Alg2 Analyzing Graphs of Polynomial Functions

Which of the following polynomials is written in standard form?

• y = 7x³ − 5x² + 2x + 7

• y = 8 − 4x² + 3x + 2x³

• y = 10x² − 5x⁵ + 2x + 20

• y = 1 + 2x² − 3x³ + 4x⁴

Answer explanation

The polynomial in standard form has terms arranged by decreasing degree. The correct choice, y = 7x³ − 5x² + 2x + 7, is in standard form, while the others are not due to mixed order of terms.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

What is another way to say "where a function crosses the x-axis"?

• x-intercept

• zero

• root

• all of these.

Answer explanation

The term "x-intercept" refers to where a function crosses the x-axis. Additionally, the terms "zero" and "root" also describe this point. Therefore, the correct answer is "all of these."

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

The degree of the polynomial determines the max number of roots.

• True

• False

Answer explanation

True. The degree of a polynomial indicates the highest power of the variable, which corresponds to the maximum number of roots (real or complex) it can have, according to the Fundamental Theorem of Algebra.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

What are the zeros of the given graph ? Choose all that apply.

Answer explanation

The zeros of the graph are the x-values where the graph intersects the x-axis. The correct choices are -3, -1, and 1, as these points correspond to where the function equals zero.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

Using the given table, determine the consecutive integer values of x between which each real zero of the function is located.

Answer explanation

The function has real zeros between -4 and -3, and between 0 and 1, as indicated by the sign changes in the table. Thus, the correct intervals are -4

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

Using the given table, determine the consecutive integer values of x between which each real zero of the function is located.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

• 0

• 1

• 4

• -1

• 2

• 5

Answer explanation

To find extrema, we calculate the derivative f'(x) = -6x^2 + 24x - 8 and set it to zero. Solving gives x = 0, 1, and 4. The relative minimum occurs between x = 0 and x = 1, and the relative maximum is at x = 4.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

• 0

• 1

• -1

• 2

• 5

• -2

• 3

• 4

• -3

Answer explanation

To find extrema, we calculate the derivative f'(x) = 6x^2 - 8x - 3 and set it to zero. Solving gives critical points at x = 2 and x = -1. The relative minimum occurs between x = -1 and x = 0, and the relative maximum between x = 1 and x = 2.

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

A polynomial of degree n has at most ​ (a)   zeros

and at most ​ (b)   extrema.

• n
• n-1
• n+1
• 2n
• n^2

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c

f(x) = 2x³ - 3x² + 4x -10

State the maximum number of turns the graph of f(x) could make.

• 3

• 2

• 1

• 0

Tags

CCSS.MATH.CONTENT.HSF.IF.B.4

CCSS.MATH.CONTENT.HSF.IF.C.7.c