Lesson 1: Basics

Property

The point $(a, b)$ lies on the graph of the function $f$ if and only if $f(a) = b$. Each point on the graph of the function $f$ has coordinates $(x, f(x))$ for some value of $x$.

Examples

• If $f(5) = 8$, the point $(5, 8)$ is on the graph of the function $f$.
• If the point $(-1, 4)$ is on the graph of a function $g$, it means that $g(-1) = 4$.
• The coordinates of any point on the graph of $h$ can be written in the form $(x, h(x))$ for some input $x$.

Explanation

A function's graph is a picture of all its input-output pairs. The horizontal position (x-coordinate) is the input, and the vertical position (y-coordinate) is the corresponding output. It's a visual map of the function's behavior.

Property

We can construct a graph for a function described by an equation by plotting points whose coordinates satisfy the equation. We choose several convenient values for $x$ and evaluate the function to find the corresponding $f(x)$ values.

Examples

• To graph $f(x) = x^2 + 1$, we can choose $x=2$. We find $f(2) = 2^2 + 1 = 5$, so we plot the point $(2, 5)$.
• For the same function, $f(x) = x^2 + 1$, we find $f(0) = 0^2 + 1 = 1$. This gives us the y-intercept at $(0, 1)$.
• After calculating points like $(-2, 5)$, $(0, 1)$, and $(2, 5)$, connecting them reveals the U-shape of the parabola.

Explanation

To draw a function, create a table of values. Pick several inputs ($x$), calculate their outputs ($f(x)$) using the function's rule, and plot each $(x, f(x))$ coordinate pair. Then, connect the dots with a smooth curve.