Graphing Of Linear Functions, Rate Of Change, Growth And Decay

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Unit: Linear Functions

Graphing of Linear Functions, Rate of Change, Growth and Decay

Linear functions are a specific type of function that create straight lines when graphed. They are fundamental in algebra and widely used to model relationships between variables.

Definition and General Form

A linear function is a function that can be written in the form: f(x)=mx+b where:

• f(x) or y is the output or dependent variable.
• x is the input or independent variable.
• m is the slope of the line.
• b is the y-intercept, the point where the line crosses the y-axis.

Characteristics of Linear Functions

1. Constant Rate of Change:
   • The slope m represents the constant rate of change of the function.
   • For every unit increase in x, y increases by m.
2. Straight Line Graph:
   • The graph of a linear function is always a straight line.
3. Intercepts:
   • Y-Intercept: The value of y when x=0, given by b.
   • X-Intercept: The value of x when y=0, found by solving mx+b=0.
4. Domain and Range:
   • The domain of a linear function is all real numbers, (−∞,∞).
   • The range of a linear function is all real numbers, (−∞,∞).

Slope and Y-Intercept

1. Slope (m):
   • Describes the steepness and direction of the line.
   • Calculated as:

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1. • Positive slope: line rises from left to right.
   • Negative slope: line falls from left to right.
   • Zero slope: horizontal line.
   • Undefined slope: vertical line.
2. Y-Intercept (b):
   • The point where the line crosses the y-axis.
   • Found by setting x=0 in the equation f(x)=mx+b.

Example

For the linear function f(x)=3x+2:

• Slope (m): 3
• Y-Intercept (b): 2

Graphing Linear Functions

To graph a linear function, follow these steps:

1. Identify the Slope and Y-Intercept:
   • From the equation f(x)=mx+b.
2. Plot the Y-Intercept:
   • Locate the point (0,b) on the graph.
3. Use the Slope to Find Another Point:
   • From the y-intercept, use the slope 𝑚m (rise over run) to find another point.
   • Example: For m=3, from (0,2), move up 3 units and right 1 unit to (1,5).
4. Draw the Line:
   • Connect the points with a straight line extending in both directions.

Example

Graph the function f(x)=−2x+4:

• Slope m=−2
• Y-Intercept b=4

1. Plot the y-intercept (0,4).
2. Use the slope to find another point: From (0,4), move down 2 units and right 1 unit to (1,2).
3. Draw the line through (0,4) and (1,2).

Linear Function Applications

Linear functions are used in various real-life scenarios, including:

1. Business and Economics:
   • Cost Functions: Representing the total cost as a function of the number of units produced.
   • Revenue Functions: Representing total revenue as a function of the number of units sold.
2. Physics:
   • Motion: Representing the relationship between distance and time for objects moving at constant speed.
3. Everyday Situations:
   • Budgeting: Modelling expenses over time.
   • Conversion: Converting units, such as temperature or currency exchange rates.

Example

A taxi company charges a flat fee of $3 plus $2 per mile driven. The cost C of a trip that covers x miles can be modelled by the linear function: C(x)=2x+3

• Slope (m): $2 per mile.
• Y-Intercept (b): $3 (flat fee).

Summary

• Definition: Linear functions are written as f(x)=mx+b.
• Characteristics: Constant rate of change, straight-line graph, intercepts, and an infinite domain and range.
• Slope and Y-Intercept: The slope measures steepness and direction, while the y-intercept indicates where the line crosses the y-axis.
• Graphing: Identify slope and y-intercept, plot points, and draw the line.
• Applications: Used in business, physics, and daily life to model linear relationships.

Understanding linear functions is crucial for solving problems and modelling relationships in various disciplines.