Questions: Using Completing the Square The function f(x)=x^2+22x+58 is translated 4 units to the right and 16 units up. What is the vertex form of the new function? (x-11)^2+58 (x+22)^2-121 (x+7)^2-47 (x-15)^2+94

Using Completing the Square
The function f(x)=x^2+22x+58 is translated 4 units to the right and 16 units up. What is the vertex form of the new function?
(x-11)^2+58
(x+22)^2-121
(x+7)^2-47
(x-15)^2+94

Transcript text: Using Completing the Square The function $f(x)=x^{2}+22 x+58$ is translated 4 units to the right and 16 units up. What is the vertex form of the new function? $(x-11)^{2}+58$ $(x+22)^{2}-121$ $(x+7)^{2}-47$ $(x-15)^{2}+94$

Solution

Solution Steps

To solve this problem, we need to first convert the given quadratic function into its vertex form by completing the square. Then, we will apply the given translations to find the new vertex form of the function.

Start with the given function $ f(x) = x^2 + 22x + 58 $.
Complete the square to convert it into vertex form.
Translate the function 4 units to the right and 16 units up.
Write the new function in vertex form.

Step 1: Completing the Square for the Original Function

We start with the function $ f(x) = x^2 + 22x + 58 $.

To complete the square, we need to rewrite the quadratic expression in the form $ (x - h)^2 + k $.

First, we take the coefficient of $ x $, which is 22, divide it by 2, and then square it: \[ \left(\frac{22}{2}\right)^2 = 11^2 = 121 \]

Next, we add and subtract this square inside the function: \[ f(x) = x^2 + 22x + 121 - 121 + 58 \]

This can be rewritten as: \[ f(x) = (x + 11)^2 - 63 \]

So, the vertex form of the original function is: \[ f(x) = (x + 11)^2 - 63 \]

Step 2: Translating the Function

The problem states that the function is translated 4 units to the right and 16 units up.

To translate the function 4 units to the right, we replace $ x $ with $ x - 4 $: \[ f(x - 4) = ((x - 4) + 11)^2 - 63 \] \[ f(x - 4) = (x + 7)^2 - 63 \]

Next, we translate the function 16 units up by adding 16 to the function: \[ f(x - 4) + 16 = (x + 7)^2 - 63 + 16 \] \[ f(x - 4) + 16 = (x + 7)^2 - 47 \]