Question

Characteristic of graphs?

Answer 1

See below.

Explanation:

Increasing and Decreasing Function

• A function is increasing when its slope is positive, meaning that as x increases, y also increases.
• A function is decreasing when its slope is negative, meaning that as x increases, y decreases.

Minimum and Maximum Points

• The minimum point on a graph represents the lowest value or local minimum of a function.
• The maximum point represents the highest value or local maximum of a function.

Continuous and Discrete Functions

• A continuous graph represents a function with smooth, uninterrupted curves or lines, enabling the function to assume any value within its domain.
• A discrete graph is a collection of data points often represented as a scatter plot.

Domain and Range

• The domain of a function is the set of all possible input values (x-values) for which the function is defined.
• The range of a function is the set of all possible output values (y-values) for which the function is defined.

`\hrulefill`

Graph 5

An absolute value function creates a V-shaped graph centered at its vertex (local maximum or minimum). It can open either upward or downward, depending on the sign of the leading coefficient.

The equation of the graphed function is y = 2|x - 3| - 2.

Graph Family: Absolute value function

Increasing: (3, ∞)

Decreasing: (-∞, 3)

Maximum or Minimum Point: Local minimum at (3, -2)

Continuous or Discrete: Continuous function

Domain: (-∞, ∞)

Range: [-2, ∞)

`\hrulefill`

Graph 6

A quadratic function is represented by a U-shaped graph known as a parabola, which can open either upward or downward, depending on the sign of the leading coefficient.

The equation of the graphed function is y = -x² + 2x + 5.

Graph Family: Quadratic function

Increasing: (-∞, 1)

Decreasing: (1, ∞)

Maximum or Minimum Point: Local maximum at (1, 6)

Continuous or Discrete: Continuous

Domain: (-∞, ∞)

Range: (-∞, 6]

`\hrulefill`

Graph 7

An exponential function is represented by a curve that either rapidly increases (exponential growth) or rapidly decreases (exponential decay) as the independent variable increases. An exponential curve does not have a maximum or minimum point within its domain. It either grows indefinitely or decreases indefinitely.

The equation of the graphed function is
`y = 2^x`.

In this case, the function grows indefinitely, and approaches zero as x approaches negative infinity.

Graph Family: Exponential function

Increasing: (-∞, ∞)

Decreasing: None

Maximum or Minimum Point: None

Continuous or Discrete: Continuous

Domain: (-∞, ∞)

Range: (0, ∞)

`\hrulefill`

Graph 8

The graph of a discrete function is a series of isolated data points or values plotted on a coordinate plane, representing specific, distinct input-output pairs.

The domain of a discrete function refers to the set of specific, distinct x-values or input values for which the function is defined.

The range of a discrete function refers to the set of specific, distinct y-values or output values for which the function is defined.

Graph Family: Linear

Increasing: None

Decreasing: Decreasing (negative slope)

Maximum or Minimum Point: Maximum (1, 8), Minimum (3, 0)

Continuous or Discrete: Discrete

Domain: {1, 2, 3}

Range: {0, 4, 8}