Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials.

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy…

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1. Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function.

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points.

Graphing Polynomial Functions We will now use the strategies from the previous sections to sketch the graphs of higher order polynomials. Follow these steps and the process is quite easy… 1.Using the Remainder / Factor theorem and synthetic division, factor the function completely. This will give you the roots of the function where it crosses the x – axis. It will also set up intervals in which we will test points to get a general picture of the function. 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions Then, using synthetic division… Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions Then, using synthetic division… Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions Then, using synthetic division… Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions The roots of the function are the x – values that create a zero… 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The intervals are set by the roots

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - Now plug each test value into the function…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x – axis Above x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x – axis Above x – axis Plot the point…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegative Graph Below x – axis Above x – axis Below x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegative Graph Below x – axis Above x – axis Below x – axis Plot the point…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositivenegativepositive Graph Below x – axis Above x – axis Below x – axis Above x _ axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph Interval Test Value ( x ) negativepositivenegativepositive Graph Below x – axis Above x – axis Below x – axis Above x _ axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Now sketch in the rest of the graph using smooth curves…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division… 1-313

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Then use synthetic division…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function The roots of the function are the x – values that create a zero…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The intervals are set by the roots

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - The test value will be some “x” within the interval or close to the root…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) Graph - Now plug each test value into the function…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negative Graph Below x - axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositive Graph Below x - axis Above x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Plot the point… Interval Test Value ( x ) negativepositive Graph Below x - axis Above x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegative Graph Below x - axis Above x – axis Below x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegative Graph Below x - axis Above x – axis Below x – axis Plot the point…

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegativepositive Graph Below x - axis Above x – axis Below x – axis Above x – axis

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Interval Test Value ( x ) negativepositiveNegativepositive Graph Below x - axis Above x – axis Below x – axis Above x – axis - The point is off the graph, no need to plot extreme values…we will use arrows to show the “ends” of the graph

Graphing Polynomial Functions 1.Factor completely 2.Plot the roots on the graph 3.Create a test table using the roots of the function 4.Test value(s) in each interval and plot the points. 5.Sketch the function Now sketch in the rest of the graph using smooth curves…

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.

Graphing Polynomial Functions One note here… I am choosing values in between the roots that is half way. The sketch then shows the resulting point as what would be called a relative maximum or minimum. There are functions where this is not the case. Some values to the left or right of the middle of two roots could be higher or lower.