Math 116: Study Guide - Chapter 3

No Calculators are allowed on this exam.

1. Complete the square to put a quadratic into standard form for a parabola (2 pts). Identify the vertex (1 pt), x-intercepts (1 pt), y-intercept (1 pt) of the function. Sketch the graph (2 pts) of the function. Look at problems 3.1.13 - 3.1.21.
2. Find the equation of the parabola with the given vertex and passing through the given point. Look at problems 3.1.31 - 3.1.35.
3. List all possible rational zeros of a polynomial function. Do not find which ones are solutions, just list the possible roots.
4. Use synthetic division to show the value given is a solution to the equation and use the result to completely factor the polynomial. Look at problems 3.3.41 - 3.3.47.
5. A polynomial is evaluated using synthetic division. The value and bottom row from the synthetic division are given. Indicate whether the value is an upper bound, lower bound, or neither. Five parts.
6. Identify the translation, and determine, if possible, the zeros of a transformed function. Look at problems 3.3.87 - 3.3.92. Three parts.
7. A polynomial function is given in both expanded and factored form. Be able to identify
1. the number of real or complex zeros, (1 pt)
2. the maximum number of extrema (maximums or minimums), (1 pt)
3. the right hand behavior of the graph, (1 pt)
4. The left hand behavior of the graph, (1 pt )
5. the form of any possible rational zeros, (1 pt)
6. the maximum number of positive real roots, (1 pt)
7. the maximum number of negative real roots, (1 pt)
8. all the real and complex zeros, (2 pts)
9. where the graph crosses the x-axis, (1 pt)
10. where the graph touches the x-axis, (1 pt)
11. the y-intercept, (1 pt)
12. the domain of the function, (1 pt)
13. Also be able to sketch the function. When you sketch, pay attention to the information above. (2 pts)
8. Same as #7.
9. A rational function is given in factored form. Be able to identify
1. the domain of the function, (1 pt)
2. simplify the function, be sure to state any restrictions that may be necessary after the simplification. (1 pt)
3. the behavior of the graph when there is a common factor between numerator and denominator (multiple choice), (1 pt)
4. for what values in the domain of the function will the graph cross the x-axis (1 pt)
5. for what values in the domain of the function will the graph touch the x-axis (1 pt)
6. the behavior at the right and left sides [ horizontal asymptote ] (multiple choice), (1 pt)
7. where the graph is asymptotic in different directions to a vertical line (1 pt)
8. where the graph is asymptotic in the same direction to a vertical line (1 pt)
9. Sketch the graph of the function. (2 pts)
10. Same as #9.
11. Write the function (in factored form) with integer coefficients which has the indicated zeros. Be aware of multiplicity and complex roots or roots with radicals. You do not need to expand the polynomial, but you do need to make sure there are no radicals, complex numbers, decimals, or fractions in the coefficients. Three Parts. Look at problems 3.4.29 - 3.4.36.
12. True or False. Know ...
1. That complex solutions involving i come in pairs.
2. When an oblique asymptote occurs.
3. Polynomials are smooth and continuous, continuous functions however do not have to be smooth.
4. What the Intermediate Value Theorem does and does not guarantee.
5. The role of the being able to sketch functions by hand when there are graphing calculators which will do it for you.
6. All polynomials are continuous.
7. Not all polynomials are one-to-one functions.
8. Rational functions aren't continuous.
13. Write the function whose graph could be shown. There are more than one possible function. Watch out for the exponents on factors to make the behavior turn out right. Don't forget about the number of extrema and its relation to the degree of the polynomial. Three parts.

Notes:

• No Calculators are allowed on this exam.
• Where specific problems are indicated to look at, the problem is similar to, but not exactly the same as, those problems in the book.
• Problems 7 - 10 each take one page. This violates my rule of 4. However, these problems have very little computation on them (factoring the difference of two squares, reducing a fraction, or counting the number of sign changes).
• There is a chance for extra credit if you really know the material.
• Do not spend too much time on any one problem or you will have difficulty getting through the entire test in the 50 minutes.