Characteristics of Polynomial Graphs

Questions and Answers

Which of the following describes a continuous function?

A function that only increases

A function with breaks in its domain

A function that only decreases

A function without holes or breaks (correct)

A vertical asymptote is a line that a curve intersects.

False

What is the definition of an inflection point?

A point where the concavity of the function changes.

The x-intercept occurs when the point is (______ , 0).

x Signup and view all the answers

What is an example of a discontinuous function?

A function with a hole at a specific point Signup and view all the answers

Match the function type with its characteristic:

Constant = A line parallel to the x-axis Linear = A straight line Cubic = A curve with potentially turning points Absolute Value = V-shape graph Signup and view all the answers

Turning points indicate where the slope changes from increasing to decreasing.

True Signup and view all the answers

What are global maxima and minima?

The highest and lowest points over the entire domain of a function. Signup and view all the answers

Which of the following statements defines a function?

A relation where every input corresponds to only one output. Signup and view all the answers

The vertical line test can be used to determine if a graph represents a function.

True Signup and view all the answers

What is the graphical representation of the function $y = |x|$?

V-shaped graph opening upwards Signup and view all the answers

The set of all possible input values for a function is called the ______.

domain Signup and view all the answers

For which of the following equations does the vertical line test apply?

y = x^2 Signup and view all the answers

Match the following types of functions with their characteristics:

Polynomial = A function involving terms of x raised to non-negative integer powers Linear = A function that creates a straight line on a graph Absolute Value = A piecewise function that outputs non-negative values Quadratic = A polynomial function of degree 2 Signup and view all the answers

In the relation {(1,2), (1,3), (2,4)}, the value '1' is considered the ______.

input Signup and view all the answers

What does the range of a function represent?

The set of all possible output values. Signup and view all the answers

Which of the following equations represents a polynomial function?

y = 10x Signup and view all the answers

The range of a polynomial function is always limited.

False Signup and view all the answers

What is the leading coefficient in the polynomial function y = -5x^4 + x^3 + 1?

-5 Signup and view all the answers

A polynomial function can have _____ as its domain.

all real numbers Signup and view all the answers

Match the type of function with its characteristic:

Polynomial Function = Has no horizontal or vertical asymptotes Trigonometric Function = Includes sine, cosine, etc. Exponential Function = Involves a variable in the exponent Logarithmic Function = The inverse of exponential functions Signup and view all the answers

What does 'n' indicate in the standard form of a polynomial expression?

The degree of the function Signup and view all the answers

The vertical line test can be used to determine if a relation is a function.

True Signup and view all the answers

Identify the constant term in the polynomial function f(x) = 3x^4 - 2x^2 + 5.

5 Signup and view all the answers

Study Notes

Continuous Functions

Functions without holes or breaks across their entire domain.
Types include Constant, Linear, Quadratic, Cubic, and Absolute Value.

Discontinuous Functions

Functions containing at least one hole or break in their domain.
Examples:
- Domain: 𝑥 ϵ (− ∞, 2) ∪ (2, + ∞); Range: 𝑦 ϵ (− ∞, 7) ∪ (7, + ∞)
- Domain: 𝑥 ϵ (− ∞, 3] ∪ (3, + ∞); Range: 𝑦 ϵ (− ∞, 1] ∪ (4, + ∞)
- Domain: 𝑥 ϵ (− ∞, 3) ∪ (3, + ∞); Range: 𝑦 ϵ (− ∞, 0) ∪ (0, + ∞)

Asymptote

Lines (vertical, horizontal, oblique) that a curve approaches but never intersects.

Intercepts

X-intercept: Occurs when y = 0 at the point (x, 0).
Y-intercept: Occurs when x = 0 at the point (0, y).

Turning Points

Points on a graph where the slope changes direction.
Minimum points indicate a transition from decreasing to increasing interval.
Maximum points indicate a transition from increasing to decreasing interval.

Inflection Points

Locations where the concavity of the function changes.
Concave Down: A line segment joining two points lies below the curve.
Concave Up: A line segment joining two points lies above the curve.

Extrema Points

Global Maxima: The highest point in the entire domain.
Global Minima: The lowest point in the entire domain.
Polynomial functions can be unbounded.

Polynomial Functions

Defined with standard forms: n is a non-negative integer, x is the variable, an are real coefficients.
Leading coefficient (an) is the coefficient of the term with the highest degree.
Domain: Real numbers (𝐷 = {𝑥 | 𝑥 ϵ 𝑅}).
Range can be all real numbers or have bounds.

Non-Polynomial Functions

Some relations are functions but not polynomial:
- Trigonometric function (e.g., 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥)
- Exponential function (e.g., 𝑔(𝑥) = 𝑎^𝑥)
- Logarithmic function (e.g., ℎ(𝑥) = 𝑙𝑜𝑔𝑎 𝑥).

Class Exercise Highlights

Identify polynomial function characteristics, such as degree, leading coefficient, and constant.
Example: For 𝑦 = −5𝑥^4 + 𝑥^3 + 1, the Degree = 4, Leading Coefficient = -5, Constant = 1.

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Description

This quiz reviews essential terminologies related to the characteristics of polynomial graphs, focusing on continuous and discontinuous functions. You'll explore various polynomial types such as constants, linear, quadratic, cubic, and absolute value functions. Test your understanding of the key features that define these graphs.

Characteristics of Polynomial Graphs

Questions and Answers

Which of the following describes a continuous function?

A vertical asymptote is a line that a curve intersects.

What is the definition of an inflection point?

The x-intercept occurs when the point is (______ , 0).

What is an example of a discontinuous function?

Match the function type with its characteristic:

Turning points indicate where the slope changes from increasing to decreasing.

What are global maxima and minima?

Which of the following statements defines a function?

The vertical line test can be used to determine if a graph represents a function.

What is the graphical representation of the function $y = |x|$?

The set of all possible input values for a function is called the ______.

For which of the following equations does the vertical line test apply?

Match the following types of functions with their characteristics:

In the relation {(1,2), (1,3), (2,4)}, the value '1' is considered the ______.

What does the range of a function represent?

Which of the following equations represents a polynomial function?

The range of a polynomial function is always limited.

What is the leading coefficient in the polynomial function y = -5x^4 + x^3 + 1?

A polynomial function can have _____ as its domain.

Match the type of function with its characteristic:

What does 'n' indicate in the standard form of a polynomial expression?

The vertical line test can be used to determine if a relation is a function.

Identify the constant term in the polynomial function f(x) = 3x^4 - 2x^2 + 5.

Study Notes

Continuous Functions

Discontinuous Functions

Asymptote

Intercepts

Turning Points

Inflection Points

Extrema Points

Polynomial Functions

Non-Polynomial Functions

Class Exercise Highlights

Studying That Suits You

Description

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