Graphing and Interpreting Linear Models

Every linear equation represents a straight line on the coordinate plane. Understanding how the numbers in an equation affect its graph is key to solving many SAT questions.

Slope–Intercept Form and Graphing For a line written as

y = mx + b

$m$ is the slope, showing how steep the line is.
$b$ is the $y$-intercept, showing where the line crosses the $y$ -axis.

To graph: 1. Plot the

y

-intercept

(0, b)

. 2. Use the slope to find another point: rise = change in

y

, run = change in

x

. 3. Connect the points with a straight line.

Example: For

y = 2x + 1

: -

m = 2

: rise 2, run 1 -

b = 1

: start at

(0, 1)

Standard Form and Intercepts A line in standard form

Ax + By = C

can be graphed quickly using intercepts:

\text{\(x\)-intercept: set } y = 0, \quad \text{\(y\)-intercept: set } x = 0.

Example: For

2x + 3y = 12

x\text{-int: } (6, 0), \quad y\text{-int: } (0, 4)

Interpreting Linear Models Linear equations often model real-world relationships, such as cost, distance, or temperature over time.

The slope represents the rate of change (how much one quantity changes per unit of another).
The $y$-intercept represents the starting value when $x = 0$ .

Example: A taxi fare is given by

C = 2.5x + 4

, where

C

is cost and

x

is miles.

\text{slope } 2.5 \Rightarrow \text{cost per mile}, \quad \text{intercept } 4 \Rightarrow \text{base fee}.

Intersections of Lines The intersection point of two lines represents the solution to both equations. It can be found by solving the system of equations simultaneously.

Example:

\begin{cases} y = 2x + 3 \\ y = -x + 9 \end{cases}

Set equal:

2x + 3 = -x + 9 \Rightarrow 3x = 6 \Rightarrow x = 2, \quad y = 7

\boxed{(2, 7)}

Interpretation: both models have the same value at

x = 2

Key Takeaways

In $y = mx + b$ , $m$ controls steepness and $b$ controls vertical position.
The intersection of two lines represents the solution to a system.
On the SAT, slope often represents a rate (like cost per mile or speed).
Understand how changes in $m$ and $b$ shift the graph:
- Increasing $m$ : steeper line
- Increasing $b$ : line shifts upward