Question

name:
parent function transformations: homework
directions: identify the parent function (quadratic, cubic, absolute value, square root) and describe the transformation for each function below.
$w(x) = -(x + 4)^2 - 1$
parent function: __________
transformation(s): __________

$h(x) = (x - 3)^3 + 1$
parent function: __________
transformation(s): __________

$f(x) = -sqrt{x + 3} - 2$
parent function: __________
transformation(s): __________

$r(x) = |x - 4|$
parent function: __________
transformation(s): __________

$g(x) = 2x^2 + 11$
parent function: __________
transformation(s): __________

$w(x) = 0.5|x| - 7$
parent function: __________
transformation(s): __________

© auroras guide to math 2022

name:
parent function transformations: homework
directions: use what you know about transformations of parent functions to write an equation for each of the following graphs.

transformation(s): __________
$f(x) = $

transformation(s): __________
$g(x) = $

transformation(s): __________
$h(x) = $

© auroras guide to math

Explanation:

---

Top Section: Identify Parent Functions & Transformations

For each function, match to the core parent form, then apply transformation rules:

• Horizontal shifts: $f(x-h)$ = right $h$, $f(x+h)$ = left $h$
• Vertical shifts: $f(x)+k$ = up $k$, $f(x)-k$ = down $k$
• Reflection: $-f(x)$ = over x-axis, $f(-x)$ = over y-axis
• Dilation: $a \cdot f(x)$ = stretch ($|a|>1$) or shrink ($0<|a|<1$) vertically

1. First graph: Matches quadratic parent $p(x)=x^2$, reflected over x-axis, shifted right 2, shifted up 3.
2. Second graph: Matches absolute value parent $p(x)=|x|$, vertically stretched by 2, shifted down 3.
3. Third graph: Matches square root parent $p(x)=\sqrt{x}$, shifted right 2, shifted up 1.

Answer:

1. $w(x) = -(x + 4)^2 - 1$

• Parent Function: $p(x)=x^2$
• Transformation(s): Reflected over x-axis, shifted left 4, shifted down 1

1. $h(x) = (x - 3)^3 + 1$

• Parent Function: $p(x)=x^3$
• Transformation(s): Shifted right 3, shifted up 1

1. $f(x) = -\sqrt{x + 3} - 2$

• Parent Function: $p(x)=\sqrt{x}$
• Transformation(s): Reflected over x-axis, shifted left 3, shifted down 2

1. $r(x) = |x - 4|$

• Parent Function: $p(x)=|x|$
• Transformation(s): Shifted right 4

1. $g(x) = 2x^2 + 11$

• Parent Function: $p(x)=x^2$
• Transformation(s): Vertically stretched by 2, shifted up 11

1. $w(x) = 0.5 |x| - 7$

• Parent Function: $p(x)=|x|$
• Transformation(s): Vertically shrunk by 0.5, shifted down 7

---

Bottom Section: Write Equations from Graphs