Functions and Graphs Chapter Notes | Mathematics for Grade 10 PDF Download

Function Notation

Definition of a Function:A function is a relationship where each input (denoted by x) has exactly one output (denoted by f(x)).

Components:

• Name of the function: Often denoted by f, g, etc.
• Variable: x is the independent variable, and f(x) is the output.
• Rule: A mathematical expression describing the relationship between x and f(x).

Example: If f(x) = 3x + 2, the rule is 3x + 2, x is the variable, and f is the name of the function.

Revision of Linear Functions

Linear Functions: These are functions of the form f(x) = mx + c, where m is the slope and c is the y-intercept.

Domain and Range:

• Domain: x ∈ ℝ
• Range: y ∈ ℝ

Graphing: Plot points corresponding to various values of x, and draw a straight line.

Quadratic Function: y = ax²

Function: Quadratic functions have the form y = ax², where a is a constant.

Graph Features:

• Shape: The graph of a quadratic function is a parabola. If a > 0, the parabola opens upwards. If a < 0, the parabola opens downwards.
• Turning Point: The vertex of the parabola, where the function changes direction.

Intercepts:

• x-intercepts: Points where the graph intersects the x-axis (y = 0).
• y-intercept: The point where the graph intersects the y-axis (x = 0).

Example: For f(x) = x², the points are:

• (-4, 16), (-3, 9), (0, 0), (3, 9), (4, 16).
• Turning Point: The vertex is at (0, 0).
• Domain: x ∈ ℝ
• Range: y ≥ 0 (if a > 0).

Quadratic Function: y = ax² + q

Function: The form y = ax² + q represents a shifted parabola.

Shift: The value q shifts the graph vertically.

• If q > 0, the graph shifts upward.
• If q < 0, the graph shifts downward.

Graphing: The vertex will be at (0, q).

Domain and Range:

• Domain: x ∈ ℝ
• Range: For a > 0, y ≥ q, and for a < 0, y ≤ q.

Exponential Graphs: y = abˣ

Exponential Function: The function y = abˣ, where a and b are constants and b > 0, is an exponential function.

Features:

• Asymptote: The graph has a horizontal asymptote at y = 0 (when b > 1).
• Domain: x ∈ ℝ
• Range: For b > 1, y > 0.

Ex​​​​ample: For y = 2ˣ, as x increases, y increases rapidly, and as x decreases, y approaches 0.

The Hyperbola

Hyperbolic Function: The equation y = a/x represents a hyperbola.

• Asymptotes: The graph has vertical and horizontal asymptotes at x = 0 and y = 0, respectively.
• Symmetry: The hyperbola is symmetric with respect to the line y = -x.

Domain and Range:

• Domain: x ≠ 0
• Range: y ≠ 0

Graphing: The graph is divided into two branches with a discontinuity at x = 0.

Reading from Graphs

Interpreting Graphs: To extract information from graphs, identify:

• Intercepts: Points where the graph intersects the axes.
• Domain: The set of all possible x-values.
• Range: The set of all possible y-values.

Examples: Given a graph of a quadratic function, identify the x- and y-intercepts and the turning point.

Points to Remember

• Function: A rule that associates every input value with exactly one output.
• Domain: All the possible values of x for which the function is defined.
• Range: All possible output values f(x) for the function.
• Intercepts: Points where the graph intersects the axes.
• Asymptotes: Lines that the graph approaches but never intersects.
• Quadratic Function: A function of the form f(x) = ax² + q, where the graph is a parabola.
• Exponential Function: A function of the form y = abˣ, where b > 1 and the graph increases rapidly.
• Hyperbola: A curve defined by y = a/x, with vertical and horizontal asymptotes.

Difficult Words

• Function Notation: A way to write a function, e.g., f(x) = 3x + 2.
• Asymptote: A line that the graph approaches but never intersects.
• Domain: The set of all possible input values for a function.
• Range: The set of all possible output values for a function.
• Quadratic: A function of the form f(x) = ax² + q, which graphs as a parabola.
• Exponential: A function of the form y = abˣ, where the rate of change is proportional to the current value.

Summary

In this chapter, we covered various types of functions, such as linear, quadratic, exponential, and hyperbolic functions. Each type of function has unique characteristics that are illustrated by its graph. The domain and range of a function define the input and output values, respectively. Understanding how to read and interpret graphs is crucial for analyzing these functions. Functions are essential tools in mathematics that model real-world relationships, from growth patterns to geometric shapes.