Question: Use the graph of each translated parent function to identify $k$ and/or $h$ and write its equation.
7.
g(x)
f(x)
8.
g(x)
f(x)
9.
g(x)
f(x)
10.
f(x)
g(x)
11.
g(x)
f(x)
12.
g(x)
f(x)
7: top-left 6 blue parabola opening upward with vertex at $(0,0)$; maroon parabola opening upward with vertex at $(0,-2)$.
8: top-right 6 blue V-shape opening upward with vertex at $(0,0)$; maroon V-shape opening upward with vertex at $(-3,0)$.
9: center-left 6 blue line with slope $1$ crossing the origin; maroon line parallel to it shifted up $1$ unit.
10: center-right 6 blue V-shape opening upward with vertex at $(0,0)$; maroon V-shape opening upward with vertex at $(1,-2)$.
11: bottom-left 6 blue V-shape opening upward with vertex at $(0,0)$; maroon V-shape opening upward with vertex at $(-3,1)$.
12: bottom-right 6 blue parabola opening upward with vertex at $(0,0)$; maroon parabola opening upward with vertex at $(-2,-5)$.
1. **Problem Statement:** Identify the translation parameters $h$ and $k$ for each pair of parent function $f(x)$ and translated function $g(x)$, then write the equation of $g(x)$.
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2. **General Formula for Translations:**
For a parent function $f(x)$, a translation by $h$ units horizontally and $k$ units vertically is given by:
$$g(x) = f(x - h) + k$$
- $h$ shifts the graph horizontally: right if $h > 0$, left if $h < 0$.
- $k$ shifts the graph vertically: up if $k > 0$, down if $k < 0$.
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### 7. Parabola
- Parent function: $f(x) = x^2$ with vertex at $(0,0)$.
- Translated function $g(x)$ has vertex at $(0,-2)$.
Since vertex moved down by 2 units, $h=0$, $k=-2$.
Equation:
$$g(x) = (x - 0)^2 + (-2) = x^2 - 2$$
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### 8. V-shape (Absolute Value)
- Parent function: $f(x) = |x|$ with vertex at $(0,0)$.
- Translated function $g(x)$ has vertex at $(-3,0)$.
Since vertex moved left by 3 units, $h = -3$, $k=0$.
Equation:
$$g(x) = |x - (-3)| + 0 = |x + 3|$$
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### 9. Line
- Parent function: $f(x) = x$ with slope $1$ and intercept $0$.
- Translated function $g(x)$ is parallel and shifted up by 1 unit.
Horizontal shift $h=0$, vertical shift $k=1$.
Equation:
$$g(x) = (x - 0) + 1 = x + 1$$
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### 10. V-shape (Absolute Value)
- Parent function: $f(x) = |x|$ with vertex at $(0,0)$.
- Translated function $g(x)$ has vertex at $(1,-2)$.
Horizontal shift $h=1$, vertical shift $k=-2$.
Equation:
$$g(x) = |x - 1| - 2$$
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### 11. V-shape (Absolute Value)
- Parent function: $f(x) = |x|$ with vertex at $(0,0)$.
- Translated function $g(x)$ has vertex at $(-3,1)$.
Horizontal shift $h=-3$, vertical shift $k=1$.
Equation:
$$g(x) = |x - (-3)| + 1 = |x + 3| + 1$$
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### 12. Parabola
- Parent function: $f(x) = x^2$ with vertex at $(0,0)$.
- Translated function $g(x)$ has vertex at $(-2,-5)$.
Horizontal shift $h=-2$, vertical shift $k=-5$.
Equation:
$$g(x) = (x - (-2))^2 - 5 = (x + 2)^2 - 5$$