Slope-Intercept Form (y = mx + b)

The slope-intercept form is expressed by the formula: $y = mx + b$

On the Digital SAT, this formula is not provided on the reference sheet. You must memorize it, but more importantly, you must understand what each component represents in different contexts.

1. The Slope ($m$)

The slope represents the steepness and direction of the line. Mathematically, it is the "rise over run." If you are given two points, $(x_1, y_1)$ and $(x_2, y_2)$, the formula for slope is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

When it appears: The SAT uses slope to describe any "unit rate." How to recognize it: Look for keywords like "per," "each," "every," "increase of," or "rate." If a taxi charges \2.50$per mile,$m = 2.5$. If a water tank drains at$5$gallons per minute,$m = -5$.

2. The Y-Intercept ($b$)

The y-intercept is the point where the line crosses the y-axis. This occurs when $x = 0$. In coordinate geometry, this point is always written as $(0, b)$.

When it appears: The SAT uses the y-intercept to describe the "initial value." How to recognize it: Look for keywords like "flat fee," "starting amount," "initial height," "deposit," or "at the beginning." If a plumber charges a \50$service fee just to show up,$b = 50$.

3. Parallel and Perpendicular Lines

The SAT frequently asks about the relationship between two lines.

• Parallel Lines: Have the same slope. If Line A has $m = 3$, any line parallel to it also has $m = 3$.
• Perpendicular Lines: Have negative reciprocal slopes. If Line A has $m = \frac{2}{3}$, a line perpendicular to it will have $m = -\frac{3}{2}$.

4. Standard Form to Slope-Intercept Form

Often, the SAT will give you an equation in Standard Form: $Ax + By = C$ To find the slope or y-intercept quickly, you should isolate $y$: $By = -Ax + C$ $y = -\frac{A}{B}x + \frac{C}{B}$ From this, we can see that the slope $m = -\frac{A}{B}$ and the y-intercept $b = \frac{C}{B}$.

5. Horizontal and Vertical Lines

These are special cases that often trip students up:

• Horizontal Lines: The equation is $y = k$. The slope is $0$.
• Vertical Lines: The equation is $x = k$. The slope is undefined. (Note: Vertical lines are not functions, but they do appear on the SAT).

6. Interpreting the Context

This is the most common "Foundational" to "Medium" skill. You will be given an equation like $H = 1.5t + 20$, where $H$ is the height of a tree in inches and $t$ is the number of months.

• The $1.5$ means the tree grows $1.5$ inches per month.
• The $20$ means the tree was $20$ inches tall at the start ($t=0$).

1. The Desmos Advantage

The Digital SAT includes an integrated Desmos graphing calculator. For any question involving $y = mx + b$, your first instinct should be to consider graphing it. If you are given a point and a slope, or two points, you can type them into Desmos to instantly see the line and its intercepts.

2. "The Slope is the Rate"

Whenever you see a word problem, immediately ask yourself: "What is changing?" That change is your $m$. Then ask: "What is the fixed cost or starting point?" That is your $b$. This simple mental framework allows you to build equations from word problems in seconds.

3. Use the "Plug-In" Method

If you are asked which equation represents a line that passes through a specific point, like $(4, 10)$, don't always do the algebra. Simply plug $x = 4$ into the answer choices. The correct answer must result in $y = 10$.

4. Watch the Units

The SAT loves to give you a rate in "miles per hour" but ask for a result in "minutes." Always check if the units in your $m$ match the units requested in the question.

5. Eliminate by Sign

If a graph is going "downhill" from left to right, the slope must be negative. If it’s going "uphill," it must be positive. You can often eliminate two out of four answer choices just by looking at the sign of $m$.