To determine the translation that maps the graph of the function [tex]\( f(x) = x^2 \)[/tex] onto the function [tex]\( g(x) = x^2 + 2x + 6 \)[/tex], let's follow these steps:
1. Step 1: Recognize the basic form:
The function [tex]\( f(x) = x^2 \)[/tex] is a basic quadratic function whose graph is a parabola with the vertex at the origin [tex]\((0,0)\)[/tex].
2. Step 2: Analyze the given function:
Our goal is to understand how [tex]\( g(x) = x^2 + 2x + 6 \)[/tex] can be derived from [tex]\( f(x) \)[/tex] through transformations, particularly translations.
3. Step 3: Complete the square for [tex]\( g(x) \)[/tex]:
Let's complete the square for [tex]\( g(x) \)[/tex]:
[tex]\[
g(x) = x^2 + 2x + 6
\][/tex]
To complete the square, we need to rewrite the quadratic expression in the form [tex]\((x-h)^2 + k\)[/tex]:
[tex]\[
x^2 + 2x + 6 = (x^2 + 2x + 1) + 5 = (x+1)^2 + 5
\][/tex]
4. Step 4: Identify the transformations:
The equation [tex]\( g(x) = (x+1)^2 + 5 \)[/tex] shows the completed square form. In this form:
- [tex]\((x+1)^2\)[/tex] indicates a shift to the left by 1 unit because the term inside the parentheses is [tex]\((x + 1)\)[/tex].
- The [tex]\( + 5 \)[/tex] outside the square indicates a vertical shift upwards by 5 units.
5. Step 5: Summarize the transformation:
Therefore, the graph of [tex]\( g(x) = x^2 + 2x + 6 \)[/tex] is obtained by taking the graph of [tex]\( f(x) = x^2 \)[/tex] and:
- Shifting it left by 1 unit.
- Shifting it up by 5 units.
In conclusion, the translation that maps the graph of the function [tex]\( f(x) = x^2 \)[/tex] onto the function [tex]\( g(x) = x^2 + 2x + 6 \)[/tex] is a translation left by 1 unit and up by 5 units.
