SAT math linear functions: understand slope, intercepts, and graphs

A linear function changes at a steady rate

A linear function has a constant rate of change: whenever x increases by the same amount, y changes by the same amount. For example, in \(\displaystyle y = 2x + 1\), every increase of 1 in x makes y increase by 2. Plot the matching points and they line up on a straight line.

Here are several points that satisfy \(\displaystyle y = 2x + 1\): when x is -1, y is -1; when x is 0, y is 1; when x is 1, y is 3; and when x is 2, y is 5. The graph below plots those points and draws the line through them.

The study lab

Graph the function y = 2x + 1

The straight line passes through the plotted points; each step right raises y by 2.

y = 2x + 1

What to notice: The line crosses the y-axis at 1. Moving one unit right raises the line two units, matching its slope of 2.

Read slope and intercept from y = mx + b

The form \(\displaystyle y = mx + b\) makes two features easy to see. The coefficient \(\displaystyle m\) is the slope, or the change in y for each one-unit change in x. The constant \(\displaystyle b\) is the y-intercept: set x to 0, and the function gives \(\displaystyle y = b\).

For \(\displaystyle y = 2x + 1\), the slope is 2 and the y-intercept is 1, so the line crosses the vertical axis at \(\displaystyle (0,1)\). A negative slope goes down as you move right; a slope of 0 makes a horizontal line.

Positive slope: the output increases as the input increases.
Negative slope: the output decreases as the input increases.
Zero slope: the output stays constant.

Find slope using two points

If you know two points instead of an equation, slope is the change in y divided by the change in x: \(\displaystyle m = \frac{y_2-y_1}{x_2-x_1}\). For \(\displaystyle (1,3)\) and \(\displaystyle (4,9)\), the change in y is 6 and the change in x is 3, so the slope is \(\displaystyle 6/3 = 2\).

Keep the subtraction order consistent in the top and bottom. Using \(\displaystyle 9-3\) on top means use \(\displaystyle 4-1\) on the bottom. Mixing the order can accidentally reverse the sign.

The study lab

Slope is rise over run

From (1, 3) to (4, 9), move 3 units right and 6 units up.

y = 2x + 1

What to notice: Rise 6 divided by run 3 gives slope 2. The order of the two points does not change the result if both differences use the same order.

Connect equations, tables, and function notation

A function can be written with \(\displaystyle f(x)\) instead of y. The notation \(\displaystyle f(x)=2x+1\) means the output is 2 times the input, plus 1. To find \(\displaystyle f(3)\), substitute 3 for x: \(\displaystyle f(3)=2(3)+1=7\).

A table is another way to show the same rule. For \(\displaystyle y=2x+1\), x-values 0, 1, and 2 produce y-values 1, 3, and 5. Notice the output increases by 2 each time the input increases by 1; that repeated difference is a clue that the relationship is linear.

Equation: \(\displaystyle y=2x+1\).
Input-output rule: multiply x by 2, then add 1.
Table pairs: \(\displaystyle (0,1), (1,3), (2,5)\).

Translate a word problem into a line

Suppose a bike rental costs a 6-dollar start fee plus 4 dollars for each hour. If h is the number of hours and C is the total cost, then \(\displaystyle C=4h+6\). The slope 4 is the cost per hour, and the intercept 6 is the starting fee when h is 0.

The units help you keep the meaning straight: slope is dollars per hour, while the intercept is dollars. If the question asks for a 3-hour rental, substitute \(\displaystyle h=3\): \(\displaystyle C=4(3)+6=18\) dollars.

Common mistakes to catch

The y-intercept is not the slope: in \(\displaystyle y=2x+1\), 1 is where the line crosses the y-axis, while 2 is how steeply it rises. In a table, check the ratio of changes rather than looking only at the y-values.

A vertical line is not a function of x because one x-value would have many y-values. Also, a line with an undefined slope is not described by \(\displaystyle y=mx+b\) in the usual way; the slope formula would require dividing by a change in x of zero.

Slope uses change in y divided by change in x.
Do not reverse just one of the two subtractions.
Use the story’s units to distinguish a rate from a starting value.

Practice: try these before checking the answers

Show enough work to explain what each number means. When possible, check an equation by substituting a point or a value from the problem.

1. For \(\displaystyle y=-3x+4\), identify the slope and y-intercept.
2. If \(\displaystyle f(x)=2x+3\), find \(\displaystyle f(2)\).
3. Find the slope between \(\displaystyle (1,2)\) and \(\displaystyle (4,8)\).
4. A line has slope 2 and passes through \(\displaystyle (1,2)\). Write its equation in \(\displaystyle y=mx+b\) form.
5. Find the x-intercept of \(\displaystyle y=2x-6\).
6. A streaming service charges 8 dollars plus 4 dollars per rental. Write a rule for n rentals and find the cost of 5 rentals.

A quick linear-function check for SAT Math

Before you calculate, identify what x and y represent. Then find the rate of change and the starting value from the equation, table, graph, or story. Use those values to build or interpret the line.

Check the result in context: Does the slope have the right units? Does the intercept match the starting condition? Does the point you found actually satisfy the equation? These checks can catch sign and substitution errors quickly.

SAT math linear functions: understand slope, intercepts, and graphs

What you'll take away