Algebra - Transformations of Linear Function Graphs

Given the function $f(x) = 3x + 2$, find the new function $g(x)$ after a vertical translation 4 units down.

$g(x) = (3x + 2) - 4 = 3x - 2$

To translate a function vertically down by $k$ units, we subtract $k$ from the entire function. Here, $k = 4$, so $g(x) = f(x) - 4$.

Let $f(x) = x$. Describe the transformations required to obtain $g(x) = 2(x - 5) + 3$.

$1. \text{ Vertical stretch by a factor of } 2$ $2. \text{ Horizontal translation } 5 \text{ units to the right}$ $3. \text{ Vertical translation } 3 \text{ units up}$

Comparing $g(x) = a(f(x - h)) + k$ to the parent function $f(x) = x$: $a=2$ (stretch), $h=5$ (right shift), and $k=3$ (upward shift).

Reflect the function $f(x) = 4x - 7$ across the x-axis.

$g(x) = -(4x - 7) = -4x + 7$

A reflection across the x-axis is achieved by multiplying the entire function by $-1$. This changes the signs of both the slope and the y-intercept.

$g(x) = f(x) - 4$ This is a vertical translation $4$ units downward. The $y$-intercept changes from $(0,0)$ to $(0,-4)$.

To transform $f(x)$ into $g(x)$, every point $(x, y)$ on the parent graph is shifted to $(x, y-4)$.

1. Start with the parent function: $f(x) = x$.
2. Apply the vertical stretch by factor $a = 3$: $h(x) = 3 \cdot f(x) = 3x$.
3. Apply the vertical translation up by $k = 2$: $g(x) = h(x) + 2 = 3x + 2$.
4. The final equation is $g(x) = 3x + 2$.

A vertical stretch by a factor of $3$ multiplies the output values of the function by $3$, making the slope steeper ($m=3$). A vertical translation of $2$ units up adds $2$ to the entire function, shifting the $y$-intercept from $(0,0)$ to $(0,2)$.