setting each factor equal to zero: \(x - 2 = 0\) gives \(x = 2\), and \(x + 3 = 0\) gives \(x = -3\). These are the x-intercepts of the graph.
3. Determine the multiplicity of each zero: The zero at \(x = 2\) has multiplicity 2 (because of the squared factor), and the zero at \(x = -3\) has multiplicity 1. This affects the shape of the graph at these points (touching vs crossing the x-axis).
4. Analyze the end behavior of the polynomial by considering the degree and leading coefficient. The degree is 3 (since \((x-2)^2\) contributes degree 2 and \((x+3)\) contributes degree 1), and the leading term will be positive, so as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
5. Plot the zeros on the x-axis, sketch the behavior near each zero based on multiplicity, and use the end behavior to draw the overall shape of the graph. Optionally, calculate \(f(0)\) to find the y-intercept for additional accuracy.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents. Understanding the degree and terms of the polynomial helps predict the general shape and behavior of its graph.

Zeros and Multiplicity

Zeros of a polynomial are the values of x that make the function equal to zero. The multiplicity of a zero indicates how many times that root is repeated, affecting the graph's behavior at that point—whether it crosses or just touches the x-axis.

End Behavior of Polynomials

End behavior describes how the graph of a polynomial behaves as x approaches positive or negative infinity. It depends on the leading term's degree and coefficient, helping to sketch the overall direction of the graph's arms.