Slope-intercept Form of a Linear Function - Algebra-2

1. Fundamental Concepts

Definition: The slope-intercept form of a linear function is an equation that explicitly shows the slope and y-intercept of a line, expressed as:\(y = mx + b\) Here:
- x and y are variables representing coordinates of points on the line,
- m is the slope (rate of change) of the line,
- b is the y-coordinate of the y-intercept (the point \((0, b)\) where the line crosses the y-axis).

Components and Meanings:
- Slope (m): Determines the steepness and direction of the line. A positive m means the line rises from left to right; a negative m means it falls from left to right; \(m = 0\) indicates a horizontal line.
- Y-intercept (b): Indicates where the line crosses the y-axis. It is the starting point when graphing the line using this form.
Relationship to Linear Functions: Any linear function (a function whose graph is a straight line) can be rewritten in slope-intercept form, making it easy to analyze and graph.

Simple: For the equation \(y = -2x + 5\), identify the slope and y-intercept, then describe the graph.
- Slope (m): \(-2\) (negative, so the line falls from left to right).
- Y-intercept (b): 5 (the line crosses the y-axis at \((0, 5)\)).
Medium: Rewrite \(3x - 6y = 12\) in slope-intercept form, then find the slope and y-intercept.
- Step 1: Isolate y: Subtract 3x from both sides: \(-6y = -3x + 12\).
- Step 2: Divide by \(-6\): \(y = \frac{-3}{-6}x + \frac{12}{-6}\), simplifying to \(y = \frac{1}{2}x - 2\).
- Slope (m): \(\frac{1}{2}\) (positive, so the line rises left to right).
- Y-intercept (b): \(-2\) (crosses the y-axis at \((0, -2)\)).
Hard: A line passes through \((3, 7)\) and \((5, 11)\). Write its equation in slope-intercept form .
- Step 1: Calculate the slope m:\(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{11 - 7}{5 - 3} = \frac{4}{2} = 2\)
- Step 2: Substitute \(m = 2\) and one point (e.g., \((3, 7)\)) into \(y = mx + b\) to find b:\(7 = 2(3) + b \implies 7 = 6 + b \implies b = 1\)
- Step 3: Write the equation: \(y = 2x + 1\).

Extracting m and b from the Form: If the equation is already \(y = mx + b\), directly read m (coefficient of x) and b (constant term).
Rewriting Equations in Slope-Intercept Form:
1. Start with a linear equation (e.g., standard form \(Ax + By = C\)).
2. Isolate y by:
  - Subtracting Ax from both sides to get \(By = -Ax + C\).
  - Dividing every term by B to get \(y = \left(-\frac{A}{B}\right)x + \frac{C}{B}\).
Writing Equations from Two Points:
1. Calculate the slope m using \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
2. Substitute m and one of the points into \(y = mx + b\), solve for b.
3. Substitute m and b back into \(y = mx + b\) to get the equation.