Linear Functions

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Type of Question:

Linear functions usually written in slope-intercept form $y = mx + b$ or standard form $Ax + By = C$ . They can involve identifying slope, intercepts, writing equations, or interpreting graphs.

How to Approach a Linear Functions Questions:

1. Identifying the form of the equation given (or needed):

Slope-intercept form :
- $m$ = slope (rise over run, or rate of change).
- $b$ = y-intercept (value of $y$ when $x = 0$ ).
Point-slope form $y - y1 = m(x-x1)$ when slope and a point are known.
Standard form $Ax + By = C$ : rearrange to slope-intercept form if needed.

2. Find the slope:

If given two points $(x1, x2)$ and $(x2, y2)$ , use: $m =\dfrac{y2-y1}{x2-x1}$
From a graph, count the rise/run between two clear points.

3. Find the intercept:

If you have slope $m$ and a point $(x, y)$ , plug them into $y = mx + b$ to solve for $b$ .

4. Write or interpret the equation:

Plug $m$ and $b$ into $y = mx + b$ .
Interpret $m$ as the rate of change and $b$ as the starting value.

5. Check units and meaning:

In word problems, relate $m$ and $b$ to the situation described.

Example (from the guide):

If the line passes through (2, 5) and (6, 9):

Slope: $m$ = $\dfrac{9-5}{6-2}=\dfrac{4}{4}=1$
Use point (2, 5) in $y = mx + b$ :
5 = 1(2) + $b$ => $b$ = 3
Equation: $y = x + 3$