Question

4.6 practice with calcchat and calcview
in exercises 1–4, describe the transformation of f represented by g. then graph each function. (see example 1.)

1. $f(x)=x^4, g(x)=x^4 + 3$
2. $f(x)=x^4, g(x)=(x - 5)^4$
3. $f(x)=x^5, g(x)=(x - 2)^5 - 1$
4. $f(x)=x^6, g(x)=(x + 1)^6 - 4$

reasoning in exercises 5–8, match the function with the correct transformation of the graph of f. explain your reasoning.

1. $y = f(x - 2)$
2. $y = f(x + 2) + 2$
3. $y = f(x - 2) + 2$
4. $y = f(x) - 2$

a.
graph a
b.
graph b
c.
graph c
d.
graph d
4.6 transformations of polynomial functions

Explanation:

(Exercises 1-4):

Step1: Identify base function & transformation

For each pair, compare $g(x)$ to $f(x)$ using function transformation rules:

• Vertical shift: $f(x)+k$ = up $k$ (if $k>0$), down $|k|$ (if $k<0$)
• Horizontal shift: $f(x-h)$ = right $h$ (if $h>0$), left $|h|$ (if $h<0$)

---

1. $f(x)=x^4$, $g(x)=x^4+3$

Step1: Classify vertical shift

$g(x)=f(x)+3$, so shift up 3 units.

Step2: Graph description

Plot $f(x)=x^4$ (U-shaped, vertex at $(0,0)$), then shift all points up 3 (new vertex $(0,3)$).

---

2. $f(x)=x^4$, $g(x)=(x-5)^4$

Step1: Classify horizontal shift

$g(x)=f(x-5)$, so shift right 5 units.

Step2: Graph description

Plot $f(x)=x^4$, then shift all points right 5 (new vertex $(5,0)$).

---

3. $f(x)=x^5$, $g(x)=(x-2)^5-1$

Step1: Classify horizontal + vertical shift

$g(x)=f(x-2)-1$, so shift right 2, down 1.

Step2: Graph description

Plot $f(x)=x^5$ (increasing, inflection at $(0,0)$), shift right 2, down 1 (new inflection $(2,-1)$).

---

4. $f(x)=x^6$, $g(x)=(x+1)^6-4$

Step1: Rewrite & classify shifts

$g(x)=f(x-(-1))-4$, so shift left 1, down 4.

Step2: Graph description

Plot $f(x)=x^6$ (U-shaped, vertex $(0,0)$), shift left 1, down 4 (new vertex $(-1,-4)$).

---

(Exercises 5-8):

Step1: Match to transformation rules

Use the same shift rules to pair each $y$ with the graph:

1. $y=f(x-2)$: Horizontal shift right 2 → Graph D (vertex moves right 2)
2. $y=f(x+2)+2$: Shift left 2, up 2 → Graph C (vertex moves left 2, up 2)
3. $y=f(x-2)+2$: Shift right 2, up 2 → Graph A (vertex moves right 2, up 2)
4. $y=f(x)-2$: Vertical shift down 2 → Graph B (vertex moves down 2)

---

Answer:

Exercises 1-4:

1. Transformation: Vertical shift up 3 units. Graph: $x^4$ shifted up to vertex $(0,3)$.
2. Transformation: Horizontal shift right 5 units. Graph: $x^4$ shifted right to vertex $(5,0)$.
3. Transformation: Horizontal shift right 2 units, vertical shift down 1 unit. Graph: $x^5$ shifted to inflection $(2,-1)$.
4. Transformation: Horizontal shift left 1 unit, vertical shift down 4 units. Graph: $x^6$ shifted to vertex $(-1,-4)$.

Exercises 5-8:

1. $\boldsymbol{y=f(x-2)

ightarrow \text{Graph D}}$ (right 2 shift)

1. $\boldsymbol{y=f(x+2)+2

ightarrow \text{Graph C}}$ (left 2, up 2 shift)

1. $\boldsymbol{y=f(x-2)+2

ightarrow \text{Graph A}}$ (right 2, up 2 shift)

1. $\boldsymbol{y=f(x)-2

ightarrow \text{Graph B}}$ (down 2 shift)