Analyzing Graphs of Polynomial Functions

Select all statements that apply to this graphed polynomial...

• The domain

  is (-∞, ∞)

• The range

  is (-∞, ∞)

• The range

  is [-4, ∞)

• There is one absolute minimum

• There are no local maximum

Answer explanation

The domain of a polynomial is always (-∞, ∞). The range is [-4, ∞) indicating a minimum value of -4. There is one absolute minimum at this point, and no local maximum exists in this polynomial.

Tags

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

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

Select all statements that apply to the graphed polynomial...

• The domain

  is (-∞, ∞)

• The range

  is (-∞, ∞)

• The range

  is (-∞, 6)

• There is 1 absolute maximum

• There are 1 local minimum, 1 local maximum

Answer explanation

The domain of a polynomial is always (-∞, ∞). The range is also (-∞, ∞) if it does not have a maximum or minimum. The graph shows 1 local minimum and 1 local maximum, confirming these statements.

Tags

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

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

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

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

Estimate the x-coordinate at which extrema occur.

• 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

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

Which of the following statement is FALSE?

• There is a relative min at the point (8, 0)

• There is another relative min at the point (0, -7.5)

• There is a relative max at the point (5, 2.5)

• There are more relative maximums than there are relative minimums.

Answer explanation

The statement 'There are more relative maximums than there are relative minimums' is false because the given points indicate there are two relative minima and only one relative maximum.

Tags

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

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