Analyzing Linear Functions from Graphs

y-intercept: The point where the graph crosses the vertical y-axis, occurring where $x = 0$ .
x-intercept: The point where the graph crosses the horizontal x-axis, occurring where $y = 0$ .
Slope ( $m$ ): The rate of change of a line, defined as the ratio of vertical change (rise) to horizontal change (run) between any two points $(x_1, y_1)$ and $(x_2, y_2)$ :
$m = \frac{y_2 - y_1}{x_2 - x_1}$
Slope-Intercept Form: A standard representation of a linear equation given by:
$y = mx + b$
where $m$ is the slope and $b$ is the value of the y-intercept.
Factored Form: A linear equation written in terms of its slope and its root (x-intercept):
$y = a(x - r)$
where $a$ is the slope ( $m$ ) and $r$ is the x-intercept value.

Linear function graph

y-intercept Determination:
- Observing the graph along the vertical y-axis, the line crosses at $y = 1$ .
- As an ordered pair coordinate, it is expressed as $(0, 1)$ .
x-intercept Determination:
- Observing the graph along the horizontal x-axis, the line crosses at $x = 3$ .
- As an ordered pair coordinate, it is expressed as $(3, 0)$ .
Slope Calculation:
- Selecting two coordinate points on the line: $(0, 1)$ and $(3, 0)$ .
- Applying the slope formula: $m = \frac{0 - 1}{3 - 0}$ $m = \frac{-1}{3}$ $m = -\frac{1}{3}$
- Verification using points $(-9, 4)$ and $(3, 0)$ : $m = \frac{0 - 4}{3 - (-9)}$ $m = \frac{-4}{12}$ $m = -\frac{1}{3}$
Equation in Slope-Intercept Form:
- Substituting the slope $m = -\frac{1}{3}$ and y-intercept value $b = 1$ into $y = mx + b$ : $y = -\frac{1}{3}x + 1$
Equation in Factored Form:
- Substituting the slope $a = -\frac{1}{3}$ and x-intercept root $r = 3$ into $y = a(x - r)$ :
  $y = -\frac{1}{3}(x - 3)$
- Expanding the expression $-\frac{1}{3}(x - 3)$ gives $-\frac{1}{3}x + 1$ , which confirms consistency with the slope-intercept form.