4. explain what transformations are occurring compared to the parent functions.
f(x)=(x + 3)² −5 f(x) = 3|x − 4| f(x) = −x² + 8

5. evaluate the following piecewise functions at x = 0 and x = 6.
(g(x)=\begin{cases}x + 4 & ext{if } x leq -2 \x^2 + 7 & ext{if } -2

Question

4. explain what transformations are occurring compared to the parent functions.
f(x)=(x + 3)² −5  f(x) = 3|x − 4|  f(x) = −x² + 8

5. evaluate the following piecewise functions at x = 0 and x = 6.
(g(x)=\begin{cases}x + 4 & 	ext{if } x leq -2 \x^2 + 7 & 	ext{if } -2 < x < 6 \5x & 	ext{if } x geq 6end{cases})  (f(x)=\begin{cases}3x - 8 & 	ext{if } x < 1 \- x + 7 & 	ext{if } x geq 1end{cases})

6. graph the following piecewise functions.
(g(x)=\begin{cases}2 & 	ext{if } -3 < x leq -1 \4 & 	ext{if } -1 < x leq 2 \6 & 	ext{if } 2 < x leq 5end{cases})  (f(x)=\begin{cases}-2 & 	ext{if } x < -1 \x + 3 & 	ext{if } -1 leq xend{cases})  (with two coordinate grid images for graphing)

Accepted Answer

### Question 4: Analyze Transformations for Each Function

#### For $ f(x) = (x + 3)^2 - 5 $ (Parent: $ y = x^2 $)
# Explanation:
## Step1: Horizontal Shift
The term $ (x + 3) $ means a shift. For $ y = (x - h)^2 $, $ h = -3 $, so shift left 3 units.
## Step2: Vertical Shift
The $ -5 $ at the end means shift down 5 units.
# Answer: Shift left 3 units, shift down 5 units.

#### For $ f(x) = 3|x - 4| $ (Parent: $ y = |x| $)
# Explanation:
## Step1: Vertical Stretch
The coefficient $ 3 $ (greater than 1) vertically stretches the graph by a factor of 3.
## Step2: Horizontal Shift
The $ (x - 4) $ means shift right 4 units (since $ h = 4 $ in $ y = |x - h| $).
# Answer: Vertically stretch by factor 3, shift right 4 units.

#### For $ f(x) = -x^2 + 8 $ (Parent: $ y = x^2 $)
# Explanation:
## Step1: Reflection
The negative sign in front of $ x^2 $ reflects the graph over the x - axis.
## Step2: Vertical Shift
The $ +8 $ at the end means shift up 8 units.
# Answer: Reflect over x - axis, shift up 8 units.

### Question 5: Evaluate Piecewise Functions

#### For $ g(x) $ at $ x = 0 $ and $ x = 6 $
# Explanation:
## Step1: Evaluate $ g(0) $
Check the domain: $ -2 < 0 < 6 $, so use $ g(x)=x^2 + 7 $. Substitute $ x = 0 $: $ 0^2+7 = 7 $.
## Step2: Evaluate $ g(6) $
Check the domain: $ x = 6 $ satisfies $ x\geq6 $, so use $ g(x)=5x $. Substitute $ x = 6 $: $ 5\times6 = 30 $.
# Answer: $ g(0)=7 $, $ g(6)=30 $

#### For $ f(x) $ at $ x = 0 $ and $ x = 6 $
# Explanation:
## Step1: Evaluate $ f(0) $
Check the domain: $ 0 < 1 $, so use $ f(x)=3x - 8 $. Substitute $ x = 0 $: $ 3\times0 - 8=-8 $.
## Step2: Evaluate $ f(6) $
Check the domain: $ 6\geq1 $, so use $ f(x)=-x + 7 $. Substitute $ x = 6 $: $ -6 + 7 = 1 $.
# Answer: $ f(0)=-8 $, $ f(6)=1 $

### Question 6: Graph Piecewise Functions

#### For $ g(x)=\begin{cases}2&-3 < x\leq - 1\4&-1 < x\leq2\6&2 < x\leq5\end{cases}$
# Explanation:
## Step1: Graph $ y = 2 $
This is a horizontal line segment. The domain is $ -3 < x\leq - 1 $. So we draw a horizontal line at $ y = 2 $ from $ x=-3 $ (open circle) to $ x = - 1 $ (closed circle).
## Step2: Graph $ y = 4 $
Domain: $ -1 < x\leq2 $. Draw a horizontal line at $ y = 4 $ from $ x=-1 $ (open circle) to $ x = 2 $ (closed circle).
## Step3: Graph $ y = 6 $
Domain: $ 2 < x\leq5 $. Draw a horizontal line at $ y = 6 $ from $ x = 2 $ (open circle) to $ x = 5 $ (closed circle).

#### For $ f(x)=\begin{cases}-2&x < - 1\x + 3&-1\leq x\end{cases}$
# Explanation:
## Step1: Graph $ y=-2 $
Domain: $ x < - 1 $. Draw a horizontal line at $ y=-2 $ for all $ x < - 1 $ (open circle at $ x=-1 $).
## Step2: Graph $ y=x + 3 $
Domain: $ x\geq - 1 $. This is a linear function. When $ x=-1 $, $ y=-1 + 3 = 2 $ (closed circle). We can also find another point, e.g., when $ x = 0 $, $ y=3 $, and draw the line from $ x=-1 $ (point $ (-1,2) $) onwards.

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QUESTION IMAGE

Question

Explanation:

Question 4: Analyze Transformations for Each Function

For \( f(x) = (x + 3)^2 - 5 \) (Parent: \( y = x^2 \))

Step1: Horizontal Shift

Step2: Vertical Shift

Step1: Vertical Stretch

Step2: Horizontal Shift

Step1: Reflection

Step2: Vertical Shift

Answer:

For \( f(x) = 3|x - 4| \) (Parent: \( y = |x| \))