Math Analysis Honors – MATH Sheets

M = Modeling A = Again T = Today’s Topic H = Homework

#14 Monday 9/16________________________________ M Ex. 1 Given f x( ) = ax2 + bx + c , find the x-coordinate of the vertex by completing the square and

converting the standard form equation into vertex form. Ex. 2 Locate the vertex and axis of symmetry of the parabola defined by f x( ) = −3x2 + 6x +1 . Graph

f x( ) . Ex. 3 Graph f x( ) = x2 − 6x + 9 . Determine where f is increasing and where it is decreasing.

A Given g x( ) = −12

x − 2 + 5 , list the transformations that occur in order to change f x( ) into g x( ) , if

f x( ) = x . T Graph a Quadratic Function in Standard Form using its Vertex and Axis of Symmetry H Worksheet 13 #15 Tuesday 9/17_______________________________ M Ex. 1 Determine whether the quadratic function, f x( ) = x2 − 4x − 5 , has a maximum or minimum. Find

the maximum or minimum value. Ex. 2 The marketing department at Texas Instruments has found that, when certain calculators are sold

at a price p dollars per unit, the revenue R (in dollars) as a function of the price p is R p( ) = −150p2 + 21,000p . What unit price should be established to maximize revenue? If this price is charged, what is the maximum revenue?

Ex. 3 A farmer has 2000 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area?

A Solve the quadratic equation by any method: 3x2 − 4x − 2 = 0 T Use the Max or Min Value of a Quadratic Function to Solve Applied Problems H Worksheet 14 #16 Wednesday 9/18 – Delayed Start________________ M Ex. 1 A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec reaches a

height of h t( ) = 24t − 0.8t 2 meters in t seconds. (a) How long did it take the rock to reach its highest point? (b) How high did the rock go? (c) When did the rock reach half its maximum height? (d) When did the rock hit the surface of the moon again? Ex. 2 A rock is thrown from the top of an 80-foot building with an initial upward velocity of 64 ft/sec.

The height of the rock is determined by h t( ) = −16t 2 + 64t + 80 , where h is measured in feet and t is measured in seconds.

(a) How long did it take the rock to reach its highest point? (b) How high did the rock go? (c) When did the rock reach half its maximum height? (d) When did the rock hit the ground?

“M” is the sample problems for the day “A” are review problems - “A” questions are ALWAYS done first!!

“T” are the objectives of the day “H” are homework problems

A Graph the function, f x( ) =2x −1,     x < 2x2 ,           x ≥ 2

⎧⎨⎩⎪

.

T Use the Max or Min Value of a Quadratic Function to Solve Applied Problems (Vertical Motion) H Worksheet 15 #17 Thursday 9/19_______________________________ M Complete the following statements for each function: As x→∞, f x( )→ ____ .      As x→−∞, f x( )→ ____ .       1. (a) f x( ) = 2x3 (b) f x( ) = −2x3 (c) f x( ) = x5 (d) f x( ) = −0.5x 2. (a) f x( ) = −3x4 (b) f x( ) = 0.6x4 (c) f x( ) = 2x6 (d) f x( ) = −0.5x2 Describe the patterns you observe. In particular, how do the values of the coefficient an and the degree

n affect the end behavior of f x( ) = anxn A Use a graphing utility to approximate (round to three decimal places) the local maxima and local minima

of f x( ) = x3 − 2x2 − 4x + 5 . T Short Quadratics Quiz!

- A polynomial function is a function in the form f x( ) = anxn + an−1xn−1 + ...+ a 1x + a0 , where an ,an−1,... a1, a0 are real number and n is a nonnegative integer. The domain of all polynomials is ALL real numbers. - End Behavior of Polynomial Functions; The end behavior of a function depends on the degree and leading coefficient of the polynomial.

H None #18 Friday 9/20 – Assembly________________________ M Ex. 1 Describe the end behavior of f x( ) = x3 + 2x2 −11x −12 . Ex. 2 Describe the end behavior of f x( ) = 2x4 + 2x3 − 22x2 −18x + 35 .

A Describe how to transform the graph of f x( ) = x into g x( ) = −12

x +1 + 2 .

T End Behavior of a Function. H Worksheet 16 #19 Monday 9/23_________________________________ M Ex. 1 Find a polynomial of degree 3 whose zeros are −3, 2, and 5 . Draw a possible graph of this

function.

Ex. 2 For the polynomial f x( ) = 5 x − 2( ) x + 3( )2 x − 12

⎛⎝⎜

⎞⎠⎟4

, state the zeros and their multiplicities. Draw a

possible graph of f x( ) . Ex. 3 For the polynomial f x( ) = x2 x − 2( )

a) Find the x- and y-intercepts of the graph of f. b) Using a graphing calculator, graph the polynomial. c) For each x-intercept, determine whether it is of odd or even multiplicity

A Graph f x( ) = x2 + 4x − 5 . T Identify the Zeros of a Polynomial Function and Their Multiplicity.

The Factor Theorem: A polynomial function f x( ) has a factor x − k if and only if f k( ) = 0 , where k is an x-intercept of the graph of f x( ) .

H Worksheet 17

#20 Tuesday 9/24_________________________________ M Ex. 1 Find the zeros (roots) of f x( ) = x3 − x2 − 6x Ex. 2 Find the zeros (roots) of f x( ) = x3 − 36x Ex. 3 Find the zeros (roots) of f x( ) = 3x3 − x2 − 2x Ex. 4 Find the zeros (roots) of f x( ) = x3 − 3x2 − 4x +12 A Find the vertex and axis of symmetry of the parabola determined by the function f x( ) = 3x2 − 4x − 7 . T Finding the Zeros of a Polynomial Function by Factoring

Finding the real-number zeros of a function f is equivalent to finding x-intercepts of the graph of y = f x( ) or the solutions to the equation f x( ) = 0 .

H Worksheet 18 #21 Wednesday 9/25 – Back to School_________________ M Ex. 1 Graph f x( ) = x + 2( )3 x −1( )2 by finding the x-intercepts, y-intercept and end behavior. Ex. 2 Graph f x( ) = 3x3 − x2 − 2x by finding the x-intercepts, y-intercept and end behavior. Ex. 3 Graph f x( ) = x4 − 5x2 + 4 A Factor 6x3 − 22x2 +12x T Graphing Polynomials using x-intercepts (with multiplicity), y-intercept and end behavior. H Worksheet 19 #22 Thursday 9/26_________________________________ M Review the following topics: Graphing Quadratics; Solve Applied Quadratic Problems; End Behavior of

a Polynomial Function; Zeroes of a Polynomial; Graphing Polynomials

A In the set {−2,− 2,0, 12, 4.5,π} , name the numbers that are integers. Which are rational numbers?

T Quadratics and Polynomials Review – Test on Monday H Worksheet 20 #23 Friday 9/27 – Minimum Day______________________ M Good luck on Today’s Test A None T Test #3 - Quadratics and Polynomials Test H None #24 Monday 9/30___________________________________ M Ex.1 List the possible rational zeros for the function, f x( ) = 3x3 + 4x2 − 5x − 2 . Find all rational zeros.

Ex. 2 List the possible rational zeros for the function, f x( ) = 2x3 − x2 − 9x + 9 . Find all rational zeros. Ex. 3 Prove that all the real zeros of f x( ) = 2x4 − 7x3 − 8x2 +14x + 8 must lie in the interval −2,5[ ] .

Ex. 4 Prove that the number k is an upper bound for the real zeros of the function, f x( ) = 2x2 − 4x2 + x − 2;k = 3

A Rewrite the expression as a polynomial in standard form: 2x3 − 5x2 − 6x2x

T Rational Zeros Theorem: Using the constant and the leading coefficient, we can develop a list of all potential rational (fractional) zeros. Bounds Test

H Worksheet 21 #25 Tuesday 10/1___________________________________ M Adding and Subtracting Complex Numbers

1) 7 − 3i( ) + 4 + 5i( ) 2) 2 − i( )− 8 + 3i( ) 3) 8i − 4 − 3i( ) Multiplying Complex Numbers

4) 2 + 3i( ) 5 − i( ) 5) 3− 5i( )4i Raising a Complex Number to a Power

6) 12+

32i

⎛

⎝⎜⎞

⎠⎟

2

Complex Conjugates and Division

7) 4 + 5i( ) 4 − 5i( ) 8) 5 + i2 − 3i

9) Solve x2 + x +1= 0 A Perform the indicated operation on the complex ( a + bi ) numbers: 1) 2 + i( ) 3− 2i( ) 2) 4 − 3i( ) 4 + 3i( ) 3) 5 + i( ) 5 − i( ) T Complex Numbers H Worksheet 22 #26 Wednesday 10/2_________________________________ M Ex. 1 One zero of f x( ) = x4 − 5x3 − 3x2 + 43x − 60 is 2 − i . Find the other zeros (real and nonreal)

Ex. 2 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros include x = 1 and x = 3+ i . Ex. 3 Write a polynomial function of minimum degree in standard form with real coefficients whose zeros and their multiplicities are x = 1 (multiplicity 2) and x = 3+ i (multiplicity 1)

A List all potential rational zeros of f x( ) = 3x4 − 5x3 + 3x2 − 7x + 2 T Fundamental Theorem of Algebra: A polynomial function of degree n has n complex zeros (real and

nonreal). Some of these zeros may be repeated. Complex Conjugate Zeros H None #27 Thursday 10/3___________________________________ M Ex. 1 The complex number z = 1− 2i is a zero of f x( ) = 4x4 +17x2 +14x + 65 . Find the remaining zeros

of f x( ) , and write it in factored form. Ex. 2 Find all of the zeros and write a linear factorization of f x( ) if z = 1+ i is a zero of f x( ) = x4 − 2x3 − x2 + 6x − 6 .

Ex. 3 Find all zeros of f x( ) = x5 − 3x4 − 5x3 + 5x2 − 6x + 8 . A A polynomial of degree 5 can have at most ____ distinct real zeros. T Finding Complex Zeros H Worksheet 23 #28 Friday 10/4_____________________________________ M Ex. 1 Write f x( ) = 3x5 − 2x4 + 6x3 − 4x2 − 24x +16 as a product of linear and irreducible quadratic

factors, each with real coefficients. Ex. 2 Write f x( ) = x3 − x2 − x − 2 as a product of linear and irreducible quadratic factors with real

coefficients. Ex. 3 Write a polynomial function of minimum degree in standard from with real coefficients whose

zeros include: −1, 2, and 1+ i . A Find the vertex and axis of symmetry of the graph of f x( ) = x − 2( ) x + 6( ) T Factoring Polynomial Functions; Review for our next Test (Tuesday, 10/8) H Worksheet 24