Study Guide: Polynomial Functions – End Behavior, Zeros, Multiplicity, and Graphing

Q1. Use the leading term to determine the end behavior of the graph of the function:

• $f(x) = 5x^5 - 4x^3 + x$

• $g(x) = (x-5)^3(x+2)$

Background

Topic: End Behavior of Polynomial Functions

This question tests your understanding of how the leading term of a polynomial determines the function's end behavior as $x$ approaches $\infty$ and $-\infty$.

Key Terms and Formulas

• Leading Term: The term with the highest degree in a polynomial, which dominates the function's behavior for large $|x|$.

• End Behavior: The direction the graph heads as $x \to \infty$ or $x \to -\infty$.

For a polynomial $f(x) = a_nx^n + \ldots$, the end behavior is determined by the degree $n$ and the leading coefficient $a_n$:

• If $n$ is even and $a_n > 0$: both ends up.

• If $n$ is even and $a_n < 0$: both ends down.

• If $n$ is odd and $a_n > 0$: left end down, right end up.

• If $n$ is odd and $a_n < 0$: left end up, right end down.

Step-by-Step Guidance

1. Identify the leading term of each polynomial. For $f(x)$, it's $5x^5$; for $g(x)$, expand to find the highest degree term.

2. Determine the degree ($n$) and the sign of the leading coefficient ($a_n$) for each function.

3. Use the rules above to predict the end behavior for each function as $x \to \infty$ and $x \to -\infty$.

4. Sketch or describe the general direction of the graph's ends based on your findings.

Try solving on your own before revealing the answer!

Q2. Determine the zeros of the function defined by $f(x) = x^3 - 16x^2 - 80x$.

Background

Topic: Zeros of Polynomial Functions

This question asks you to find the values of $x$ for which $f(x) = 0$. These are the x-intercepts of the graph.

Key Terms and Formulas

• Zero (Root): A value of $x$ where $f(x) = 0$.

• Factoring: Expressing the polynomial as a product of simpler polynomials to find zeros.

Step-by-Step Guidance

1. Set $f(x) = 0$ to get $x^3 - 16x^2 - 80x = 0$.

2. Factor out the greatest common factor from all terms.

3. Factor the remaining quadratic (if possible) to find all real zeros.

4. Set each factor equal to zero and solve for $x$.

Try solving on your own before revealing the answer!

Q3. Determine the zeros and their multiplicities for $f(x) = (x-1)^2(x+3)^3$.

Background

Topic: Multiplicity of Zeros

This question tests your ability to identify zeros and their multiplicities, which affect how the graph behaves at each zero.

Key Terms and Formulas

• Multiplicity: The number of times a particular zero appears as a factor.

• If the multiplicity is odd, the graph crosses the x-axis at that zero; if even, it touches but does not cross.

Step-by-Step Guidance

1. Identify the factors of the polynomial and their exponents.

2. For each factor $(x-c)^k$, $c$ is a zero with multiplicity $k$.

3. List each zero and its multiplicity.

4. Describe whether the graph crosses or touches the x-axis at each zero based on the multiplicity.

Try solving on your own before revealing the answer!

Q4. Given $f(x) = (x-3)^2(x+4)^3(x-7)$, identify the zeros of the function and whether there is a cross point or a touch point at each zero.

Background

Topic: Zeros and Graph Behavior

This question asks you to connect the concept of multiplicity to the graphical behavior at each zero.

Key Terms and Formulas

• Zero of Odd Multiplicity: Graph crosses the x-axis.

• Zero of Even Multiplicity: Graph touches the x-axis but does not cross.

Step-by-Step Guidance

1. Identify each zero and its multiplicity from the factored form.

2. For each zero, determine if the multiplicity is odd or even.

3. State whether the graph crosses or touches the x-axis at each zero.

Try solving on your own before revealing the answer!

Q5. Give the maximum number of turning points for $f(x) = x^3 - 21x + 20$.

Background

Topic: Turning Points of Polynomials

This question tests your knowledge of how the degree of a polynomial relates to the number of turning points (relative maxima and minima) it can have.

Key Terms and Formulas

• Turning Point: A point where the graph changes direction from increasing to decreasing or vice versa.

• Maximum Number of Turning Points: For a polynomial of degree $n$, the maximum is $n-1$.

Step-by-Step Guidance

1. Identify the degree $n$ of the polynomial.

2. Apply the formula: maximum number of turning points is $n-1$.

Try solving on your own before revealing the answer!

Q6. Sketch the function $f(x) = (x-1)^2(x+3)(x-4)$.

Background

Topic: Graphing Polynomial Functions

This question asks you to combine your knowledge of end behavior, intercepts, and multiplicity to sketch a polynomial.

Key Terms and Formulas

• End Behavior: Determined by the leading term.

• Intercepts: Where the graph crosses the axes.

• Multiplicity: Affects whether the graph crosses or touches the x-axis at each zero.

Step-by-Step Guidance

1. Determine the degree and leading coefficient to predict end behavior.

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

3. List the real zeros and their multiplicities.

4. Decide for each zero if the graph crosses or touches the x-axis.

5. Plot the intercepts and sketch the general shape, connecting the points according to the end behavior and multiplicity rules.

Try solving on your own before revealing the answer!

Q7. Apply the Intermediate Value Theorem: Show that $f(x) = x^3 - 2x^2 + 13x + 35$ has a zero on the interval $[-1, 0]$.

Background

Topic: Intermediate Value Theorem (IVT)

This question tests your ability to use the IVT to prove the existence of a zero in a given interval for a continuous function.

Key Terms and Formulas

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then $f$ has at least one zero in $[a, b]$.

Step-by-Step Guidance

1. Evaluate $f(-1)$ and $f(0)$.

2. Check the signs of $f(-1)$ and $f(0)$.

3. If the signs are opposite, state that by IVT, there is at least one zero in $[-1, 0]$.

Try solving on your own before revealing the answer!