Question

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1. graph $f(x) = (x - 1)^3 + 3$
2. the graph of $f(x)$ is a transformation of $g(x) = x^3$

write the equation of $f(x)$.

1. if $g(x) = x^3$ and $h(x)=- (x - 2)^3 - 6$, describe the transformation from $g(x)$ to $h(x)$.

Explanation:

Step1: Identify parent function

Parent function: $g(x)=x^3$

Step2: Analyze Q5: Transform parent for $f(x)$

Base point of $x^3$ is $(0,0)$. For $f(x)=(x-1)^3+3$:

• Shift right 1: $(0+1,0)=(1,0)$
• Shift up 3: $(1,0+3)=(1,3)$

Plot points: e.g., $x=0$: $f(0)=(0-1)^3+3=-1+3=2$; $x=2$: $f(2)=(2-1)^3+3=1+3=4$; $x=-1$: $f(-1)=(-1-1)^3+3=-8+3=-5$. Connect points to match the given graph shape.

Step3: Analyze Q6: Find $f(x)$ from graph

Parent $g(x)=x^3$, base point $(0,0)$ moves to $(-2,-5)$:

• Shift left 2: $x\to x+2$
• Shift down 5: $+(-5)$

Equation: $f(x)=(x+2)^3-5$

Step4: Analyze Q7: Describe $g(x)\to h(x)$

Given $g(x)=x^3$, $h(x)=-(x-2)^3-6$:

1. Reflect over x-axis: $-x^3$
2. Shift right 2: $-(x-2)^3$
3. Shift down 6: $-(x-2)^3-6$

Answer:

1. The graph has a critical point at $(1,3)$, passes through $(0,2)$, $(2,4)$, $(-1,-5)$, matching the shape of $y=x^3$ shifted right 1 and up 3 (matches the provided graph).
2. $f(x)=(x+2)^3-5$
3. Reflect the graph of $g(x)$ over the x-axis, shift it 2 units to the right, and shift it 6 units downward.