SAT Math Lesson Plan 21: Polynomial and Their Graphs

Lesson Summary

In this lesson, students will learn to analyze and graph polynomials, focusing on key characteristics such as degree, leading coefficient, and roots. The lesson explores how polynomials can be factored and how the factored form helps identify the graph’s x-intercepts. Students will also study the behavior of polynomials at both ends of the graph (end behavior). By the end of the lesson, students will be able to recognize and graph polynomial functions and understand their key properties, which are essential for solving problems in the SAT Math section.

This lesson will equip students with the necessary skills to handle questions related to polynomials and their graphs, including factoring and identifying roots, as well as understanding end behavior. Polynomials and their graphs make up an essential portion of algebra questions on the SAT Math section.

Approximately 10-15% of the SAT Math questions focus on polynomials and their graphs, with topics like factoring, identifying the degree of polynomials, and interpreting their behavior on the graph. This lesson aligns well with these topics and will help students navigate these common types of questions effectively.

Lesson Objectives

• Identify the key characteristics of a polynomial, including degree, leading coefficient, and roots.
• Factor polynomials to find the roots (solutions) of polynomial equations.
• Understand end behavior of polynomial functions.
• Graph polynomial functions based on their characteristics.
• Apply knowledge of polynomial functions to solve SAT-style questions involving polynomials and their graphs.

Common Core Standards

• CCSS.MATH.CONTENT.HSA.CED.A.2: Create equations that describe numbers or relationships.
• CCSS.MATH.CONTENT.HSA.SSE.B.3: Write expressions in equivalent forms to solve problems.
• CCSS.MATH.CONTENT.HSA.REI.B.4: Solve quadratic equations in one variable.
• CCSS.MATH.CONTENT.HSF.IF.C.7: Analyze functions using different representations.

Prerequisite Skills

• Understanding of basic algebraic operations (addition, subtraction, multiplication, division).
• Ability to factor polynomials of the form \( ax^2 + bx + c \).
• Understanding of function notation and basic graphing principles.
• Familiarity with quadratic functions and their graphs.
• Knowledge of end behavior for linear and quadratic functions.

Key Vocabulary

• Polynomial: An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
• Degree: The highest power of the variable in a polynomial. For example, the degree of \( 3x^2 + 2x - 5 \) is 2.
• Leading Coefficient: The coefficient of the term with the highest degree in a polynomial. For example, in \( 3x^2 + 2x - 5 \), the leading coefficient is 3.
• Root (or Zero): A solution to a polynomial equation, where the polynomial equals zero.
• End Behavior: The direction the graph of a polynomial function moves as \( x \) approaches positive or negative infinity.

Warm Up

Directions: Review key concepts related to polynomials, their graphs, and transformations.

Activity 1: Review of Polynomial Terminology

Before diving into the graphs of polynomials, let's review some important terminology. A polynomial is an expression that involves sums and products of variables raised to non-negative integer powers. The degree of a polynomial is determined by the highest power of the variable, and the leading coefficient is the coefficient of the highest degree term. Here are some examples:

Activity 2: Graphing Polynomial Functions

Polynomials have different types of graphs depending on their degree and leading coefficient. To warm up, let's graph a simple polynomial function. Consider the function \( f(x) = x^2 - 4 \). This is a quadratic function, so its graph will be a parabola. You can use Desmos or a graphing calculator to see the curve. Notice that the graph opens upward because the leading coefficient is positive, and the vertex is at (0, -4).

Try graphing the following polynomials and observing their behavior:

• \( f(x) = x^2 + 2x - 3 \)
• \( f(x) = -x^2 + 5 \)

Activity 3: Review of Function Notation

Function notation is an essential concept when working with polynomials. In this activity, we’ll review how to read and use function notation. For example, if \( f(x) = x^2 + 2x + 1 \), then \( f(3) \) means that we substitute \( 3 \) for \( x \) in the function:

Teach

Introduction to Polynomials and Their Graphs

In this lesson, we will explore the key concepts of polynomials and how to analyze and graph them. These topics are crucial for understanding various types of questions you'll encounter on the SAT Math test. Specifically, we will cover:

• Identifying and working with different types of polynomial functions.
• Understanding the degree and leading coefficient of a polynomial, and how they affect the graph.
• Graphing polynomials by identifying key features such as roots, end behavior, and turning points.
• Recognizing how the behavior of the graph corresponds to the polynomial's factored form.

We will begin by reviewing the structure of polynomials, followed by examples of how to graph them based on their properties. We will then look at how polynomials can be factored, and how these factored forms relate to the graph of the function. This understanding will help you tackle SAT Math questions related to polynomial functions and their graphs.

By the end of this lesson, you'll be comfortable with graphing polynomials and identifying key features that are essential for solving SAT-style problems.

Now, let’s go deeper into how polynomials are graphed and analyzed. In this section, we will explore different types of polynomials, their behavior, and how to graph them.

Example 1: Analyzing a Polynomial Function

Consider the polynomial function \( f(x) = 3x^3 - 2x^2 + 4x - 5 \). 

Question: What are the following characteristics of the polynomial function \( f(x) = 3x^3 - 2x^2 + 4x - 5 \)?

• The leading coefficient
• The degree
• The end behavior of the function

Use the polynomial function to analyze and answer the questions.

Solution:

1. The leading coefficient is the coefficient of the highest degree term in the polynomial. In this case, the highest degree term is \( 3x^3 \), so the leading coefficient is 3.

2. The degree of the polynomial is the highest exponent in the expression. Since the highest power of \( x \) is \( x^3 \), the degree of this polynomial is 3.

3. The end behavior is determined by the degree and the sign of the leading coefficient. 
Since the degree is odd and the leading coefficient is positive, the polynomial will behave as follows:

• As \( x \to \infty \), \( f(x) \to \infty \).
• As \( x \to -\infty \), \( f(x) \to -\infty \).

This means that as the values of \( x \) increase in the positive direction, the function's values will increase to positive infinity. As \( x \) decreases, the function's values will decrease to negative infinity.

Example 2: Factoring and Graphing a Polynomial

Let’s factor the polynomial \( f(x) = x^2 - 5x + 6 \). Factoring will help us find the roots of the polynomial, which are key for graphing the function.

The factored form is \( (x - 2)(x - 3) \), so the roots are \( x = 2 \) and \( x = 3 \). This means that the graph of this quadratic will intersect the x-axis at \( x = 2 \) and \( x = 3 \), and it will be a parabola opening upwards because the leading coefficient is positive.

Example 3: Analyzing the Inflection Point of a Polynomial Function

Consider the graph of the polynomial function \( f(x) = x^3 - 6x^2 + 11x - 6 \). Given the graph, analyze its behavior in terms of roots, inflection points, and end behavior.

Problem: Use the graph of this polynomial function to analyze its behavior. 

Solution:

• Roots: The graph intersects the x-axis at \( x = 1 \), \( x = 2 \), and \( x = 3 \). These are the roots of the polynomial, which means that \( f(x) = 0 \) at these points. Therefore, the values of \( x \) that satisfy the equation \( f(x) = 0 \) are \( x = 1 \), \( x = 2 \), and \( x = 3 \). These represent the solutions to the equation \( f(x) = 0 \).
• Inflection Point: The graph has an inflection point at \( x = 2 \), where the graph changes concavity. This means the graph transitions from concave down to concave up at \( x = 2 \).
• End Behavior: Based on the graph, we observe that as \( x \to \infty \), \( f(x) \to \infty \) (the graph rises to the right), and as \( x \to -\infty \), \( f(x) \to -\infty \) (the graph falls to the left). This is consistent with the behavior of a cubic polynomial with a positive leading coefficient.

Example 4: Analyzing the End Behavior of Polynomial Functions

Consider the following three polynomial functions. Analyze their end behavior based on their graphs and equations.

Problem: Given the graphs of these polynomial functions, analyze their end behavior. Specifically, observe the direction of the graph as \( x \to \infty \) and as \( x \to -\infty \).

Solution: In the table below, we analyze the end behavior of three polynomial functions. The first row shows their graphs, and the second row displays their equations. You will observe different types of end behavior based on the degree and leading coefficient of each polynomial, and whether the function is even or odd.

Equation: \( f(x) = x^2 + 4x + 3 \)	Equation: \( g(x) = -x^2 + 3x - 2 \)	Equation: \( h(x) = x^3 - 6x^2 + 9x \)

End Behavior Analysis:

• For \( f(x) = x^2 + 4x + 3 \): 
  As \( x \to \infty \), \( f(x) \to \infty \) (the graph rises to the right). As \( x \to -\infty \), \( f(x) \to \infty \) (the graph rises to the left). This is a quadratic function with a positive leading coefficient. 
  More specifically, since the degree is even and the leading coefficient is positive, the graph approaches positive infinity on both ends: \[ \lim_{x \to \infty} f(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty \]
• For \( g(x) = -x^2 + 3x - 2 \): 
  As \( x \to \infty \), \( g(x) \to -\infty \) (the graph falls to the right). As \( x \to -\infty \), \( g(x) \to -\infty \) (the graph falls to the left). This is a quadratic function with a negative leading coefficient. 
  More specifically, since the degree is even and the leading coefficient is negative, the graph approaches negative infinity on both ends: \[ \lim_{x \to \infty} g(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} g(x) = -\infty \]
• For \( h(x) = x^3 - 6x^2 + 9x \): 
  As \( x \to \infty \), \( h(x) \to \infty \) (the graph rises to the right). As \( x \to -\infty \), \( h(x) \to -\infty \) (the graph falls to the left). This is a cubic function with an odd degree. 
  More specifically, since the degree is odd and the leading coefficient is positive, the graph approaches positive infinity on the right end and negative infinity on the left end: \[ \lim_{x \to \infty} h(x) = \infty \quad \text{and} \quad \lim_{x \to -\infty} h(x) = -\infty \] 
  Key Concept: Odd degree functions have opposite end behaviors. The sign of the leading coefficient determines whether the graph rises or falls on the right side.

The graphs and corresponding equations illustrate how the degree and leading coefficient of a polynomial function affect its end behavior. These concepts are important for understanding the long-term trends of polynomial functions.

Example 5: Identifying Local Maxima and Minima

Given the graph of the polynomial function \( f(x) = x^3 - 6x^2 + 9x - 2 \), we want to identify the local maxima and local minima and analyze the behavior.

Graph:

Solution:

1. Roots: The graph intersects the x-axis at \( x = 1 \) and \( x \approx 3.75 \), which are the roots of the function. These points do not represent maxima or minima but help locate the regions where the function changes direction.
2. Local Maximum: The graph has a local maximum at \( x = 1 \), where the function reaches its highest point in this region. The corresponding y-coordinate for this maximum is \( f(1) = 2 \). Thus, the local maximum occurs at the coordinate \( (1, 2) \).
3. Local Minimum: The graph has a local minimum at \( x = 3 \), where the function reaches its lowest point in this region. The corresponding y-coordinate for this minimum is \( f(3) = -2 \). Thus, the local minimum occurs at the coordinate \( (3, -2) \).
4. End Behavior: The end behavior for this cubic function is as follows:
   • As \( x \to \infty \), \( f(x) \to \infty \).
   • As \( x \to -\infty \), \( f(x) \to -\infty \).

Key Takeaways:

• Local Maximum: The function reaches a local maximum where the graph changes direction from increasing to decreasing. For this example, the local maximum occurs at \( (1, 2) \).
• Local Minimum: The function reaches a local minimum where the graph changes direction from decreasing to increasing. For this example, the local minimum occurs at \( (3, -2) \).
• End Behavior: As the degree of the polynomial is odd, the end behavior is opposite at the two ends. The graph rises to the right and falls to the left.

Example 6: Analyzing the Graph of a Polynomial Function

Consider the polynomial function:

f(x) = \( x^4 - 8x^3 + 17x^2 + 2x - 24 \)

1. Roots of the Polynomial (x-intercepts):

The graph intersects the x-axis at the points where the function equals zero. These are the roots of the polynomial, which can be found from the factored form. The roots are at x = -1, x = 2, x = 3, and x = 4.

2. Turning Points:

The graph of this polynomial has local maxima and minima. The turning points occur at the locations where the graph changes direction. Specifically:

• The local minimum occurs at roughly x = 0.
• The local maximum occurs at roughly x = 2.5.

3. End Behavior:

As \( x \to -\infty \), the polynomial's value approaches \( f(x) \to +\infty \).

As \( x \to +\infty \), the polynomial's value approaches \( f(x) \to +\infty \).

This means the graph rises on both the left and right sides, indicating that the polynomial behaves like an even-degree function (since the leading term is \( x^4 \), which has a positive leading coefficient).

4. Inflection Points:

Inflection points occur where the concavity of the graph changes. In this case, there is an inflection point at \( x = 2 \). This is where the graph changes from concave down to concave up. The concavity changes at this point, which is why it represents an inflection point.

Example 7: Solving for Roots of a Polynomial

Consider the polynomial \( f(x) = x^3 - 3x^2 - 4x + 12 \). We can solve for the roots of this polynomial by factoring it.

Step 1: Look for possible rational roots using the Rational Root Theorem

The Rational Root Theorem suggests that we look at factors of the constant term (12) and divide them by factors of the leading coefficient (1). The factors of 12 are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \).

Step 2: Test possible roots

We can test these possible rational roots by substituting them into the polynomial to check which ones make \( f(x) = 0 \).

• For \( x = 2 \): \( f(2) = 2^3 - 3(2^2) - 4(2) + 12 = 8 - 12 - 8 + 12 = 0 \). So, \( x = 2 \) is a root.

Step 3: Factor the polynomial

Since \( x = 2 \) is a root, we know that \( (x - 2) \) is a factor of the polynomial. We can divide \( f(x) \) by \( (x - 2) \) using synthetic division or polynomial long division:

Step 4: Write the factored form

The factored form of the polynomial is:

\( f(x) = (x - 2)(x - 3)(x + 2) \).

Step 5: Find the roots

The roots are the values of \( x \) that make the factored expression equal to zero:

• Set \( (x - 2) = 0 \), so \( x = 2 \).
• Set \( (x - 3) = 0 \), so \( x = 3 \).
• Set \( (x + 2) = 0 \), so \( x = -2 \).

Conclusion:

The roots of the polynomial \( f(x) = x^3 - 3x^2 - 4x + 12 \) are \( x = 2, x = 3, \) and \( x = -2 \).

Review

In this lesson, we have covered the following topics:

• Graphing polynomials and analyzing their behavior
• Identifying and interpreting the roots of polynomials
• Understanding the end behavior of polynomials
• Analyzing turning points (local maxima/minima)
• Using the Factor Theorem to find factors of polynomials
• Identifying inflection points and analyzing polynomial graphs

Example 1: Graphing a Polynomial

Problem: Given the polynomial function \( f(x) = x^3 - 6x^2 + 9x \), graph it and analyze its behavior.

Solution:

1. Roots: The roots are \( x = 0 \) and \( x = 3 \) (with multiplicity 2). The graph intersects the x-axis at these points.
2. Turning Points: There are local maxima and minima. Use a graphing tool to identify them.
3. End Behavior: Since the degree is odd and the leading coefficient is positive, as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).

Example 2: Identifying Local Maxima and Minima from a Graph

Problem: For the function \( f(x) = x^3 - 3x^2 - 4x + 12 \), analyze the graph and find the local maxima and minima.

Solution:

1. Step 1: Identify the turning points from the graph: Examine the graph of the polynomial. Look for the highest and lowest points (the peaks and valleys) on the graph. These points represent local maxima (highest point) and local minima (lowest point).
2. Step 2: Analyze the local maximum and minimum: From the graph, the local maximum is around \( x = -0.5 \), where the graph reaches its peak. The local minimum is around \( x = 2.5 \), where the graph reaches its lowest point.

Note: The graph visually shows the behavior of the polynomial and helps us identify the local maxima and minima directly. This process does not require calculus and can be completed by analyzing the shape of the graph.

Example 3: Analyzing the End Behavior of a Polynomial

Problem: Given the polynomial function \( f(x) = 2x^4 - 3x^3 + x^2 - 2x + 5 \), analyze the end behavior of the function.

Solution:

The degree is even, and the leading coefficient is positive, so the graph rises to positive infinity as \( x \to \infty \) and rises to positive infinity as \( x \to -\infty \).

Example 4: Using the Factor Theorem

Problem: Find the factors of the polynomial \( f(x) = x^3 - 6x^2 + 11x - 6 \) using the Factor Theorem.

Solution:

1. Identify a possible root using the Rational Root Theorem. Test \( x = 1 \), and we find \( f(1) = 0 \).
2. Use polynomial division to divide \( f(x) \) by \( (x - 1) \). This gives \( x^2 - 5x + 6 \).
3. Factor the quadratic \( x^2 - 5x + 6 \) to get \( (x - 2)(x - 3) \).
4. The factored form is \( f(x) = (x - 1)(x - 2)(x - 3) \).

Multimedia Resources

See a collection of Media4Math resources that support this topic. Click on this link (link to come).

📌 These resources include definitions, examples, and explanations to reinforce understanding.

Quiz

Directions: Solve each problem. Choose the correct answer from the options provided.

1. What is the end behavior of the polynomial function \( f(x) = x^3 - 4x^2 + 3x - 2 \)?
   a) As \( x \to \infty \), \( f(x) \to -\infty \); As \( x \to -\infty \), \( f(x) \to \infty \)
   b) As \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to -\infty \)
   c) As \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to -\infty \)
   d) As \( x \to \infty \), \( f(x) \to 0 \); As \( x \to -\infty \), \( f(x) \to 0 \)
   
    
2. What are the roots of the polynomial \( f(x) = x^2 - 5x + 6 \)?
   a) \( x = 1, x = 6 \)
   b) \( x = 2, x = 3 \)
   c) \( x = -1, x = -6 \)
   d) \( x = -2, x = -3 \)
   
    
3. Which of the following describes the end behavior of the polynomial \( f(x) = -x^4 + 3x^3 + 5x + 2 \)?
   a) As \( x \to \infty \), \( f(x) \to -\infty \); As \( x \to -\infty \), \( f(x) \to \infty \)
   b) As \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to \infty \)
   c) As \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to -\infty \)
   d) As \( x \to \infty \), \( f(x) \to 0 \); As \( x \to -\infty \), \( f(x) \to 0 \)
   
    
4. Which of the following is the correct factorization of the polynomial \( f(x) = x^2 - 4x - 12 \)?
   a) \( (x - 6)(x + 2) \)
   b) \( (x + 6)(x - 2) \)
   c) \( (x - 4)(x + 3) \)
   d) \( (x - 3)(x + 4) \)
   
    
5. What is the leading coefficient of the polynomial \( f(x) = -2x^3 + 5x^2 - 3x + 1 \)?
   a) -2
   b) 2
   c) 5
   d) -5
   
    
6. Which of the following describes the graph of the polynomial \( f(x) = 2x^3 - x^2 - 3x + 2 \)?
   a) A cubic function with a local minimum
   b) A cubic function with a local maximum
   c) A cubic function with an inflection point
   d) A quadratic function with a local minimum
   
    
7. What is the degree of the polynomial \( f(x) = 4x^3 - 2x^2 + 5x - 7 \)?
   a) 3
   b) 2
   c) 5
   d) 4
   
    
8. For the polynomial \( f(x) = 5x^4 + 3x^3 - 2x^2 + 6 \), what is the end behavior as \( x \to \infty \)?
   a) \( f(x) \to \infty \)
   b) \( f(x) \to -\infty \)
   c) \( f(x) \to 0 \)
   d) \( f(x) \to 6 \)
   
    
9. What is the range of the quadratic function \( f(x) = -x^2 + 4x + 1 \)?
   a) \( y \leq 6 \)
   b) \( y \geq 6 \)
   c) \( y \leq 2 \)
   d) \( y \geq 2 \)
   
    
10. What are the real roots of the polynomial \( f(x) = x^2 - 6x + 9 \)?
    a) \( x = 3 \)
    b) \( x = -3 \)
    c) \( x = 6 \)
    d) \( x = 0 \)
    
     

Answer Key

1. Answer: b) \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to -\infty \)
   Explanation: The degree of the polynomial is odd, and the leading coefficient is positive, meaning the graph falls to the left and rises to the right.
2. Answer: b) \( x = 2, x = 3 \)
   Explanation: The roots of the polynomial can be found by factoring. \( f(x) = (x - 2)(x - 3) \).
3. Answer: b) \( x \to \infty \), \( f(x) \to \infty \); As \( x \to -\infty \), \( f(x) \to -\infty \)
   Explanation: The degree of the polynomial is odd, and the leading coefficient is negative, so the graph rises to the left and falls to the right.
4. Answer: b) \( (x - 2)(x - 3) \)
   Explanation: Factoring the quadratic gives \( f(x) = (x - 2)(x - 3) \).
5. Answer: a) -2
   Explanation: The leading coefficient is the coefficient of the highest degree term, which is -2.
6. Answer: c) A cubic function with an inflection point
   Explanation: The function has a change in concavity at \( x = 0 \), creating an inflection point.
7. Answer: a) 3
   Explanation: The degree of the polynomial is determined by the highest exponent, which is 3.
8. Answer: a) \( f(x) \to \infty \)
   Explanation: As \( x \to \infty \), the degree of the polynomial is even, and the leading coefficient is positive, meaning the graph rises as \( x \to \infty \).
9. Answer: a) \( y \leq 6 \)
   Explanation: The vertex form of the quadratic shows the maximum value is at \( y = 6 \).
10. Answer: a) \( x = 3 \)
    Explanation: The quadratic can be factored as \( f(x) = (x - 3)(x - 3) \), which gives a double root at \( x = 3 \).