Question

no calc, no notes part

1. graph ( f(x) = x^2 ), ( f(x) = x^3 ), ( f(x) = sqrt{x} ), ( f(x) = sqrt3{x} )
2. graph by transformations (shifts) and list the transformations

a. ( f(x) = -2sqrt3{x - 1} + 2 )
b. ( f(x) = -\frac{1}{2}x^2 + 5 )
c. ( f(x) = 2|x + 2| - 1 )
of those found on hw lesson 6 hando

Explanation:

Step 1: Identify parent function (2a)

Parent: $f(x)=\sqrt[3]{x}$

Step 2: List transformations (2a)

1. Horizontal shift right 1: $\sqrt[3]{x-1}$
2. Vertical stretch by 2: $2\sqrt[3]{x-1}$
3. Reflect over x-axis: $-2\sqrt[3]{x-1}$
4. Vertical shift up 2: $-2\sqrt[3]{x-1}+2$

Step 1: Identify parent function (2b)

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

Step 2: List transformations (2b)

1. Vertical shrink by $\frac{1}{2}$: $\frac{1}{2}x^2$
2. Reflect over x-axis: $-\frac{1}{2}x^2$
3. Vertical shift up 5: $-\frac{1}{2}x^2+5$

Step 1: Identify parent function (2c)

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

Step 2: List transformations (2c)

1. Horizontal shift left 2: $|x+2|$
2. Vertical stretch by 2: $2|x+2|$
3. Vertical shift down 1: $2|x+2|-1$

Step 1: Sketch basic parent graphs (1)

• $f(x)=x^2$: U-shaped parabola, vertex at $(0,0)$, symmetric over y-axis.
• $f(x)=x^3$: Cubic curve, passes through $(0,0)$, increasing everywhere, symmetric over origin.
• $f(x)=\sqrt{x}$: Right-opening curve, domain $x\geq0$, passes through $(0,0),(1,1)$.
• $f(x)=\sqrt[3]{x}$: Cubic root curve, domain all real numbers, passes through $(0,0),(1,1),(-1,-1)$, symmetric over origin.

Answer:

1.

• $f(x)=x^2$: Parabola with vertex at $(0,0)$, opening upward, symmetric across the y-axis.
• $f(x)=x^3$: Cubic curve passing through $(0,0)$, strictly increasing, symmetric about the origin.
• $f(x)=\sqrt{x}$: Non-negative curve defined for $x\geq0$, starting at $(0,0)$ and increasing slowly through $(1,1)$.
• $f(x)=\sqrt[3]{x}$: Curve defined for all real $x$, passing through $(0,0),(1,1),(-1,-1)$, increasing across its domain.

2.
a. Transformations of $f(x)=\sqrt[3]{x}$:

1. Horizontal shift 1 unit to the right
2. Vertical stretch by a factor of 2
3. Reflection across the x-axis
4. Vertical shift 2 units upward

Graph: Cubic root curve shifted right 1, stretched vertically, flipped over x-axis, shifted up 2.

b. Transformations of $f(x)=x^2$:

1. Vertical shrink by a factor of $\frac{1}{2}$
2. Reflection across the x-axis
3. Vertical shift 5 units upward

Graph: Downward-opening parabola, wider than $x^2$, vertex at $(0,5)$.

c. Transformations of $f(x)=|x|$:

1. Horizontal shift 2 units to the left
2. Vertical stretch by a factor of 2
3. Vertical shift 1 unit downward

Graph: V-shaped curve, vertex at $(-2,-1)$, opening upward, narrower than $|x|$.