1. Problem 57: Given the function $y = x^2 - 1$, we stretch it vertically by a factor of 3. 2. Vertical stretching means multiplying the entire function by 3: $$ y = 3(x^2 - 1) = 3x^2 - 3 $$ 3. The original graph: $y = x^2 - 1$ 4. The vertically stretched graph: $y = 3x^2 - 3$ 1. Problem 58: Given $y = x^2 - 1$, compressed horizontally by a factor of 2. 2. Horizontal compression by a factor of 2 means replacing $x$ with $2x$: $$ y = (2x)^2 - 1 = 4x^2 - 1 $$ 3. The original graph: $y = x^2 - 1$ 4. The horizontally compressed graph: $y = 4x^2 - 1$ 1. Problem 59: Given $y = 1 + \frac{1}{x^2}$, compressed vertically by a factor of 2. 2. Vertical compression means multiplying the function's vertical distance from the x-axis by $\frac{1}{2}$: $$ y = 1 + \frac{1}{2} \cdot \frac{1}{x^2} = 1 + \frac{1}{2x^2} $$ 3. The original graph: $y = 1 + \frac{1}{x^2}$ 4. The vertically compressed graph: $y = 1 + \frac{1}{2x^2}$ 1. Problem 60: Given $y = 1 + \frac{1}{x^2}$, stretched horizontally by a factor of 3. 2. Horizontal stretching by 3 means replacing $x$ by $\frac{x}{3}$: $$ y = 1 + \frac{1}{(\frac{x}{3})^2} = 1 + \frac{1}{\frac{x^2}{9}} = 1 + \frac{9}{x^2} $$ 3. The original graph: $y = 1 + \frac{1}{x^2}$ 4. The horizontally stretched graph: $y = 1 + \frac{9}{x^2}$ 1. Problem 61: Given $y = \sqrt{x+1}$, compressed horizontally by a factor of 4. 2. Horizontal compression by 4 means replace $x$ with $4x$: $$ y = \sqrt{4x + 1} $$ 3. Original graph: $y = \sqrt{x+1}$ 4. Horizontally compressed graph: $y = \sqrt{4x + 1}$ 1. Problem 62: Given $y = \sqrt{x+1}$, stretched vertically by a factor of 3. 2. Vertical stretch means multiply $y$ values by 3: $$ y = 3 \sqrt{x+1} $$ 3. Original graph: $y = \sqrt{x+1}$ 4. Vertically stretched graph: $y = 3 \sqrt{x+1}$ 1. Problem 63: Given $y = \sqrt{4 - x^2}$, stretched horizontally by a factor of 2. 2. Horizontal stretch by 2 means replace $x$ with $\frac{x}{2}$: $$ y = \sqrt{4 - (\frac{x}{2})^2} = \sqrt{4 - \frac{x^2}{4}} = \sqrt{\frac{16 - x^2}{4}} = \frac{1}{2} \sqrt{16 - x^2} $$ 3. Original graph: $y = \sqrt{4 - x^2}$ 4. Horizontally stretched graph: $y = \frac{1}{2} \sqrt{16 - x^2}$ 1. Problem 64: Given $y = \sqrt{4 - x^2}$, compressed vertically by a factor of 3. 2. Vertical compression by 3 means multiply by $\frac{1}{3}$: $$ y = \frac{1}{3} \sqrt{4 - x^2} $$ 3. Original graph: $y = \sqrt{4 - x^2}$ 4. Vertically compressed graph: $y = \frac{1}{3} \sqrt{4 - x^2}$ 1. Problem 65: Given $y = 1 - x^3$, compressed horizontally by a factor of 3. 2. Horizontal compression by 3 means replace $x$ with $3x$: $$ y = 1 - (3x)^3 = 1 - 27 x^3 $$ 3. Original graph: $y = 1 - x^3$ 4. Horizontally compressed graph: $y = 1 - 27 x^3$ 1. Problem 66: Given $y = 1 - x^3$, stretched horizontally by a factor of 2. 2. Horizontal stretch by 2 means replace $x$ with $\frac{x}{2}$: $$ y = 1 - \left( \frac{x}{2} \right)^3 = 1 - \frac{x^3}{8} $$ 3. Original graph: $y = 1 - x^3$ 4. Horizontally stretched graph: $y = 1 - \frac{x^3}{8}$