Stretching and reflecting functions

Shifting moves a graph without changing its shape.
Stretching and reflecting are different: they change how tall the graph is or which way it faces.
There are two things to look for — a number multiplying the function, and a minus sign.

Multiplying the whole function by a number, $g(x) = a \cdot f(x)$, is a vertical stretch.
Every $y$-value is multiplied by $a$, so the points move away from or toward the $x$-axis.
If $a > 1$, the graph is stretched taller, which makes a parabola look narrower.
If $0 < a < 1$, the graph is shrunk flatter, which makes a parabola look wider.

Putting a minus sign in front, $g(x) = -f(x)$, reflects the graph over the $x$-axis.
It flips upside down: every $y$-value changes sign, so a low point becomes a high point.
For a line like $y = 2x + 6$, reflecting over the $x$-axis gives $y = -2x - 6$ — both the slope and the intercept change sign.

Putting the minus sign inside, on the $x$, gives $g(x) = f(-x)$, which reflects the graph over the $y$-axis.
It flips left-to-right, like a mirror image.
A point that was at $x = 2$ moves to $x = -2$.

These can combine.
For example, $g(x) = -f(x + 1)$ means flip $f$ upside down (the minus sign) and shift it left by $1$ (the $+1$ inside).
Read the transformations one piece at a time: handle the stretch or flip, then the shift.

How does the graph of $g(x) = 3f(x)$ compare with $f(x)$?
Multiplying by $3$ is a vertical stretch, since every $y$-value becomes $3$ times as large.
So $g(x)$ is $f(x)$ stretched taller (three times as far from the $x$-axis).

The line $y = 3x - 4$ is reflected over the $x$-axis. What is the new equation?
Reflecting over the $x$-axis replaces $y$ with $-y$, so $-y = 3x - 4$.
Solving for $y$ gives $y = -3x + 4$: both the slope and the intercept change sign.

A function $f(x)$ crosses the $x$-axis at $x = 2$ and $x = -3$. Where does $g(x) = f(-x)$ cross the $x$-axis?
$g(x) = f(-x)$ reflects the graph over the $y$-axis, so each $x$-intercept flips to the opposite sign.
So $g(x)$ crosses at $x = -2$ and $x = 3$.