Question

1.3 homework - transformations
for each - a) identify the parent function; b) write an equation for the function shown.

1. parent:

function:

1. parent:

function:

1. parent:

function:

1. parent:

function:
describe the transformations necessary to take f(x) to g(x).

1. $f(x) = |x|$

$g(x) = -|x| - 1$

1. $f(x) = \sqrt{x}$

$g(x) = \sqrt{x} - 2$

1. $f(x) = |x|$

$g(x) = |x - 3| + 3$

1. $f(x) = \sqrt{x}$

$g(x) = \sqrt{x + 1} - 1$

Explanation:

1. (First Graph)

Step1: Identify parent function

The graph is a downward-opening parabola with vertex on the y-axis, so parent is $f(x)=x^2$.

Step2: Find transformed equation

Vertex at $(0,-3)$, vertical stretch by 3, reflected over x-axis: $g(x)=-3x^2-3$

2. (Second Graph)

Step1: Identify parent function

The graph is a V-shaped graph, parent is $f(x)=|x|$.

Step2: Find transformed equation

Reflected over x-axis, vertex at $(0,2)$: $g(x)=-|x|+2$

3. (Third Graph)

Step1: Identify parent function

The graph is a cubic curve, parent is $f(x)=x^3$.

Step2: Find transformed equation

Shifted right 2, up 1, vertical stretch: $g(x)=(x-2)^3+1$

4. (Fourth Graph)

Step1: Identify parent function

The graph is a V-shaped graph, parent is $f(x)=|x|$.

Step2: Find transformed equation

Shifted left 2, down 1: $g(x)=|x+2|-1$

5. $f(x)=|x|$, $g(x)=-|x|-1$

Step1: Identify reflection

Negative sign outside $|x|$: reflect over x-axis.

Step2: Identify vertical shift

Subtract 1: shift down 1 unit.

6. $f(x)=\sqrt{x}$, $g(x)=\sqrt{x}-2$

Step1: Identify vertical shift

Subtract 2: shift down 2 units.

7. $f(x)=|x|$, $g(x)=|x-3|+3$

Step1: Identify horizontal shift

$x-3$: shift right 3 units.

Step2: Identify vertical shift

Add 3: shift up 3 units.

8. $f(x)=\sqrt{x}$, $g(x)=\sqrt{x+1}-1$

Step1: Identify horizontal shift

$x+1$: shift left 1 unit.

Step2: Identify vertical shift

Subtract 1: shift down 1 unit.

Answer:

1. Parent: $f(x)=x^2$

Function: $g(x)=-3x^2-3$

1. Parent: $f(x)=|x|$

Function: $g(x)=-|x|+2$

1. Parent: $f(x)=x^3$

Function: $g(x)=(x-2)^3+1$

1. Parent: $f(x)=|x|$

Function: $g(x)=|x+2|-1$

1. Reflect $f(x)$ over the x-axis, then shift down 1 unit.
2. Shift $f(x)$ down 2 units.
3. Shift $f(x)$ right 3 units, then shift up 3 units.
4. Shift $f(x)$ left 1 unit, then shift down 1 unit.