The prototype for a roller coaster is represented by the equation \begin{align*}y = x^5 - 8x^3 + 10x + 6\end{align*}

### Parts of a Polynomial Graph

By now, you should be familiar with the general idea of what a polynomial function graph does. It should cross the \begin{align*}x-\end{align*}

Notice that in both the cubic (third degree, on the left) and the quartic (fourth degree, on the right) functions, there is no vertex. We now have minimums and maximums. If there are more than one minimum or maximum, there will be an **absolute maximum/minimum**, which is the lowest/highest point of the graph. A **local maximum/minimum** is a maximum/minimum relative to the points around it. The places where the function crosses the \begin{align*}x-\end{align*}**solutions** (also called \begin{align*}x-\end{align*}**critical values.**

Another important thing to note is **end behavior.** It is exactly what it sounds like; how the “ends” of the graph behaves or points. The cubic function above has ends that point in the opposite direction. We say that from left to right, this function is *mostly increasing.* The quartic function’s ends point in the same direction, both positive, just like a quadratic function. When considering end behavior, look at the leading coefficient and the degree of the polynomial.

Let's use a table to graph \begin{align*}y=x^3\end{align*}

Draw a table and pick at least 5 values for \begin{align*}x\end{align*}

\begin{align*}x\end{align*} |
\begin{align*}x^3\end{align*} |
\begin{align*}y\end{align*} |
---|---|---|

-2 | \begin{align*}(-2)^3\end{align*} |
-8 |

-1 | \begin{align*}(-1)^3\end{align*} |
-1 |

0 | \begin{align*}0^3\end{align*} |
0 |

1 | \begin{align*}1^3\end{align*} |
1 |

2 | \begin{align*}2^3\end{align*} |
8 |

Plot the points and connect. This particular function is the **parent graph** for cubic functions. Recall from quadratic functions, that the parent graph has a leading coefficient of 1, no other \begin{align*}x-\end{align*}

Now, let's analyze the graph below. Find the critical values, end behavior, and find the domain and range.

First, find the solutions. They appear to be (-2, 0), (1, 0), and (2, 0). Therefore, this function has a minimum degree of 3. However, look at the \begin{align*}y-\end{align*}

maximum: (-1.1, 10)

minimum: (1.5, -1.3)

\begin{align*}y-\end{align*}

In general, this function is mostly increasing and the ends go in opposite directions. The domain and range are both all real numbers.

When describing critical values, you may approximate their location. You will later use the graphing calculator to find these values exactly.

Sometime it can be tricky to see if a function has imaginary solutions from the graph. Compare the graph in the previous problem to the cubic function above it. Notice that it is smooth between the maximum and minimum. As was pointed out earlier, the graph from the previous problem bends. Any function with imaginary solutions will have a slightly irregular shape or bend like this one does.

Finally, let's sketch a graph of a function with roots \begin{align*}-4, -3, \frac{1}{2}\end{align*}

There are several possible answers for this graph because we are only asking for a sketch. You would need more information to get an exact answer. Because this function has negative end behavior and four roots, we know that it will pass through the \begin{align*}x-\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find the maximum height the coaster will reach over the domain [-1, 2].

Use a table to graph \begin{align*}x^5 - 8x^3 + 10x + 6\end{align*}.

Draw a table and pick at least 5 values for \begin{align*}x\end{align*}. Remember that we are dealing only with *x* values between and including -1 and 2.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

-1 | 3 |

0 | 6 |

0.5 | 10.03125 |

1 | 9 |

2 | -16 |

Plot the points and connect.

From your graph you can see that the maximum height the roller coaster reaches is just slightly over 10.

#### Example 2

Use a table to graph \begin{align*}f(x)=-(x+2)^2(x-3)\end{align*}.

This function is in intercept form. Because the factor, \begin{align*}(x + 2)\end{align*} is squared, we know it is a repeated root. Therefore, the function should just touch at -2 and not pass through the \begin{align*}x-\end{align*}axis. There is also a zero at 3. Because the function is negative, it will be generally decreasing. Think of the slope of the line between the two endpoints. It would be negative. Select several points around the zeros to see the behavior of the graph.

\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|

-4 | 14 |

-2 | 0 |

0 | 12 |

2 | 16 |

3 | 0 |

4 | -36 |

#### Example 3

Analyze the graph. Find all the critical values, domain, range and describe the end behavior.

There are three real zeros at approximately -3.5, 1, and 7. Notice the curve between the zeros 1 and 7. This indicated there are two imaginary zeros, making this at least a fifth-degree polynomial. Think about an imaginary horizontal line at \begin{align*}y = 3\end{align*}. This line would touch the graph five times, so there should be five solutions. Next, there is an absolute minimum at (-0.5, -7.5), a local maximum at (2.25, 5), a local minimum at (2.25, 2.25) and an absolute maximum at (5, 6). The \begin{align*}y-\end{align*}intercept is at (0, -6). The domain and range are both all real numbers and the end behavior is mostly decreasing.

#### Example 4

Draw a graph of the cubic function with solutions of -6 and a repeated root at 1. This function is generally increasing and has a maximum value of 9.

To say the function is “mostly increasing” means that the slope of the line that connects the two ends (arrows) is positive. Then, the function must pass through (-6, 0) and touch, but not pass through (1, 0). From this information, the maximum must occur between the two zeros and the minimum will be the double root.

### Review

Use the given \begin{align*}x-\end{align*}values to make a table and graph the functions below.

- \begin{align*}f(x) &= x^3-7x^2+15x-2\\ x &= -2, -1, 0, 1, 2, 3, 4\end{align*}
- \begin{align*}g(x) &= -2x^4 - 11x^3 - 3x^2+37x+35\\ x &= -5, -4, -3, -2, -1, 0, 1, 2\end{align*}
- \begin{align*}y &=2x^3+25x^2+100x+125\\ x &= -7,-6,-5,-4,-3,-2,-1,0\end{align*}

Make your own table and graph the following functions.

- \begin{align*}f(x)=(x+5)(x+2)(x-1)\end{align*}
- \begin{align*}y=x^4\end{align*}
- \begin{align*}y=x^5\end{align*}
- Analyze the graphs of \begin{align*}y=x^2, y=x^3, y=x^4\end{align*}, and \begin{align*}y=x^5\end{align*}. These are all parent functions. What do you think the graph of \begin{align*}y=x^6\end{align*} and \begin{align*}y=x^7\end{align*} will look like? What can you say about the end behavior of all even functions? Odd functions? What are the solutions to these functions?
**Writing**How many repeated roots can one function have? Why?

Analyze the graphs of the following functions. Find all critical values, the domain, range, and end behavior.

For questions 13-15, make a sketch of the following real-solution functions.

- Draw
**two**different graphs of a cubic function with zeros of -1, 1, and 4.5 and a minimum of -4. - A fourth-degree polynomial with roots of -3.2, -0.9, 1.2, and 8.7, positive end behavior, and a local minimum of -1.7.
- A fourth-degree function with solutions of -7, -4, 1, and 2, negative end behavior, and an absolute maximum at \begin{align*}\left(-\frac{11}{2}, \frac{1755}{128}\right)\end{align*}.
**Challenge**Find the equation of the function from #15.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.13.