- Learn how to Graph Transformations of Monomials
- Learn about Graphs of Polynomial Functions: End Behavior
- Learn about Graphs of Polynomial Functions: Shape of the Graph Near a Zero
- Learn about Graphs of Polynomial Functions: Local Maxima and Minima
- Learn how to Create a Rough Sketch of the Graph of a Polynomial Function

## How to Sketch the Graph of a Polynomial Function

### Definition of a Polynomial Function of Degree n

Let's begin with a basic definition of a polynomial function of degree n: $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$- n is a non-negative integer and represents the degree of the polynomial
- a
_{n}≠ 0 - a
_{0}, a_{1},..., a_{n}are known as the coefficients of the polynomial - a
_{0}is known as the constant term or the constant coefficient - a
_{n}is known as the leading coefficient- The leading coefficient is the coefficient of the term where x is raised to the highest power
- If the polynomial function is written in standard form (descending powers of x), it will be the leftmost coefficient

- a
_{n}x^{n}is known as the leading term- The leading term is the term where x is raised to the highest power
- If the polynomial function is written in standard form (descending powers of x), it will be the leftmost term

Polynomial Function | Degree | Leading Coefficient | Leading Term | Constant Term |
---|---|---|---|---|

$$f(x) = -9x^4 + x^2 - 5$$ | 4 | -9 | -9x^{4} | -5 |

$$f(x) = 2x^3 - x^2 + x + 1$$ | 3 | 2 | 2x^{3} | 1 |

$$f(x) = 3x^2 - x - 1$$ | 2 | 3 | 3x^{2} | -1 |

$$f(x) = -x + 7$$ | 1 | -1 | -x | 7 |

### Transformations of Monomials

We know that a monomial is a polynomial with one term only. For example: $$4x^3, -2x^2, -\frac{1}{2}x^9$$ Earlier in this section, we studied the graphs of parabolas. The simplest parabola is that of f(x) = x^{2}. Desmos Link for More Detail

^{n}, where n is even has the same general shape as the graph of f(x) = x

^{2}. As the degree of the polynomial n gets larger, the graphs will be more flat around the origin and steeper everywhere else. We can observe this with f(x) = x

^{4}. Desmos Link for More Detail

^{6}. Desmos Link for More Detail

^{3}. Desmos Link for More Detail

^{n}, where n is odd has the same general shape as the graph of f(x) = x

^{3}. Again, as the degree of the polynomial n gets larger, the graphs will be more flat around the origin and steeper everywhere else. We can observe this with f(x) = x

^{5}. Desmos Link for More Detail

^{7}. Desmos Link for More Detail

Example #1: Describe the transformation from f(x) to g(x) and then graph g(x). $$f(x) = x^4$$ $$g(x) = \frac{1}{2}(x - 1)^4 - 2$$ Compared to the graph of f(x), g(x) has been vertically compressed by a factor of 2 (could also say vertically shrunk by a factor of 1/2), shifted right by 1 unit, and shifted down by 2 units. A given (x, y) on f(x) will now occur at (x + 1, y/2 - 2) on g(x). For example, the point (0, 0) on the graph of f(x) will become (1, -2) on the graph of g(x). Additionally, a point such as (2, 16) will become (3, 6), and (-2, 16) will become (-1, 6). Desmos Link for More Detail

### Smooth and Continuous Definition

Polynomial functions of degree 2 or more have graphs that are smooth and continuous. When a graph is referred to as being "smooth", it means the graph has only smooth, rounded turns. In other words, the graph of a polynomial function can't have a sharp turn, which is a turn where the direction is changed suddenly. When a graph is referred to as being "continuous", it means the function can be graphed without lifting the pencil from the page. In other words, there are no breaks or holes in the graph.#### Graphs of Non-Polynomial Functions

#### Graphs of Polynomial Functions

### Graphs of Polynomial Functions: End Behavior

The end behavior of a polynomial function describes what happens to the function as x approaches both positive and negative infinity. $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$ $$g(x) = a_nx^n$$ The two functions given above, f(x) and g(x), will have the same end behavior. This is due to the influence of the leading term, which is commonly referred to as the "dominating term". When x is large in terms of absolute value, the other terms are going to be insignificant in size. Let's take a look at an actual example. Below, we have f(x) and g(x) with the same leading term of x^{3}. $$f(x) = x^3 - 2x^2 - x + 1$$ $$g(x) = x^3$$ Factor out x

^{3}from f(x): $$f(x) = x^3\left(1 - \frac{2}{x} - \frac{1}{x^2} + \frac{1}{x^3}\right)$$ If we think about what's inside of the parentheses: -2/x, -1/x

^{2}, and 1/x

^{3}are all going to approach 0 as x becomes very large in terms of absolute value. This means what's inside of the parentheses is going to be very insignificant and the leading term x

^{3}is going to dominate. This can also be shown with a simple table of values like we have below.

x | f(x) | g(x) |
---|---|---|

10 | 791 | 1000 |

50 | 119,951 | 125,000 |

100 | 979,901 | 1,000,000 |

^{3}. Desmos Link for More Detail

n is odd | n is even | |
---|---|---|

a_{n} > 0 | ↙ ↗ | ↖ ↗ |

a_{n} < 0 | ↖ ↘ | ↙ ↘ |

_{n}. Some books refer to the process of finding the end behavior as the leading coefficient test since we are only analyzing the leading coefficient and nothing else. We will cover the four different cases below.

#### Odd Degree with a Positive Leading Coefficient

For our first case, the polynomial function will have an odd degree and a positive leading coefficient. $$y → ∞ \, \text{as} \, x → ∞$$ y approaches positive infinity as x approaches positive infinity. $$y → -∞ \, \text{as} \, x → -∞$$ y approaches negative infinity as x approaches negative infinity.In other words, the graph falls to the left and rises to the right.

#### Odd Degree with a Negative Leading Coefficient

For our second case, the polynomial function will have an odd degree and a negative leading coefficient. $$y → ∞ \, \text{as} \, x → -∞$$ y approaches positive infinity as x approaches negative infinity. $$y → -∞ \, \text{as} \, x → ∞$$ y approaches negative infinity as x approaches positive infinity.In other words, the graph rises to the left and falls to the right.

#### Summary Images for End Behavior of Polynomials when n is odd

$$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$#### Even Degree with a Positive Leading Coefficient

For our third case, the polynomial function will have an even degree and a positive leading coefficient. $$y → ∞ \, \text{as} \, x → -∞$$ y approaches positive infinity as x approaches negative infinity. $$y → ∞ \, \text{as} \, x → ∞$$ y approaches positive infinity as x approaches positive infinity.In other words, the graph rises to the left and rises to the right.

#### Even Degree with a Negative Leading Coefficient

For our fourth and final case, the polynomial function will have an even degree and a negative leading coefficient. $$y → -∞ \, \text{as} \, x → -∞$$ y approaches negative infinity as x approaches negative infinity. $$y → -∞ \, \text{as} \, x → ∞$$ y approaches negative infinity as x approaches positive infinity.In other words, the graph falls to the left and falls to the right.

#### Summary Images for End Behavior of Polynomials when n is even

$$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$### Shape of the Graph Near a Zero

At this point, we know how to find the zeros of a polynomial function. Recall that a zero or root of a polynomial function is a solution to the equation: $$f(x) = 0$$ The real solutions to the above equation are the x-intercepts of the graph. Let's look at an example.Example #6: Find the x-intercepts for f(x). $$f(x) = 2\left(x - \frac{5}{2}\right)(x - 1)(x - 3)$$ Replace f(x) with 0: $$2\left(x - \frac{5}{2}\right)(x - 1)(x - 3) = 0$$ $$x = 1, \frac{5}{2}, 3$$ This tells us that the graph of our function f(x) will have x-intercepts of (1, 0), (5/2, 0), and (3, 0). Desmos Link for More Detail

#### Multiplicity and x-Intercepts

When discussing how to find the zeros of polynomial functions, we learned that a zero may occur more than once. For example: $$f(x) = (x - 2)^2(x - 1)$$ Here both 1 and 2 are zeros, however, 2 is a zero of multiplicity 2, which comes from the exponent of 2. To make this clear, we could rewrite the function as: $$f(x) = (x - 2)(x - 2)(x - 1)$$ When we have a zero of even multiplicity, the graph will touch the x-axis and turn around. In other words, it touches but does not cross the x-axis there. When we have a zero of odd multiplicity, the graph will cross the x-axis. Additionally, graphs will flatten out near zeros where the multiplicity is greater than one. Desmos Link for More Detail(-∞, 1) | (1, 2) | (2, ∞) | |
---|---|---|---|

(x - 1) | - | + | + |

(x - 2)^{2} | + | + | + |

(x - 1)(x - 2)^{2} | - | + | + |

^{2}will always be non-negative. It will be 0 at an x-value of 2 and then positive everywhere else. We also know that (x - 1) will be negative when x is less than 1, 0 at an x-value of 1, and then positive when x is greater than 1.

Let's think about what happens on the graph working left to right. For the interval (-∞, 1), we have y-values that are negative. In other words, the graph is below the x-axis. Then at an x-value of 1, we have an x-intercept, so the y-value is 0. Moving on to the interval (1, 2), the y-values are now positive. Again, this means the graph is now above the x-axis. Notice the sign changed from (-) in the interval (-∞ , 1) to (+) in the interval (1, 2). When we move from (1, 2) to (2, ∞) the y-value is 0 at an x-value of 2 but positive everywhere else. This is why the function touches the x-axis at (2, 0) but then turns around and heads back up.

In summary, when you have a zero of even multiplicity, the sign of your polynomial function is not going to change from one side of your zero to the other side of your zero. This means that the graph is going to turn around at that zero of even multiplicity. If the zero is of odd multiplicity, then the sign of the polynomial function is going to change from one side of the zero to the other side of your zero, which causes the graph to cross the x-axis at that zero.

#### Repeated Zeros

A factor (x - k)^{m}, m > 1, yields a repeated zero x = k of multiplicity m.

- When m is odd, the graph crosses the x-axis at x = k
- When m is even, the graph touches but does not cross the x-axis at x = k

Example #7: State whether the graph crosses the x-axis or touches and turns around at each zero. $$f(x) = \frac{1}{8}(x - 1)^3(x + 2)^2(x - 3)$$ The zeros here are 1 (mult. 3), -2 (mult. 2), and 3 (mult. 1). The x-intercepts will be (1, 0), (-2, 0), and (3, 0). The graph will cross through at (1, 0) and (3, 0) since those zeros of 1 and 3 have odd multiplicities, 3 and 1 respectively. Then at (-2, 0), the graph will touch the x-axis and turn around. We can use the graph below for confirmation. Desmos Link for More Detail

### Turning Points | Local Maxima and Minima

Polynomial functions often have turning points, where the function changes from increasing to decreasing or from decreasing to increasing. These turning points will be referred to as a local maximum point (also known as a relative maximum point) when the function takes on a greater value than any nearby points and a local minimum point (also known as a relative minimum point) when the function takes on a lesser value than any nearby points. When you hear the terms maxima and minima, these are just the plural forms of the words maximum and minimum, respectively, coming from Latin. The local maximum and minimum points on the graph of a polynomial function are called its local extrema. Note: Without the use of some basic Calculus, it is usually not possible or at least extremely impractical to find the actual turning points for a polynomial function when the degree is greater than 2.Example #8: Use the given graph to find the local extrema for the given polynomial function. Desmos Link for More Detail

### Creating a Rough Sketch of the Graph of a Polynomial Function

At this point, we have enough information to create a rough sketch of the graph of a polynomial function. I would again note that this process is just about understanding the concept, creating a perfect graph can be done instantly with graphing calculators such as Desmos. $$f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \cdots + a_1x + a_0$$- Use the Leading Coefficient Test to find the graph's end behavior
- Find the x-intercepts by solving the equation f(x) = 0
- If k is a zero of even multiplicity, the graph touches the x-axis at x = k and then turns around
- If k is a zero of odd multiplicity, the graph crosses the x-axis at x = k
- If the multiplicity of k is greater than 1, the graph flattens out near (k, 0)

- Find and plot the y-intercept
- Find the y-intercept as: (0, f(0))

- Use symmetry when available to sketch the graph
- y-axis symmetry: f(-x) = f(x)
- origin symmetry: f(-x) = -f(x)

- Use test points within the intervals formed by the x-intercepts to find the sign of f(x) in the interval
- Since the graph of a polynomial function is continuous, the values of f(x) are either always positive or always negative in a given interval
- When the values are positive, we are above the x-axis
- When the values are negative, we are below the x-axis

- Use the maximum number of turning points of n - 1 to check the graph

^{3}, which means the degree 3 is odd and the leading coefficient is positive. This means the graph falls to the left and rises to the right.

2) Find the x-intercepts. $$(x-1)^2(x - 4) = 0$$ $$x = 1, 4$$ The x-intercepts will occur at (1, 0) and (4, 0). The zero of 1 has a multiplicity of 2, therefore, the graph will touch the x-axis at (1, 0) and turn around. The zero of 4 has a multiplicity of 1, therefore, the graph will cross through the x-axis at (4, 0).

3) Find and plot the y-intercept. $$f(0) = -4$$ The y-intercept occurs at (0, -4).

4) This function is not odd or even.

5) Use test points within the intervals formed by the x-intercepts to find the sign of f(x) in the interval. Since we have x-intercepts of (1, 0) and (4, 0), this gives us three intervals to work with: (-∞, 1), (1, 4), and (4, ∞).

Interval | x | f(x) |
---|---|---|

(-∞, 1) | 0.5 | -0.875 |

(1, 4) | 2 | -2 |

(4, ∞) | 4.5 | 6.125 |

#### Skills Check:

Example #1

$$f(x) = x^5$$ $$g(x) = 2(x - 1)^5 + 1$$ A given point (x, y) on the graph of f(x) will be transformed to what point on the graph of g(x)?

Please choose the best answer.

Example #2

Determine the end behavior. $$f(x) = 2x^4 - 15x^3 + 29x^2 - 21x + 5$$

Please choose the best answer.

Example #3

Determine which statement is true. $$f(x) = 2(3x - 5)^3(x - 1)^2(x + 4)$$ 1) The x-intercepts occur at (5, 0), (1, 0), and (-4, 0).

2) The graph will touch the x-axis at (-4, 0) and turn around.

3) The graph will touch the x-axis at (5/3, 0) and turn around.

4) The graph will touch the x-axis at (1, 0) and turn around.

5) The y-intercept will occur at (0, 2).

Please choose the best answer.

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