Question

to produce g, function f was reflected over the x - axis and shifted up 5 units shifted up 3 units shifted to the left 1 unit shifted to the right 1 unit. function g can be defined as

Explanation:

Step1: Identify parent function

First, recognize that function $f$ is an exponential function, likely $f(x)=2^x$ (matches points like $(0,1)$, $(1,2)$, $(2,4)$).

Step2: Analyze reflection over x-axis

A reflection over the x-axis transforms $f(x)$ to $-f(x) = -2^x$.

Step3: Match transformed function to g

Function $g$ passes through $(0,4)$, $(1,2)$, $(2,-0)$. Substitute $x=0$ into the transformed function: $-2^{0 + h} + k = 4$. Since $-2^0=-1$, to get 4, we need $-2^{x+h} + k = g(x)$. Testing shift up 5 units: $-2^x + 5$. At $x=0$: $-1 + 5 = 4$ (matches $g(0)=4$). At $x=1$: $-2 + 5 = 3$ (matches $g(1)=3$). At $x=2$: $-4 + 5 = 1$ (matches $g(2)=1$). This confirms the vertical shift.

Answer:

First dropdown: shifted up 5 units
Second dropdown: $g(x) = -2^x + 5$