Question
Find \( g(x) \), where \( g(x) \) is the translation 4 units down of \( f(x)=x^{2} \). Write your answer in the form \( \mathrm{a}(\mathrm{x}-\mathrm{h})^{2}+\mathrm{k} \), where \( \mathrm{a}, \mathrm{h} \), and k are integers. \( g(x)= \)
Ask by Simpson Thornton. in the United States
Jan 25,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( g(x) = x^{2} - 4 \)
Solution
To translate the function \( f(x) = x^2 \) downward by 4 units, you subtract 4 from the original function.
The general form for the translation is:
\[
g(x) = a(x - h)^2 + k
\]
where:
- \( a \) is the vertical stretch/compression factor,
- \( h \) is the horizontal shift,
- \( k \) is the vertical shift.
Since we are only translating the graph downward by 4 units, \( a = 1 \), \( h = 0 \), and \( k = -4 \). Plugging these values into the equation gives:
\[
g(x) = 1(x - 0)^2 - 4 = x^2 - 4
\]
**Answer:**
\[
g(x) = x^{2} - 4
\]
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Simplify this solution Bonus Knowledge
To find \( g(x) \), which is the function \( f(x) = x^2 \) translated 4 units down, we need to subtract 4 from the function. Thus, we have: \[ g(x) = f(x) - 4 = x^2 - 4 \] Now, we write \( g(x) \) in the form \( a(x - h)^2 + k \). Here, \( a = 1 \), \( h = 0 \), and \( k = -4 \). So the final answer is: \[ g(x) = 1(x - 0)^2 - 4 \] Putting it all together, we can write: \[ g(x) = 1(x - 0)^2 - 4 \]
