Graph of Polynomial Functions

A polynomial function is a mathematical function that is represented by a polynomial expression. A polynomial is a sum of terms, each consisting of a constant multiplied by a non-negative integer power of the variable.

A polynomial function in one variable x has the following general form:

f(x) = a_n​xⁿ + a_n−1​xⁿ⁻¹ + … + a₁​x + a₀​)

Where:

• n is a non-negative integer called the degree of the polynomial,
• a_n, a_n−1, …, a₁, a₀​ are coefficients (real or complex numbers),
• x is the variable.

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

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 the real uses of graphs are in a polynomial.

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, f(x) = a_n​xⁿ + a_n−1​xⁿ⁻¹ + … + a₁​x + a₀​) 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 concerning 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: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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,

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(2)}}{2(2)}

x = \frac{5 \pm \sqrt{25 - 16}}{4}

x = \frac{5 \pm \sqrt{9}}{4}

x = \frac{5 \pm 3}{4}

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)

Learn, Roots of a Polynomial

Real-Life Uses of Graph of the Polynomial

Some real-life uses of graphs of polynomials 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

Solved 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

Solution:

Practice Questions of Graph of Polynomial

Question 1: Solve the quadratic equation: x² + 2x - 4 = o for x.

Question 2: A polynomial of degree n has:
a) Only one zero,
b) At least n zeroes,
c) More than n zeroes,
d) At most n zeroes.

Question 3: 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.

Question 4: Draw the graphs of the polynomial f(x) = x³ - 5.