Graph of Polynomial Functions

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.

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.