Asked by chatijustfarted🤕
The graph of f(x) = - x ^ 2 is given below. Write the equation of a function g(x) which would shift the graph of f(x) left 3 units and down 1 unit.
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Answered by
GPT-4o mini
To shift the graph of \( f(x) = -x^2 \) left by 3 units and down by 1 unit, we follow these steps:
1. **Left Shift**: To shift a function \( f(x) \) left by \( h \) units, we replace \( x \) with \( x + h \). In this case, since we want to shift left by 3 units, we replace \( x \) with \( x + 3 \):
\[
f(x) = -x^2 \quad \text{becomes} \quad f(x + 3) = -(x + 3)^2.
\]
2. **Down Shift**: To shift a function down by \( k \) units, we subtract \( k \) from the function. Since we want to shift down by 1 unit, we subtract 1:
\[
f(x + 3) - 1 = - (x + 3)^2 - 1.
\]
Combining these transformations, the function \( g(x) \) that represents the graph shifted left by 3 units and down by 1 unit is:
\[
g(x) = - (x + 3)^2 - 1.
\]
Thus, the final equation for the function \( g(x) \) is:
\[
g(x) = - (x + 3)^2 - 1.
\]
1. **Left Shift**: To shift a function \( f(x) \) left by \( h \) units, we replace \( x \) with \( x + h \). In this case, since we want to shift left by 3 units, we replace \( x \) with \( x + 3 \):
\[
f(x) = -x^2 \quad \text{becomes} \quad f(x + 3) = -(x + 3)^2.
\]
2. **Down Shift**: To shift a function down by \( k \) units, we subtract \( k \) from the function. Since we want to shift down by 1 unit, we subtract 1:
\[
f(x + 3) - 1 = - (x + 3)^2 - 1.
\]
Combining these transformations, the function \( g(x) \) that represents the graph shifted left by 3 units and down by 1 unit is:
\[
g(x) = - (x + 3)^2 - 1.
\]
Thus, the final equation for the function \( g(x) \) is:
\[
g(x) = - (x + 3)^2 - 1.
\]
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