d)) Find $$ g(x) $$, where $$ g(x) $$ is the translation 4 units left of $$ f(x)=x^{2} $$. $$ g(x)= $$ Write your answer in the form $$ \mathrm{a}(\mathrm{x}-\mathrm{h})^{2}+\mathrm{k} $$, where a , h , and k are integers. (D)

Question

UpStudy AI · Accepted Answer

To find $$ g(x) $$, which is the translation of $$ f(x) = x^2 $$ 4 units to the left, we need to apply the transformation for horizontal shifts.

1. **Understanding the translation**: 
   - Translating a function $$ f(x) $$ to the left by $$ h $$ units involves replacing $$ x $$ with $$ x + h $$. 
   - In this case, since we are translating 4 units to the left, we will replace $$ x $$ with $$ x + 4 $$.

2. **Applying the transformation**:
   - Start with the original function: 
     $$
     f(x) = x^2
     $$
   - Apply the translation:
     $$
     g(x) = f(x + 4) = (x + 4)^2
     $$

3. **Expanding the expression**:
   - Now, we need to expand $$ (x + 4)^2 $$:
     $$
     g(x) = (x + 4)(x + 4) = x^2 + 8x + 16
     $$

4. **Rewriting in the required form**:
   - We want to express $$ g(x) $$ in the form $$ a(x - h)^2 + k $$. 
   - To do this, we can complete the square:
     $$
     g(x) = x^2 + 8x + 16 = (x + 4)^2
     $$
   - Here, we can see that $$ a = 1 $$, $$ h = -4 $$, and $$ k = 0 $$.

Thus, the final answer is:
$$
g(x) = 1(x - (-4))^2 + 0
$$
or simply:
$$
g(x) = (x + 4)^2
$$

In the required form:
$$
g(x) = 1(x - (-4))^2 + 0
$$
where $$ a = 1 $$, $$ h = -4 $$, and $$ k = 0 $$.
$$ g(x) = (x + 4)^2 $$

d)) Find \( g(x) \), where \( g(x) \) is the translation 4 units left of \( f(x)=x^{2} \). \( g(x)= \) Write your answer in the form \( \mathrm{a}(\mathrm{x}-\mathrm{h})^{2}+\mathrm{k} \), where a , h , and k are integers. (D)

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Algebra Questions