Topics
3. Polynomial Functions & Expressions
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United States of AmericaFL
Grade 912

3.04 Graphing polynomial functions

Lesson
Practice
Lesson

Concept summary

A polynomial function is a function that involves variables raised to non-negative integer powers.

There are two special types of polynomial functions based on the degree of each term of the polynomial.

Even function

A function is even if f(-x)=f(x).

The graph of an even function will have either both sides up or both sides down. It is symmetric about the y-axis.

Example:

f(x) = x^2

Odd function

A function is odd if f(-x)=-f(x).

The graph of an odd function will have one side up and one side down. It is symmetric about the origin.

Example:

f(x) = x^3

Note: If -x is substituted into the function and some but not all of the signs change, the function is neither even nor odd.

Steps in constructing the graph of a polynomial function:

  1. Determine the graph’s end behavior using the Leading Coefficient Test. The end behavior is the behavior of the graph as x approaches positive infinity or negative infinity.

    DegreeLeading CoefficientEnd BehaviorGraph of the function
    \text{even}\text{positive}f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty\text{rises to the left and} \\ \text{to the right}
    \text{even}\text{negative}f(x) \to - \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to +\infty\text{falls to the left and} \\ \text{to the right}
    \text{odd}\text{positive}f(x) \to -\infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty\text{falls to the left and} \\ \text{rises to the right}
    \text{odd}\text{negative}f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to + \infty\text{rises to the left and} \\ \text{falls to the right}

  2. Find the x-intercepts by determining all zeros of the function.

    Note: Depending on the degree of the function, there may be quite a few x-intercepts to find.

  3. Find the y-intercept of the function.

  4. Find the number of maximum turning points.

    A turning point of a function is a point where the graph of the function changes from sloping upwards (positive to negative slope) to sloping downwards (negative to positive slope), or vice versa. Generally, the graph of a polynomial function of degree n has at most n − 1 turning points.

  5. Find extra points, if needed.

  6. Sketch the graph.

Worked examples

Example 1

State whether f\left(x\right)= x^{2} + 2x^{\frac{1}{2}} + 3 is a polynomial function or not.

Approach

To know whether a function is a polynomial function, check whether all variables are raised to non-negative integer powers.

Solution

The variables in the function are raised to the following powers: 2, \dfrac{1}{2} and 0. Since \dfrac{1}{2} is not a positive integer, f\left(x\right)= x^{2} + 2x^{\frac{1}{2}} + 3 is not a polynomial function.

Example 2

Determine whether the given graph is that of an odd function, even function or neither.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

Approach

Determine the symmetry of the graph. If the graph is symmetrical about the y-axis, it is that of an even function. On the other hand, if it is symmetrical about the origin, it is that of an odd function.

Solution

Since the curve is symmetrical about the y-axis, the graph is that of an even function.

Example 3

Determine whether the given table of values is that of an odd function, even function or neither.

x-2-1012
y740-4-7

Approach

Check whether f(-x) is equal to f(x) or -f(x). If f(-x)=f(x), the function is even. On the other hand, if f(-x)=-f(x), the function is odd.

Solution

From the table of values, we can see that f(-2)=-f(2), f(-1)=-f(1) and f(-0)=-f(0). Since f(-x)=-f(x), the table of values is that of an odd function.

Example 4

Consider the table of values for the function f \left( x \right), and the transformed function g \left( x \right) shown in the graph:

x-2-1012
f(x)-523411
-2
-1
1
2
x
-24
-20
-16
-12
-8
-4
4
8
12
16
20
24
y

Express g \left( x \right) in terms of f \left( x \right).

Approach

Determine the values of g(x) for the values of x shown in the table and think how the original function f(x) can be transformed to get these values.

Solution

Determining the values of g(x) for the given values of x, we have the following table of values:

x-2-1012
g(x)-1046822

From the table of values above, we can see that the values of g(x) are twice that of f(x). Thus, we can express g(x) in terms of f(x) as g \left( x \right) = 2 f \left( x \right).

Example 5

Consider the graph of y = x^{3}:

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y

Describe how to shift the graph of y = x^{3} and sketch the graph of y = \left(x + 2\right)^{3} - 1.

Approach

Check the form of the new function to determine its horizontal and vertical translation.

Horizontal Translation: If the function is in the form y=f(x+a), the graph of y=f(x) is moved to the left by a units when a is positive, and to the right by a units when a is negative.

Vertical Translation: If the function is in the form y=f(x+b), the graph of y=f(x) is moved upward by b units when b is positive, and downward by b units when b is negative.

Solution

Since a=2 and b=-1 in y = \left(x + 2\right)^{3} - 1, we can obtain the graph of y = \left(x + 2\right)^{3} - 1 by moving the graph of y = x^{3} to the left by 2 units and down by 1 unit.

-5
-4
-3
-2
-1
1
2
3
4
5
x
-5
-4
-3
-2
-1
1
2
3
4
5
y

Outcomes

MA.912.F.1.1

Given an equation or graph that defines a function, determine the function type. Given an input-output table, determine a function type that could represent it.

MA.912.F.1.7

Compare key features of two functions each represented algebraically, graphically, in tables or written descriptions.

MA.912.F.1.9

Determine whether a function is even, odd or neither when represented algebraically, graphically or in a table.

MA.912.F.2.2

Identify the effect on the graph of a given function of two or more transformations defined by adding a real number to the x- or y- values or multiplying the x- or y- values by a real number.

MA.912.F.2.3

Given the graph or table of f(x) and the graph or table of f(x)+k,kf(x), f(kx) and f(x+k), state the type of transformation and find the value of the real number k.

MA.912.F.2.5

Given a table, equation or graph that represents a function, create a corresponding table, equation or graph of the transformed function defined by adding a real number to the x- or y-values or multiplying the x- or y-values by a real number.

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