Questions: Let the graph of g be a translation 2 units up and 2 units right, followed by a reflection in the y-axis of the graph of f(x)=-(x+3)^2-2. Write a rule for g.

Let the graph of g be a translation 2 units up and 2 units right, followed by a reflection in the y-axis of the graph of f(x)=-(x+3)^2-2. Write a rule for g.

Transcript text: 3. Let the graph of $g$ be a translation 2 units up and 2 units right, followed by a reflection in the $y$-axis of the graph of $f(x)=-(x+3)^{2}-2$. Write a rule for $g$.

Solution

Solution Steps

To find the rule for $ g $, we need to apply the given transformations to the function $ f(x) $. First, translate the function 2 units up and 2 units right. Then, reflect the resulting function in the $ y $-axis.

Step 1: Define the Original Function

The original function is given by \[ f(x) = -(x + 3)^2 - 2. \]

Step 2: Translate 2 Units Right

To translate the function 2 units to the right, we replace $ x $ with $ x - 2 $: \[ f(x - 2) = -((x - 2) + 3)^2 - 2 = -(x + 1)^2 - 2. \]

Step 3: Translate 2 Units Up

Next, we translate the function 2 units up by adding 2: \[ f(x - 2) + 2 = -(x + 1)^2 - 2 + 2 = -(x + 1)^2. \]

Step 4: Reflect in the $ y $-Axis

Finally, we reflect the function in the $ y $-axis by replacing $ x $ with $ -x $: \[ g(x) = -((-x) + 1)^2 = -(1 - x)^2. \]

Step 5: Simplify the Function

We can simplify the expression: \[ g(x) = -(1 - x)^2 = -(x - 1)^2. \]