Basic Linear Functions - Math Antics

Introduction to Linear Functions

• Linear functions are fundamental in algebra and widely used.

• Familiarity with graphing and functions is recommended before continuing.

Basic Linear Function: y = x

• The equation y = x indicates that the output (y) is equal to the input (x).

• Graphing y = x results in a diagonal line through the origin, dividing quadrants 1 and 3.

• This basic linear function has the same values for x and y.

• Variables:

  • y: Output of the function

  • x: Input of the function

Versatile Linear Function: y = mx

• The equation y = mx introduces a slope (m) affecting the line's steepness.

• Choosing Values for m:

  • m = 1 gives y = 1x, same as y = x.

  • m = 2 results in y = 2x: outputs are doubled.

  • Each line passes through (0,0) because multiplying by 0 gives 0.

  • Increasing m increases the slope of the line, making it steeper, akin to climbing a hill.

Understanding Slope

• The slope measures how steep the line is:

  • m = 3 triples output, steeper than y = 2x.

  • m = 10 or m = 100 results in very steep lines, approaching vertical without ever reaching it.

• A perfectly vertical line is not a function, as it fails the vertical line test.

Decreasing Slope: Values of m Less Than 1

• For less steep lines, values of m can be fractions:

  • m = 1/2 (0.5): graph shows a less steep line; passes through (0,0).

  • m = 1/4 (0.25) and smaller fractions further flatten the line towards the x-axis.

  • As m decreases to 0, the slope approaches a perfectly horizontal line.

• The equation y = 0 represents a horizontal line with a slope of zero, akin to flat ground.

Adding the y-intercept: y = mx + b

• To shift lines away from (0,0), include a new variable b to create the equation y = mx + b. -Effect of b:

  • Increasing b shifts the line upward (e.g., b = 1, b = 2).

  • Decreasing b shifts the line downward (e.g., b = -1, b = -2).

• The y-intercept is determined by b; when x = 0, then y = b.

Importance of y = mx + b

• Allows representation of any linear function on the coordinate plane.

• m indicates slope and b indicates y-intercept.

• y = mx + b is pivotal for understanding linear graphs.

Vertical vs Horizontal Shifts

• While a negative b shifts down, shifting side to side is not possible with the linear equation form.

• Lines are parallel, meaning moving up and down equates to left and right movement on the graph, maintaining the same slope.

Identifying Linear Functions

• True linear functions can only contain first-order variables (no powers greater than 1).

• Examples of linear equations (e.g., y = mx + b) and non-linear equations (e.g., x^2, y^3) were discussed.

Rearranging Linear Equations into y = mx + b

• Example: Rearranging x - 4 = 2(y - 3):

  • Step 1: Divide by 2 => x/2 - 2 = y - 3.

  • Step 2: Add 3 to both sides => y = (1/2)x + 1.

  • Conclusion: The slope is 1/2 and the y-intercept is 1.

Conclusion

• Understanding linear functions enables graphing any linear function using y = mx + b.

• Practicing problem-solving is essential to mastering linear functions.

• Encouragement to continue learning through exercises and further exploration.