Graph of the Linear Function $y = x$

$y = x$ is the simplest linear function: slope $1$, intercept $0$¹. It is the general linear function $y = ax + b$ with $a = 1$ and $b = 0$, and because it returns whatever value it is given, it is also called the identity function². It is likewise the power function $y = x^n$ with $n = 1$.

Domain and range

• The domain is all real numbers
• The range is all real numbers
• Every real value is attained exactly once

Slope and intercepts

The slope is $1$: each time $x$ increases by $1$, $y$ increases by $1$ as well, so the rate of change $\dfrac{\Delta y}{\Delta x}$ equals $1$ everywhere on the graph. The line makes a $45^\circ$ angle with the $x$-axis, and both its $y$-intercept and its $x$-intercept are the origin.

Symmetry

Since $f(-x) = -x = -f(x)$, it is an odd function with point symmetry about the origin, running straight through the first and third quadrants.

Monotonicity

There is no maximum, no minimum and no inflection point. The rate of growth never changes, which makes this the simplest possible form of increase.

Notable points

The graph is precisely the condition that $x$ and $y$ have the same value.

Relationships with other functions

The line $y = x$ acts as a mirror between a function and its inverse: the graphs of $y = f(x)$ and $y = f^{-1}(x)$ are reflections of each other across it.

The general linear function $y = ax + b$ can be read as this same line with its slope changed to $a$ and shifted vertically by $b$.

Applications

$y = x$ represents the operation that changes nothing. In a scatter plot of predicted against observed values, the points lying on this line are exactly the ones the prediction got right, so it is drawn as the reference line for judging goodness of fit. In models of proportionality it is the starting point: the plainest case, with constant of proportionality $1$.

1. Linear function, Wikipedia
2. Identity function, Wikipedia