Algebra 2 Transformations Practice – Step-by-Step Guidance

Q1. Describe the transformations necessary to transform the graph of $f(x) = x^2$ into that of $g(x) = -x^2 - 3$.

Background

Topic: Transformations of Quadratic Functions

This question tests your understanding of how to apply transformations (reflections, translations) to the parent function $f(x) = x^2$ to obtain $g(x)$.

Key Terms and Formulas:

• Reflection across the x-axis: Changes $f(x)$ to $-f(x)$.

• Vertical translation: $f(x) + c$ moves the graph up/down by $c$ units.

Step-by-Step Guidance

1. Start with the parent function $f(x) = x^2$.

2. Apply a reflection across the x-axis: $-x^2$.

3. Apply a vertical translation downward by 3 units: $-x^2 - 3$.

Try solving on your own before revealing the answer!

Final Answer:

Reflect across the x-axis, then translate down 3 units.

This means you first flip the parabola upside down, then shift it downward by 3 units.

Q2. Describe the transformations necessary to transform $f(x) = \sqrt{x}$ into $g(x) = \sqrt{x - 3} - 2$.

Background

Topic: Transformations of Radical Functions

This question tests your ability to identify horizontal and vertical shifts for square root functions.

Key Terms and Formulas:

• Horizontal translation: $f(x - h)$ shifts right by $h$ units.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a horizontal shift right by 3 units: $\sqrt{x - 3}$.

3. Apply a vertical shift down by 2 units: $\sqrt{x - 3} - 2$.

Try solving on your own before revealing the answer!

Final Answer:

Translate right 3 units, then down 2 units.

The function moves right and then down.

Q3. Describe the transformations necessary to transform $f(x) = x$ into $g(x) = x - 2 - 1$.

Background

Topic: Transformations of Linear Functions

This question tests your understanding of horizontal and vertical translations for linear functions.

Key Terms and Formulas:

• Horizontal translation: $f(x - h)$ shifts right by $h$ units.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = x$.

2. Apply a horizontal shift right by 2 units: $x - 2$.

3. Apply a vertical shift down by 1 unit: $x - 2 - 1$.

Try solving on your own before revealing the answer!

Final Answer:

Translate right 2 units, then down 1 unit.

The line moves right and then down.

Q4. Describe the transformations necessary to transform $f(x) = \sqrt{x}$ into $g(x) = \sqrt{-x} - 1$.

Background

Topic: Transformations of Radical Functions

This question tests your understanding of reflections and vertical translations for square root functions.

Key Terms and Formulas:

• Reflection across the y-axis: $f(-x)$ flips the graph horizontally.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a reflection across the y-axis: $\sqrt{-x}$.

3. Apply a vertical shift down by 1 unit: $\sqrt{-x} - 1$.

Try solving on your own before revealing the answer!

Final Answer:

Reflect across the y-axis, then translate down 1 unit.

The graph is flipped horizontally and shifted downward.

Q5. Describe the transformations necessary to transform $f(x) = x^2$ into $g(x) = -(2x)^2$.

Background

Topic: Transformations of Quadratic Functions

This question tests your understanding of horizontal compression and reflection for quadratic functions.

Key Terms and Formulas:

• Horizontal compression: $f(ax)$ compresses the graph horizontally by a factor of $\frac{1}{|a|}$.

• Reflection across the x-axis: $-f(x)$ flips the graph vertically.

Step-by-Step Guidance

1. Start with $f(x) = x^2$.

2. Apply a horizontal compression by a factor of 2: $f(2x) = (2x)^2$.

3. Apply a reflection across the x-axis: $-(2x)^2$.

Try solving on your own before revealing the answer!

Final Answer:

Compress horizontally by a factor of 2, then reflect across the x-axis.

The parabola becomes narrower and flips upside down.

Q6. Describe the transformations necessary to transform $f(x) = \sqrt{x}$ into $g(x) = 3\sqrt{x} + 1$.

Background

Topic: Transformations of Radical Functions

This question tests your understanding of vertical stretching and translation for square root functions.

Key Terms and Formulas:

• Vertical stretch: $a f(x)$ stretches the graph vertically by a factor of $a$.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a vertical stretch by a factor of 3: $3\sqrt{x}$.

3. Apply a vertical shift up by 1 unit: $3\sqrt{x} + 1$.

Try solving on your own before revealing the answer!

Final Answer:

Stretch vertically by a factor of 3, then translate up 1 unit.

The graph is taller and shifted upward.

Q7. Describe the transformations necessary to transform $f(x) = \sqrt{x}$ into $g(x) = \frac{1}{2}\sqrt{x + 3}$.

Background

Topic: Transformations of Radical Functions

This question tests your understanding of vertical compression and horizontal translation for square root functions.

Key Terms and Formulas:

• Vertical compression: $a f(x)$ with $0 < a < 1$ compresses the graph vertically.

• Horizontal translation: $f(x + h)$ shifts left by $h$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a horizontal shift left by 3 units: $\sqrt{x + 3}$.

3. Apply a vertical compression by a factor of $\frac{1}{2}$: $\frac{1}{2}\sqrt{x + 3}$.

Try solving on your own before revealing the answer!

Final Answer:

Compress vertically by a factor of $\frac{1}{2}$, then translate left 3 units.

The graph is shorter and shifted to the left.

Q8. Describe the transformations necessary to transform $f(x) = \sqrt{x}$ into $g(x) = -\frac{1}{3}\sqrt{x} - 3$.

Background

Topic: Transformations of Radical Functions

This question tests your understanding of horizontal stretching, reflection, and vertical translation for square root functions.

Key Terms and Formulas:

• Horizontal stretch: $f(ax)$ with $0 < a < 1$ stretches the graph horizontally.

• Reflection across the x-axis: $-f(x)$ flips the graph vertically.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a horizontal stretch by a factor of 3: $\sqrt{3x}$.

3. Apply a reflection across the x-axis: $-\sqrt{3x}$.

4. Apply a vertical shift down by 3 units: $-\sqrt{3x} - 3$.

Try solving on your own before revealing the answer!

Final Answer:

Stretch horizontally by a factor of 3, reflect across the x-axis, then translate down 3 units.

The graph is wider, flipped upside down, and shifted downward.

Q9. Transform $f(x) = \sqrt{x}$ as described: reflect across the y-axis, stretch vertically by a factor of 2, translate down 3 units. Write the resulting function as an equation.

Background

Topic: Function Transformations

This question tests your ability to combine multiple transformations and write the resulting function.

Key Terms and Formulas:

• Reflection across the y-axis: $f(-x)$.

• Vertical stretch: $a f(x)$.

• Vertical translation: $f(x) + k$.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Reflect across the y-axis: $\sqrt{-x}$.

3. Stretch vertically by a factor of 2: $2\sqrt{-x}$.

4. Translate down 3 units: $2\sqrt{-x} - 3$.

Try solving on your own before revealing the answer!

Final Answer:

$g(x) = 2\sqrt{-x} - 3$

This combines all the described transformations into a single equation.

Q10. Transform $f(x) = \sqrt{x}$ as described: stretch horizontally by a factor of 3, translate left 1 unit. Write the resulting function as an equation.

Background

Topic: Function Transformations

This question tests your ability to apply horizontal stretching and translation to a square root function.

Key Terms and Formulas:

• Horizontal stretch: $f(\frac{x}{a})$ stretches by a factor of $a$.

• Horizontal translation: $f(x + h)$ shifts left by $h$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Apply a horizontal stretch by a factor of 3: $\sqrt{\frac{x}{3}}$.

3. Translate left 1 unit: $\sqrt{\frac{x + 1}{3}}$.

Try solving on your own before revealing the answer!

Final Answer:

$g(x) = \sqrt{\frac{x + 1}{3}}$

This equation reflects the described transformations.

Q11. Transform $f(x) = \sqrt{x}$ as described: reflect across the y-axis, reflect across the x-axis, translate right 2 units. Write the resulting function as an equation.

Background

Topic: Function Transformations

This question tests your ability to combine reflections and translations for a square root function.

Key Terms and Formulas:

• Reflection across the y-axis: $f(-x)$.

• Reflection across the x-axis: $-f(x)$.

• Horizontal translation: $f(x - h)$ shifts right by $h$ units.

Step-by-Step Guidance

1. Start with $f(x) = \sqrt{x}$.

2. Reflect across the y-axis: $\sqrt{-x}$.

3. Reflect across the x-axis: $-\sqrt{-x}$.

4. Translate right 2 units: $-\sqrt{-(x - 2)}$.

Try solving on your own before revealing the answer!

Final Answer:

$g(x) = -\sqrt{-(x - 2)}$

This equation combines all the described transformations.

Q12. Transform $f(x) = x^2$ as described: translate right 1 unit, translate down 3 units. Write the resulting function as an equation.

Background

Topic: Function Transformations

This question tests your ability to apply horizontal and vertical translations to a quadratic function.

Key Terms and Formulas:

• Horizontal translation: $f(x - h)$ shifts right by $h$ units.

• Vertical translation: $f(x) + k$ shifts up/down by $k$ units.

Step-by-Step Guidance

1. Start with $f(x) = x^2$.

2. Translate right 1 unit: $(x - 1)^2$.

3. Translate down 3 units: $(x - 1)^2 - 3$.

Try solving on your own before revealing the answer!

Final Answer:

$g(x) = (x - 1)^2 - 3$

This equation reflects the described translations.