Question

determine the appropriate key features for each function.

for each function (y = 2√(x + 3) - 2, y = -4∛(x - 1), y = 2√(x + 1) - 1, y = -√x + 2), match with the given key features:

• end behavior: as x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞
• cube root with turning point at (1,0)
• square root with starting point at (0,2)
• end behavior: as x → ∞, f(x) → ∞ and as x → -1, f(x) → -1

Explanation:

Step1: Analyze $y=2\sqrt[3]{x+3}-2$

This is a cube root function (odd root, domain all real numbers). As $x\to\infty$, $\sqrt[3]{x+3}\to\infty$, so $f(x)\to\infty$. As $x\to-\infty$, $\sqrt[3]{x+3}\to-\infty$, so $f(x)\to-\infty$.

Step2: Analyze $y=-4\sqrt[3]{x-1}$

This is a cube root function. The horizontal shift gives a turning point (inflection point) at $x=1$, $y=0$ (set $x-1=0$, $y=0$).

Step3: Analyze $y=2\sqrt{x+1}-1$

This is a square root function (domain $x\geq-1$). As $x\to\infty$, $\sqrt{x+1}\to\infty$, so $f(x)\to\infty$. As $x\to-1^+$, $\sqrt{x+1}\to0$, so $f(x)\to-1$.

Step4: Analyze $y=-\sqrt{x}+2$

This is a square root function. The starting point is when $x=0$, $y=-\sqrt{0}+2=2$, so $(0,2)$.

Answer:

• $y=2\sqrt[3]{x+3}-2$: End behavior: as $x \to \infty, f(x) \to \infty$ and as $x \to -\infty, f(x) \to -\infty$
• $y=-4\sqrt[3]{x-1}$: Cube root with turning point at $(1,0)$
• $y=2\sqrt{x+1}-1$: End behavior: as $x \to \infty, f(x) \to \infty$ and as $x \to -1, f(x) \to -1$
• $y=-\sqrt{x}+2$: Square root with starting point at $(0,2)$