Characteristics of Graphs of Polynomials (End behavior and multiplicity)

What is the largest number of real zeros (x-intercepts) possible for the function f(x) = -12x¹⁰+ 4x⁵ + 13?

10

What is the maximum number of turning points possible in the graph of f(x) = x¹² - 5x⁴ - 6?

11

What is the largest number of real zeros (x-intercepts) possible for the function f(x) = x⁷+ x + 1?

7

What is the maximum number of turning points possible in the graph of f(x) = -12x¹⁰+ 4x⁵ + 13?

9

As x goes to negative infinity, y goes to negative infinity. As x goes to positive infinity, y goes to negative infinity.

In words only, describe the end behavior of an even degree polynomial with a negative leading coefficient.

As x goes to negative infinity, y goes to positive infinity. As x goes to positive infinity, y goes to positive infinity.

In words only, describe the end behavior of an even degree polynomial with a positive leading coefficient.

As x goes to negative infinity, y goes to positive infinity. As x goes to positive infinity, y goes to negative infinity.

In words only, describe the end behavior of an odd degree polynomial with a negative leading coefficient.

As x goes to negative infinity, y goes to negative infinity. As x goes to positive infinity, y goes to positive infinity.

In words only, describe the end behavior of an odd degree polynomial with a positive leading coefficient.

zero at x = -½ with multiplicity of 1, crosses the x-axis zero at x = 1 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (2x+1)(x-1)

zero at x = ½ with multiplicity of 1, crosses the x-axis zero at x = 1 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (2x-1)(x-1)

zero at x = -¼ with multiplicity of 1, crosses the x-axis zero at x = 1 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (4x+1)(x-1)

zero at x = ¼ with multiplicity of 1, crosses the x-axis zero at x = 1 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (4x-1)(x-1)

What happens when a zero has an even multiplicity?

The graph bounces off the x-axis at the zero.

What happens when a zero has an odd multiplicity?

The graph crosses the x-axis at the zero.

List each zero, its multiplicity and the effect on the graph for the function f(x) = x(x-1)²(x+2)²

zero at x = 0 with a multiplicity of 1, crosses the x-axis zero at x = 1 with multiplicity of 2, bounces off the x-axis zero at x = -2 with a multiplicity of 2, bounces off the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = x²(x-1)(x+2)²

zero at x = 0 with a multiplicity of 2, bounces off the x-axis zero at x = 1 with multiplicity of 1, crosses the x-axis zero at x = -2 with a multiplicity of 2, bounces off the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (x-1)(x+2)

zero at x = 1 with multiplicity of 1, crosses the x-axis zero at x = -2 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (x-1)²(x+2)

zero at x = 1 with multiplicity of 2, bounces off the x-axis zero at x = -2 with a multiplicity of 1, crosses the x-axis

List each zero, its multiplicity and the effect on the graph for the function f(x) = (x-1)²(x+2)²

zero at x = 1 with multiplicity of 2, bounces off the x-axis zero at x = -2 with a multiplicity of 2, bounces off the x-axis

Which graph has a minimum degree of 4?

Which graph has a minimum degree of 4?

What function in factored form has a zero at x = 2 with a multiplicity of 1 and a zero at -3 with a multiplicity of 2?

f(x) = (x-2)(x+3)²

What function in factored form has a zero at x = 2 with a multiplicity of 2 and a zero at -3 with a multiplicity of 1?

f(x) = (x-2)²(x+3)