Graph of Polynomial Functions (original) (raw)

Last Updated : 5 Mar, 2026

A polynomial function is a function made by adding and subtracting powers of a variable (like x) multiplied by numbers (called coefficients).

**General form: f(x) = anxn + an−1xn−1 + ⋯ + a1x + a0

A polynomial of graphs is shown on x-y coordinate planes. 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_​__x n + _a n_−1​__x _n_−1 + … + _a_1​__x + _a0, where n (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 (an).

**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).

For Example: y = 2

Graph of Constant Function

Graph of Linear Polynomial

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

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

Graph of Linear Polynomia

Graph of Quadratic Polynomial

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

**For example, y = 3x2 + 2x - 7

Graph of Quadratic Polynomial

Graph of Cubic Polynomial Function

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

**For Example, p(x)=x−3

Graph of Cubic Polynomial

**How to Find Roots Using the 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:

**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 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)=2x−

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)
How to find roots of polynomial from graph

**Real-Life Uses of Graph of the Polynomial

Some real-life uses of graphs of polynomials are:

**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:

Graph of Polynomial Example 2

Practice Questions of Graph of Polynomial

**Question 1: Solve the quadratic equation x2 + 2x - 4 = 0 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 values of p and q.

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