Q1. The following is a graph of a polynomial, p(x):

Background

Topic: Polynomial Functions and Their Graphs

This question tests your understanding of how to interpret the graph of a polynomial, including identifying zeros, y-intercepts, and end behavior.

Key Terms and Formulas

• Zero of a function: A value of x where p(x) = 0.

• Y-intercept: The point where the graph crosses the y-axis (x = 0).

• End behavior: The behavior of the graph as x approaches positive or negative infinity, determined by the leading term of the polynomial.

Step-by-Step Guidance

1. Identify the x-values where the graph crosses the x-axis. These are the zeros of the polynomial.

2. Find the y-intercept by locating the point where the graph crosses the y-axis (where x = 0).

3. Analyze the ends of the graph as x approaches positive and negative infinity to determine the end behavior. Consider whether the degree is odd or even and the sign of the leading coefficient.

4. Write the general form of the end behavior based on your analysis (e.g., "as x → ∞, p(x) → ...").

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Q2. Sketch the graph of the polynomial function f(x) = x^3 - 2x^2 + x + 6. By first:

Background

Topic: Graphing Polynomial Functions

This question tests your ability to analyze and sketch the graph of a cubic polynomial by finding its zeros, sign chart, and intercepts.

Key Terms and Formulas

• Factoring: Expressing the polynomial as a product of its linear factors to find zeros.

• Sign chart: A table showing where the function is positive or negative based on the zeros.

• Intercepts: Points where the graph crosses the axes.

Step-by-Step Guidance

1. Factor the polynomial to find its real zeros (roots).

2. Set up a sign chart using the zeros to determine where the function is positive or negative.

3. Find the y-intercept by evaluating f(0).

4. Sketch the graph, marking the zeros, y-intercept, and general shape based on the sign chart and end behavior.

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Q3. Graph the function f(x) = 1/(x-2). Label all asymptotes. List the domain, asymptotes, and x- and y-intercepts.

Background

Topic: Rational Functions

This question tests your understanding of rational functions, including how to find and interpret vertical and horizontal asymptotes, domain, and intercepts.

Key Terms and Formulas

• Vertical asymptote: A line x = a where the function approaches infinity as x approaches a.

• Horizontal asymptote: A line y = b that the function approaches as x goes to infinity or negative infinity.

• Domain: All real x-values except where the denominator is zero.

Step-by-Step Guidance

1. Set the denominator equal to zero to find the vertical asymptote.

2. Analyze the degrees of the numerator and denominator to determine the horizontal asymptote.

3. Find the domain by excluding values that make the denominator zero.

4. Find the x- and y-intercepts by setting y = 0 and x = 0, respectively, and solving for the other variable.

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Q4. Find a polynomial of degree 4 with the following properties. Leave in factored form.

Background

Topic: Constructing Polynomials from Zeros

This question tests your ability to construct a polynomial given its zeros and degree, and to write it in factored form.

Key Terms and Formulas

• Factored form: Expressing the polynomial as a product of linear factors corresponding to its zeros.

• Degree: The highest power of x in the polynomial.

Step-by-Step Guidance

1. List the given zeros and write each as a factor of the form (x - zero).

2. Multiply the factors together, ensuring the total degree matches the required degree.

3. If a zero is repeated, include its factor the appropriate number of times.

4. Write the polynomial in factored form, and check that the degree is correct.

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