SAT Math 2: Linear Equations in Two Variables and Linear Functions

Affine (calculus-style) linear function

A function whose graph is a straight line and whose polynomial degree is at most one. For one variable, it has the form $f(x)=ax+b$, including constant functions and the zero polynomial.

What do the parameters $a$ and $b$ represent in $f(x)=ax+b$?

$a$ is the slope, describing the change in $f(x)$ per unit increase in $x$, and $b$ is the $y$-intercept, equal to $f(0)$. The graph crosses the $y$-axis at $(0,b)$.

How does the sign of the slope affect the graph of $f(x)=ax+b$?

If $a>0$, the graph rises from left to right; if $a<0$, it falls from left to right; and if $a=0$, the function is constant and its graph is horizontal.

How can the slope of a nonvertical line be calculated from two points?

For points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_1\ne x_2$, the slope is $a=\dfrac{y_2-y_1}{x_2-x_1}$. It represents the ratio of change in output to change in input.

Constant function in the context of polynomial linear functions

A function of the form $f(x)=b$, equivalent to $f(x)=0x+b$. It is considered linear in the polynomial-function convention because its degree is zero, and its graph is horizontal.

How can a linear relationship be identified from data or a graph?

The rate of change must be constant: equal changes in the independent variable produce equal changes in the dependent variable. Graphically, the data lie on a straight line and the slope remains the same between any two points.

How does adding a constant affect a linear function?

Replacing $f(x)=ax$ with $f(x)=ax+b$ translates the graph vertically by $b$. The constant $b$ changes the $y$-intercept while leaving the slope unchanged.