Graphs of polynomials provide a visual representation of polynomial functions. The graphs of polynomials play a vital role in some applications like science, finance engineering, etc.

In this article, we will cover what a polynomial is, what is a graph of a polynomial, what are the types of polynomial functions, how to make a graph of different types of polynomials, what are real-life uses of the graph of a polynomial function and conclusion of the polynomial.

What is a Polynomial?

A polynomial is an algebraic expression in which the power of a variable is a non-negative integer. A polynomial is represented as p(x) which means that the polynomial expression is in terms of variable x.

For Example,

• p(x) = 5x – 2 is a polynomial in a variable x.
• t(y) = 6x² + 8y – 4 is polynomial in variable y.

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Graph of a Polynomial

A polynomial of graphs is shown on x y coordinate plans. We can represent the polynomial in the form of a graph. In graphs of a polynomial, we should know how to draw different types of polynomials on a graph and what real uses of graphs are in a polynomial. 

Types of Polynomial Functions

Polynomials are fundamentals in algebra expression. The different types of polynomial functions:

• Constant Function
• Linear Function
• Quadratic Function
• Cubic Function
• Bi-Quadratic Function

Check Types of Polynomials

1. Constant Polynomial- A polynomial whose degree is zero is called a constant polynomial.

For examples:

• g(x) = 5
• p(y) = -8
• n(t) = 5/9

2. Linear Polynomial– A polynomial whose degree is 1 is called a linear polynomial.

For examples:

• j(x) = 6x + 2
• p(y) = 2y
• h(z) = 9z

3. Quadratic Polynomial- A polynomial whose degree is 2 is called a quadratic polynomial.

For examples:

• f(x) = 4x² + 5
• g(y) = 8/5y² – 3y

4. Cubic Polynomial– A polynomial whose degree is 3 is called a cubic polynomial.

For examples: 

• p(x) = 4x³ + 5x – 8
• h(y) = 9/5y³ + 2y² – 7

5. Bi-Quadratic Polynomial– A polynomial whose degree is 4 is called bi-quadratic polynomial.

For examples:

• p(x) = 4x⁴ + 8
• f(z) = 2z⁴ – 6z² + 7

How to Draw a Graph of a Polynomial?

Drawing the graph of a polynomial involves several steps.

Step 1: Know the form of the polynomial, , where ( n ) is the degree of the polynomial.

Step 2: Determine the degree of the polynomial to understand the overall shape and behavior of the graph. Note the leading coefficient (a_n).

Step 3: Calculate and mark the x-intercepts by setting f(x) = 0 and solving for ( x ). Also, find the y-intercept by setting (x = 0).

Step 4: Identify the end behavior by looking at the degree and leading coefficient. For even-degree polynomials, the ends go in the same direction; for odd-degree polynomials, they go in opposite directions.

Step 5: Determine turning points (where the graph changes direction) by finding the critical points where f'(x) = 0 or is undefined. Use these points to sketch the curve.

Step 6: Even-degree polynomials may exhibit symmetry about the y-axis, while odd-degree polynomials may show symmetry about the origin.

Step 7: Plot the identified points, including intercepts, turning points, and any additional points of interest. Connect the points smoothly to sketch the graph.

Graph of Constant Polynomial

The graph of a constant polynomial is a horizontal line parallel to the x-axis. A constant polynomial has the form f(x) = c, where (c) is a constant. The graph represents a straight line that does not slope upward or downward; it remains at a constant height across all values of (x).

• Horizontal Line: The graph is a horizontal line at the height corresponding to the constant term (c).

• No Slope: Since the function is a constant, there is no change in the y-values as (x) varies. The line is perfectly level.

• No Intercepts: Unless the constant term is zero (c = 0), there are no x-intercepts, and the line intersects the y-axis at the constant value (c).

 For Example: y = 2

Graph of Linear Polynomial

The graph of a linear polynomial, which is a polynomial of degree 1, has the following features:

• Straight Line: The graph is a straight line.

• One Root/Zero: It has exactly one root or x-intercept.

• Constant Slope: The slope of the line remains constant.

For example: y = 2x + 5, a = 2 and b = 5

Graph of Quadratic Polynomial

The graph of a quadratic polynomial, which is a polynomial of degree 2, has some features:

• Symmetry: The parabola is symmetric with respect to its axis of symmetry.

• Intercepts: The quadratic polynomial may have two x-intercepts, one x-intercept or no x-intercepts.

• Parabolic Shape: The graph is a parabola, which can either be open upwards or downwards.

For example, y = 3x² + 2x – 7

Graph of Cubic Polynomial Function

The graph of a cubic polynomial, which is a polynomial of degree 3, has some features:

• Cubic Shape: The graph will exhibit an “S” shape.

• Turning Points: It may have up to two turning points.

• Intercepts: It can have up to three real roots and intercepts with the x-axis.

For Example, p(x)=x³−3x²−4x+12

How to Find Roots using Graph of Polynomial Function

Finding the roots (or zeros) of a polynomial function from its graph involves identifying the x-values where the graph intersects the x-axis. The roots are the values of x for which the function equals zero. Here’s a step-by-step guide:

Step 1: Start with the given polynomial function in standard form. For example, (ax² + bx + c).

Step 2: Identify the coefficients (a), (b), and (c) in the polynomial. These coefficients are crucial for using the quadratic formula.

Step 3: Apply the quadratic formula:

Step 4: Evaluate the discriminant (b² – 4ac). The discriminant determines the nature of the roots:

• If (Δ > 0), there are two distinct real roots.
• If (Δ = 0), there is one real root (a repeated root).
• If (Δ < 0), there are two complex (conjugate) roots.

Step 5: Simplify the square root part of the formula. If the discriminant is positive, take the square root. If it’s negative, express it in terms of (i), the imaginary unit.

Step 6: Use the ∓ symbol to represent both the positive and negative square root solutions.

Step 7: Plug in the values of (a), (b), and (c) into the quadratic formula and perform the calculations.

For Example, p(x)=2x²−5x+2 

To find the roots of the polynomial function p(x) = 2x² – 5x + 2 , use the quadratic formula. The quadratic formula is given by:

In the equation (ax² + bx + c = 0), the coefficients are:

a = 2, b = -5, c = 2

put these values of a, b, and c in the formula,

This gives two solutions:

1. For the positive square root: x = (5+3)/4 = 2
2. For the negative square root: x = (5-3)/4 = 1/2

So, the roots of the polynomial function p(x) = (2x² – 5x + 2) are (x = 2) and (x = 0.5)

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Real-Life Uses of Graph of the Polynomial

Some real-life uses of graph of polynomial are:

• Profit Analysis: Businesses use polynomial graphs to analyze profit functions, helping them understand how changes in factors like pricing and production affect overall profit.

• Budget Planning: Governments and organizations use polynomial models to create budget plans. The graphs show the financial impact of different variables, aiding in effective resource allocation.

• Engineering Designs: Engineers use polynomial graphs to model and optimize designs. This is crucial in fields like structural engineering to ensure stability and efficiency.

• Economic Trends: Economists use polynomial functions to model economic trends. Graphs help predict changes in factors like inflation and employment over time.

• Medical Research: In medical research, polynomial graphs assist in modeling the growth of diseases or the effectiveness of treatments, providing valuable insights for healthcare planning.

• Environmental Studies: Environmental scientists use polynomial functions to model and predict ecological changes. This aids in understanding the impact of human activities on the environment.

• Physics Experiments: Physicists use polynomial equations to model physical phenomena. The resulting graphs help visualize and analyze experimental data, enhancing our understanding of the natural world.

• Population Studies: Demographers use polynomial models to study population growth. Graphs assist in predicting population changes and planning for future needs.

• Stock Market Analysis: Investors and financial analysts use polynomial graphs to analyze stock market trends. This helps in making informed investment decisions based on historical data.

• Criminal Justice Planning: Polynomial models are applied in criminal justice to analyze crime rates over time. This information is vital for planning law enforcement strategies and resource allocation.

Also, Check

• Degree of Polynomial
• Relationship between Zeros and Coefficient of Polynomial

Examples on Graph of Polynomial

Example 1. Find the value of a, if x – a is a factor of x³ – ax² + 5x + a – 3.

Solution:

Let p(x) = x³ – ax² + 5x + a – 3

Given that x – a is a factor of p(x).

⇒ p(a) = 0

 a³ – a(a)² + 5a + a – 3 = 0

 a³ – a³ + 5a + a – 3= 0

( a³ – a³ = 0)

6a – 3 = 0

6a = 3, a = 2

Therefore, a = 2.

Example 2. Graph the polynomial function.

f(x) = 5x⁴ – x² + 3

Graph of Polynomial – Practice Questions

Q1. Solve the quadratic equation: x² + 2x – 4 = o for x.

Q2. A polynomial of degree n has:

• only one zero
• At least n zeroes
• More than n zeroes
• At most n zeroes

Q3. If the zeroes of the polynomial x² + px + q are double in value to the zeroes of 5x² – 6 – 4. Find the value of p and q.

Q4. Draw the graphs of the polynomial f(x) = x³ – 5.

Graph of Polynomial – FAQs

1. Define Polynomial.

A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers, combined with coefficients and addition or subtraction operations.

2. What are Types of Polynomials?

 The different types of polynomials are as follow:

• Constant Polynomial 
• Linear Polynomial
• Quadratic Polynomial
• Cubic Polynomial
• Bi-Quadratic polynomial

3. What are Key Features in Graphs of Polynomials?

Some key features in graphs of polynomials are as follow:

• Turning Points- The position where the graph changes direction.
• Intercepts- Points where x-axis and y-axis intersect in a graph.
• Symmetry- Such as odd and even symmetry.

4. How to Find Zero of a Polynomial Function?

To find the zeros of a polynomial function:

• Step 1: Set the function to zero: f(x) = 0.
• Step 2: Factorize the polynomial if possible.
• Step 3: Use the Zero-Product Property: Set each factor equal to zero and solve for \( x \).
• Step 4: Solve for ( x ) to find the zeros.
• Step 5: Check for repeated roots.
• Step 6: Use the quadratic formula for quadratic polynomials (ax² + bx + c): (x = )

5. What is Significance of x-Intercepts on a Polynomial Graph?

The x-intercepts represent the points where the polynomial function equals zero. In practical terms, these points indicate the real-life values where the phenomenon modeled by the polynomial has a specific impact or occurrence.

6. How to identify Degree of a Polynomial by looking its Graph?

By examining the graph, you can count the number of times the graph intersects the x-axis to identify the degree, as each intercept corresponds to a factor of the polynomial.

7. What is Graph of a Quadratic Polynomial?

The graph of a quadratic polynomial, which has the form (ax² + bx + c), is a parabola. Depending on the sign of the coefficient (a), the parabola can open upwards (a > 0) or downwards (a < 0).

8. What is Nature of Curve of a Cubic Polynomial?

The nature of the curve can include S-shaped bends, loops, or multiple turning points, and it extends in both the positive and negative directions.

9. Which Polynomial has Line graph?

The graph of a linear polynomial is characterized by a straight line with a constant slope, and it does not show the curvature seen in higher-degree polynomials.