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Quadratic Equations and Functions

Quadratic Equations and Functions
1. For f(x) = –(x + 3)2 + 4, answer the following questions.
a. Find the values of the x-intercepts.
b. State the coordinates of the y -intercept.
c. State the vertex .
d. State the axis of symmetry.
e. State the direction of the parabola .
f. Decide whether there is a relative maximum or minimum, then state it.
g. Graph the function accurately using the above information.

2. For g(x) = 3x2 – 12x + 6, answer the following questions.
a. Find the values of the x-intercepts.
b. State the coordinates of the y-intercept.
c. Write the function in standard form by completing the square.
d. State the vertex.
e. State the axis of symmetry.
f. State the direction of the parabola.
g. Decide whether there is a relative maximum or minimum, then state it.
h. Graph the function accurately using the above information.
i. State the domain.
j. State the range.
k. State the interval of x where the graph is increasing.
l. State the interval of x where the graph is decreasing.

Applications of Quadratic Functions
1. Laura owns and operates Aunt Linda’s Pecan Pies. She has learned that
her profits, P(x), from the sale of x cases of pies, are given by P(x) = 150x – x2.
a. The company will “break-even” when the profit is zero. How many cases of pies
should Laura sell in order to break -even? (Solve for x when P(x) = 0.)
b. How many cases of pies should she sell in order to maximize profit?
c. What is the maximum profit?

2. John wants to build a corral next to his barn. He has 300 feet of fencing to enclose
three sides of his rectangular yard.
a. What is the largest area that can be enclosed?
b. What dimensions will result in the largest yard?

3. The function V(t) = –3t2 + 140t + 824 models the number (in thousands), V(t), of
murders committed in a certain state in the United States t years after 1960.
Let t = 0 represent 1960, t = 1 represent 1961, and so on.
a. Determine the year in which the most violent crimes were committed.
b. Approximately how many violent crimes were committed during this year?
c. Using a graphing calculator, graph V(t). Were the number of violent crimes
increasing or decreasing during the years 2000 to 2005?

4. A ball is thrown vertically upward from the top of a building 1600 feet tall with an initial
velocity of 80 feet per second. The distance d(t), in feet, of the ball from the ground
after t seconds is
d(t) = 1600 + 80t – 16t2.
a. After how many seconds is the ball at its maximum height?
b. Find the maximum height of the ball.

Graphs of Polynomial Functions
For the following polynomials:
a. State the degree and the sign of the leading coefficient . Then use the
information to determine the left and right behavior of the graph (the end
behavior).
b. List all of the real zeros with their multiplicity and state the behavior at the x-axis
of each zero (crosses or touches).
c. Based on parts (a) and (b), draw a rough sketch of the graph without using a
calculator.
d. Check part (c) with a graphing calculator.

Example: p(x) = 3(x + 4)3(x + 1)(x – 5)2,

1. f(x) = (x + 3)3(x – 2)2(x – 4)
2. g(x) = –x(x – 5)(x + 2)3
3. h(x) = –3(x – 1)2(x + 1)2
4. j(x) = 6(x + 5)(x + 4)2(x – 1)4(x – 3)2
5. j(x) = –x3 + 4x
6. k(x) = x3 – 4x2

Finding Real Zeros of Polynomial Functions
1. For f(x) = x3 – 7x2 + 8x + 16,
a. Find f(10) using synthetic division.
b. Is 10 a zero of the function? Explain.
c. Use synthetic division to determine if x + 1 is a factor of f(x).
d. Is –1 a zero of the function? Explain.
e. List all of the zeros of the polynomial.
f. Write the polynomial function as a product of linear factors.

2. For the function , f(x)=6x4 - 7x3 - 10x2 + 17x - 6
a. State the degree of the polynomial.
b. Use the Rational Zero Theorem to list all of the possible rational zeros.
c. Use a graphing calculator to determine which numbers in the list of possible rational
zeros are probable rational zeros (indicated by the x-intercepts of the
graph).
d. Use synthetic division to determine one rational zero.
e. Use synthetic division or other algebraic methods to find all remaining
zeros. List all of the zeros of the polynomial function.
f. Write the polynomial function as a product of linear factors .

Finding Complex Zeros of Polynomial Functions
OMIT 1. For the function f(x) = 3x4 - 4x3 + x2 + 6x - 2
a. State the degree of the polynomial
b. State the number of zeros the polynomial function will have.
c. Use the Rational Zero Theorem to list all of the possible rational zeros.
d. Use your calculator to determine which numbers in the list of rational zeros
are probable rational zeros.
e. Use synthetic division to verify one rational zero.
f. Use synthetic division or other algebraic methods to find all remaining
zeros. List all of the zeros of the polynomial function.

2. For the function g(x) = x4 - 2x3 + 14x2 - 8x + 40
a. State the degree of the polynomial
b. State the number of zeros the polynomial function will have.
c. Given that 2i is a zero, find all remaining zeros. List all of the zeros of the 2i
polynomial function.

3. Given a cubic polynomial function p(x) = ax3 + bx2 + cx + d, (a, b, c, d ≠ 0), answer the
following questions. Justify each answer.
a. How many x-intercepts can there be?
b. Does the degree of this polynomial function guarantee any x-intercepts?
c. Will the graph pass through the origin?
d. Could the graph “touch” the x- axis in two different places ?
e. Identify the end behavior of the graph.
f. If it is known that one zero is real and another zero is imaginary, what can be
determined about the remaining zeros?

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