In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x) = x^3 - x^2 - 9x + 9

Verified step by step guidance

1. **Step 1: Use the Leading Coefficient Test**: Identify the leading term of the polynomial function. For \( f(x) = x^3 - x^2 - 9x + 9 \), the leading term is \( x^3 \). Since the degree is odd and the leading coefficient is positive, the end behavior of the graph is: as \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to \infty \), \( f(x) \to \infty \).
2. **Step 2: Check for y-axis symmetry**: A function has y-axis symmetry if \( f(-x) = f(x) \) for all \( x \). Substitute \( -x \) into the function: \( f(-x) = (-x)^3 - (-x)^2 - 9(-x) + 9 = -x^3 - x^2 + 9x + 9 \). Since \( f(-x) \neq f(x) \), the function does not have y-axis symmetry.
3. **Step 3: Check for origin symmetry**: A function has origin symmetry if \( f(-x) = -f(x) \) for all \( x \). Using the expression for \( f(-x) \) from Step 2, compare it to \( -f(x) = -(x^3 - x^2 - 9x + 9) = -x^3 + x^2 + 9x - 9 \). Since \( f(-x) \neq -f(x) \), the function does not have origin symmetry.
4. **Step 4: Determine symmetry**: Since the function has neither y-axis symmetry nor origin symmetry, it has neither symmetry.
5. **Step 5: Graph the function**: Plot the function \( f(x) = x^3 - x^2 - 9x + 9 \) using the end behavior and any critical points or intercepts to guide the shape of the graph.

Key Concepts

Leading Coefficient Test

The Leading Coefficient Test is a method used to determine the end behavior of polynomial functions based on the sign and degree of the leading term. For a polynomial of the form f(x) = ax^n, where 'a' is the leading coefficient and 'n' is the degree, if 'n' is even, the ends of the graph will either both rise or both fall, depending on the sign of 'a'. If 'n' is odd, one end will rise while the other falls. This test helps predict how the graph behaves as x approaches positive or negative infinity.

Symmetry in Graphs

Symmetry in graphs refers to the property of a function where its graph remains unchanged under certain transformations. A function has y-axis symmetry if f(-x) = f(x), indicating it is even, and it has origin symmetry if f(-x) = -f(x), indicating it is odd. Identifying symmetry can simplify the graphing process and provide insights into the function's behavior, particularly in determining the presence of certain features like intercepts and turning points.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points based on the function's values and analyzing its features such as intercepts, turning points, and end behavior. Key steps include finding the roots of the polynomial, determining the y-intercept, and using the Leading Coefficient Test to understand the end behavior. Additionally, understanding the degree of the polynomial helps predict the number of turning points, which are critical for sketching an accurate graph.