Write your answer in a(x-h)^2 + k

Question

Find g(x) where g(x) is the translation 4 units left of f(x) =x^2. g(x) = ___
Find g(x) where g(x) is the translation 5 units up of f(x) = x^2. g(x)= __
Find g(x) where g(x) is the reflection across the x-axis of f(x) = x^2. g(x) = ___
Find g(x) where g(x) is the translation 2 units right and 1 unit down of f(x)=x^2. g(x) = ___

Answer 1

To find the function $g(x)$ after applying the specified translations and transformations to the function $f(x)=x^2$, we will use the vertex form of a quadratic function, which is given by $a(x-h{)}^2+k$, where $(h,k)$ is the vertex of the parabola.

1. Translation 4 units left of $f(x)=x^2$:

   • Translating 4 units left means we replace $x$ with $x+4$. $$g(x)=(x+4{)}^2=1(x+4{)}^2+0$$ So, $g(x)=(x+4{)}^2$.

2. Translation 5 units up of $f(x)=x^2$:

   • Translating 5 units up means we add 5 to the entire function. $$g(x)=x^2+5=1(x-0{)}^2+5$$ So, $g(x)=x^2+5$.

3. Reflection across the x-axis of $f(x)=x^2$:

   • Reflecting across the x-axis means we take the negative of the function. $$g(x)=-x^2=-1(x-0{)}^2+0$$ So, $g(x)=-x^2$.

4. Translation 2 units right and 1 unit down of $f(x)=x^2$:

   • Translating 2 units right means we replace $x$ with $x-2$ and translating 1 unit down means we subtract 1 from the entire function. $$g(x)=(x-2{)}^2-1=1(x-2{)}^2-1$$ So, $g(x)=(x-2{)}^2-1$.

To summarize:

1. $g(x)=(x+4{)}^2$
2. $g(x)=x^2+5$
3. $g(x)=-x^2$
4. $g(x)=(x-2{)}^2-1$