Math Snap

STEP 1

What is this asking?
We're given two graphs, one with $f(x)$ and $g(x)$ as lines, and another with $f(x)$ as a V and $g(x)$ also as a V, and we need to figure out what transformations happened to $f(x)$ to get to $g(x)$ in each graph.
Watch out!
Remember that changes inside the parentheses of $f(x)$ affect the x-values, and changes outside affect the y-values.
Also, changes inside the parentheses do the opposite of what you might expect!

STEP 2

1. Analyze the first graph (lines).
2. Analyze the second graph (V's).

STEP 3

$f(x)$ passes through $(0, 0)$ and $(1, 5)$.

STEP 4

$g(x)$ passes through $(-6, 0)$ and $(-5, 5)$.

STEP 5

The x-value of the corresponding points are 6 units apart, so $g(x)$ is a horizontal shift of $f(x)$ by $-6$.
This means $g(x) = f(x+6)$.
Remember, changes inside the parentheses do the opposite of what you might expect!

STEP 6

The vertex of $f(x)$ is at $(0,0)$.

STEP 7

The vertex of $g(x)$ is at $(0,-5)$.

STEP 8

The vertex of $g(x)$ is 5 units down from the vertex of $f(x)$, so $g(x)$ is a vertical shift of $f(x)$ by $-5$.
This means $g(x) = f(x) - 5$.

STEP 9

$f(1) = 5$, and $g(1) = 0$.
Since $0 = 5 - 5$, this confirms that $g(x) = f(x) - 5$.

For the first graph (lines), $g(x) = f(x+6)$.
For the second graph (V's), $g(x) = f(x) - 5$.