1. Synthetic Substitution
1. Synthetic substitution is a way to quickly evaluate a function for a given value.
2. In synthetic substitution, you write the value you are evaluating for in in a box to the left.
3. Next to it, carry down the coefficients of each term of the polynomial. Remember to write down a zero for any term that is "absent."
4. Then, skip a line and draw a line. Carry the leading coefficient down below the line.
5. Multiply the number below the line by the number in the box, and write the product in below the next above the line and below the next coefficient.
6. Add the number you just wrote and the coefficient above it and write the sum below the line.
7. To see an animated example problem worked through step by step, click here.

2. End Behavior of Graphs of Polynomial Functions
1. End Behavior is how a graph appears as the x-value approaches positive infinity or negative infinity.
2. For any polynomial in standard form, there are 4 possible types of end behavior.
1. When the leading coefficient is positive, and the degree of the polynomial is even, f(x) approaches positive infinity as x approaches negative infinity and f(x) approaches positive infinity as x approaches positive infinity. (Bowl)
2. When the leading coefficient is positive, and the degree of the polynomial is odd, f(x) approaches negative infinity as x approaches negative infinity, and f(x) approaches positive infinity as x approaches positive infinity. (Noodle*)
3. When the leading coefficient is negative and the degree is even, f(x) approaches negative infinity as x approaches negative infinity, and f(x) approaches negative infinity as x approaches positive infinity. (Upside-down bowl)
4. When the leading coefficient is negative and the degree is odd, f(x) approaches positive infinity as x approaches negative infinity, and f(x) approaches negative infinity as x approaches positive infinity. (Noodle*)
   
   *One way to tell the "noodles" apart is to think of a linear graph. Linear graphs (which technically have an odd degree) with negative coefficients go from the top left to the bottom right like polynomial graphs with negative coefficients and odd degrees. Likewise, graphs of linear functions with positive coefficients go from bottom left to top right just like graphs of polynomial functions with positive coefficients and odd degrees.

3. Graphing Polynomial Functions
1. Polynomial functions are graphed just like quadratic functions.
2. Make an list of x values, and use direct or synthetic substitution to find corresponding y values.
3. Plot the points from the x-y list.
4. Connect the points with a smooth curve
5. Make sure the end behavior of your graph is what it should be. If it doesn't match, you either did something wrong or need to expand your x-y list to include more values until the end behavior becomes apparent in your graph.

Fill in the blank with the appropriate end behavior for the given polynomial function.

1. f(x) = 2x⁴ + 77x³ + 4x - 19
   f(x) --> ______ as x --> negative infinity and f(x) --> ______ as x --> positive infinity
2. f(x) = 3x⁴ - 2x⁵ - 7x² + 4x + 2
   f(x) --> ______ as x --> negative infinity and f(x) --> ______ as x --> positive infinity
3. f(x) = x² - 2x + 88
   f(x) --> ______ as x --> negative infinity and f(x) --> ______ as x --> positive infinity
4. f(x) = -3x⁴ - 8x³ + 14x
   f(x) --> ______ as x --> negative infinity and f(x) --> ______ as x --> positive infinity
5. f(x) = 8x³ + 12x² + 11x - 1
   f(x) --> ______ as x --> negative infinity and f(x) --> ______ as x --> positive infinity

Graph the following polynomial functions.

6. f(x) = x³ + 2x² - 3x + 1
7. f(x) = x⁴ - 2x³ + x - 2

1. positive infinity; positive infinity
2. positive infinity; negative infinity
3. positive infinity; positive infinity
4. negative infinity; negative infinity
5. negative infinity; positive infinity
6. 
   
7.