Q1. Suppose that the graph of a function f is known. Then the graph of y = f(x - 2) may be obtained by a ___ shift of the graph of y = f(x), a distance of 2 units.

Background

Topic: Transformations of Functions

This question tests your understanding of horizontal shifts of function graphs. Specifically, it asks how the graph of a function changes when the input variable x is replaced by (x - h).

Key Terms and Formulas:

• Horizontal Shift: shifts the graph of horizontally.

• If , the shift is to the right; if , the shift is to the left.

Step-by-Step Guidance

1. Identify the transformation inside the function: means x is replaced by (x - 2).

2. Recall that shifts the graph of horizontally by units.

3. Determine the direction: If the sign is negative, the shift is to the right; if positive, to the left.

4. State the distance of the shift, which is the absolute value of .

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Q2. Suppose that the graph of a function f is known. Then the graph of y = -f(x) may be obtained by a reflection about the ___ axis of the graph of the function y = f(x).

Background

Topic: Reflections of Functions

This question tests your understanding of how multiplying a function by -1 affects its graph, specifically which axis the reflection occurs over.

Key Terms and Formulas:

• Reflection about the x-axis:

• Reflection about the y-axis:

Step-by-Step Guidance

1. Identify the transformation: means every output value of is multiplied by -1.

2. Recall that multiplying the function by -1 reflects the graph over the x-axis.

3. Compare with , which reflects over the y-axis.

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Q3. Determine whether the following statement is true or false: The graph of y = f(6x) is the graph of y = f(x) vertically stretched by a factor of 6.

Background

Topic: Function Transformations – Horizontal and Vertical Stretches

This question tests your understanding of how multiplying the input variable x by a constant affects the graph of a function, and whether this results in a vertical or horizontal stretch/compression.

Key Terms and Formulas:

• Vertical Stretch: stretches the graph vertically by a factor of .

• Horizontal Compression: compresses the graph horizontally by a factor of .

Step-by-Step Guidance

1. Identify the transformation: means the input x is multiplied by 6.

2. Recall that multiplying x inside the function affects the graph horizontally, not vertically.

3. Determine if this is a stretch or compression, and in which direction.

4. Compare with , which would be a vertical stretch.

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Q4. The graph of y = -f(x) is the reflection about the x-axis of the graph of y = f(x). Is the statement true or false?

Background

Topic: Reflections of Functions

This question tests your understanding of how multiplying a function by -1 affects its graph, and asks you to identify the correct reasoning for the reflection.

Key Terms and Formulas:

• Reflection about the x-axis:

• Reflection about the y-axis:

Step-by-Step Guidance

1. Recall that reflects the graph over the x-axis.

2. Check the reasoning: multiplying the output (y-values) by -1 flips the graph over the x-axis.

3. Compare with reflecting over the y-axis, which would involve .

4. Review the answer choices for the correct explanation.

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Q5. Which of the following functions has a graph that is the graph of y = \sqrt{x} shifted up 1 unit?

Background

Topic: Vertical Shifts of Functions

This question tests your ability to recognize how adding or subtracting a constant outside the function affects the graph vertically.

Key Terms and Formulas:

• Vertical Shift: shifts the graph up by units if , down if .

Step-by-Step Guidance

1. Identify the base function: .

2. Recall that adding a constant outside the function shifts the graph vertically.

3. Look for the function in the answer choices that matches .

4. Check that the shift is up (not down or sideways).

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Q6. Write the function whose graph is the graph of y = |x| but is shifted to the right 6 units.

Background

Topic: Horizontal Shifts of Absolute Value Functions

This question tests your ability to apply horizontal shifts to the absolute value function.

Key Terms and Formulas:

• Horizontal Shift: shifts the graph right by units if .

• For , a shift right by units is .

Step-by-Step Guidance

1. Start with the base function: .

2. To shift right by 6 units, replace with inside the absolute value.

3. Write the new function in terms of .

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Q7. Write the function whose graph is the graph of y = x^2 but is shifted up 2 units.

Background

Topic: Vertical Shifts of Quadratic Functions

This question tests your ability to apply vertical shifts to the basic quadratic function.

Key Terms and Formulas:

• Vertical Shift: shifts the graph up by units if .

• For , a shift up by units is .

Step-by-Step Guidance

1. Start with the base function: .

2. Add 2 to the entire function to shift it up by 2 units.

3. Write the new function in terms of .

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Q8. Write the function whose graph is the graph of y = \sqrt[3]{x} but is reflected about the y-axis.

Background

Topic: Reflections of Functions

This question tests your ability to reflect a function about the y-axis, which involves replacing with inside the function.

Key Terms and Formulas:

• Reflection about the y-axis:

• For , the reflection is .

Step-by-Step Guidance

1. Start with the base function: .

2. To reflect about the y-axis, replace with inside the radical.

3. Write the new function in terms of .

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Q9. Write the function whose graph is the graph of y = \sqrt{x}, but is vertically stretched by a factor of 8.

Background

Topic: Vertical Stretches of Functions

This question tests your ability to apply a vertical stretch to a function, which involves multiplying the entire function by a constant.

Key Terms and Formulas:

• Vertical Stretch: stretches the graph vertically by a factor of .

• For , a vertical stretch by 8 is .

Step-by-Step Guidance

1. Start with the base function: .

2. Multiply the entire function by 8 to stretch it vertically.

3. Write the new function in terms of .

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Q10. Find the function that is finally graphed after the following transformations are applied to the graph of y = \sqrt{x}, in the order listed: (1) Shift up 7 units, (2) Reflect about the y-axis.

Background

Topic: Multiple Transformations of Functions

This question tests your ability to apply a sequence of transformations (vertical shift and reflection) to a function, and to write the resulting function.

Key Terms and Formulas:

• Vertical Shift:

• Reflection about the y-axis:

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with .

2. First, shift up 7 units: .

3. Next, reflect about the y-axis: replace with to get .

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Q11. Find the function that is finally graphed after the following transformations are applied to the graph of y = \sqrt{x}, in the order listed: (1) Reflect about the x-axis, (2) Shift down 4 units, (3) Shift right 6 units.

Background

Topic: Multiple Transformations of Functions

This question tests your ability to apply a sequence of transformations (reflection, vertical shift, horizontal shift) to a function, and to write the resulting function.

Key Terms and Formulas:

• Reflection about the x-axis:

• Vertical Shift:

• Horizontal Shift:

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with .

2. First, reflect about the x-axis: .

3. Next, shift down 4 units: .

4. Finally, shift right 6 units: replace with to get .

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Q12. Find the function that is finally graphed after the following transformations are applied to the graph of y = \sqrt{x}, in the order listed: (1) Vertical stretch by a factor of 3, (2) Shift up 4 units, (3) Shift right 5 units.

Background

Topic: Multiple Transformations of Functions

This question tests your ability to apply a sequence of transformations (vertical stretch, vertical shift, horizontal shift) to a function, and to write the resulting function.

Key Terms and Formulas:

• Vertical Stretch:

• Vertical Shift:

• Horizontal Shift:

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with .

2. First, apply the vertical stretch: .

3. Next, shift up 4 units: .

4. Finally, shift right 5 units: replace with to get .

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Q13. Use a transformation of the graph of y = x^4 to graph the function f(x) = (x + 4)^4.

Background

Topic: Horizontal Shifts of Polynomial Functions

This question tests your ability to recognize how adding a constant inside the function argument shifts the graph horizontally.

Key Terms and Formulas:

• Horizontal Shift: shifts the graph left by units if .

Step-by-Step Guidance

1. Start with the base function: .

2. Adding 4 inside the parentheses shifts the graph left by 4 units.

3. Look for the graph that matches this transformation among the choices.

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Q14. Use a transformation of the graph of y = x^2 to graph the function f(x) = x^2 - 8.

Background

Topic: Vertical Shifts of Quadratic Functions

This question tests your ability to recognize how subtracting a constant outside the function shifts the graph vertically.

Key Terms and Formulas:

• Vertical Shift: shifts the graph down by units if .

Step-by-Step Guidance

1. Start with the base function: .

2. Subtracting 8 shifts the graph down by 8 units.

3. Look for the graph that matches this transformation among the choices.

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Q15. Use a transformation of the graph of y = \frac{1}{x} to graph the function f(x) = \frac{2}{x}.

Background

Topic: Vertical Stretches of Rational Functions

This question tests your ability to recognize how multiplying a function by a constant outside affects the graph vertically.

Key Terms and Formulas:

• Vertical Stretch: stretches the graph vertically by a factor of .

Step-by-Step Guidance

1. Start with the base function: .

2. Multiplying by 2 stretches the graph vertically by a factor of 2.

3. Look for the graph that matches this transformation among the choices.

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Q16. Use a transformation of the graph of y = x^3 to graph the function h(x) = -x^3.

Background

Topic: Reflections of Polynomial Functions

This question tests your ability to recognize how multiplying a function by -1 reflects the graph over the x-axis.

Key Terms and Formulas:

• Reflection about the x-axis:

Step-by-Step Guidance

1. Start with the base function: .

2. Multiplying by -1 reflects the graph over the x-axis.

3. Look for the graph that matches this transformation among the choices.

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Q17. Which graph shown below is the graph of the following function? f(x) = |x| - 6

Background

Topic: Vertical Shifts of Absolute Value Functions

This question tests your ability to recognize how subtracting a constant outside the function shifts the graph vertically.

Key Terms and Formulas:

• Vertical Shift: shifts the graph down by units if .

Step-by-Step Guidance

1. Start with the base function: .

2. Subtracting 6 shifts the graph down by 6 units.

3. Look for the graph that matches this transformation among the choices.

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Q18. Match the function f(x) = x^2 - 4x + 3 to one of the given graphs.

Background

Topic: Graphing Quadratic Functions

This question tests your ability to recognize the graph of a quadratic function in standard form by identifying its vertex, axis of symmetry, and intercepts.

Key Terms and Formulas:

• Standard Form:

• Vertex:

• Y-intercept:

• X-intercepts: Solve

Step-by-Step Guidance

1. Identify the vertex using .

2. Find the y-intercept by evaluating .

3. Find the x-intercepts by solving .

4. Match these features to the correct graph among the choices.

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Q19. Match the function f(x) = x^2 - 6x to one of the given graphs.

Background

Topic: Graphing Quadratic Functions

This question tests your ability to recognize the graph of a quadratic function in standard form by identifying its vertex, axis of symmetry, and intercepts.

Key Terms and Formulas:

• Standard Form:

• Vertex:

• Y-intercept:

• X-intercepts: Solve

Step-by-Step Guidance

1. Identify the vertex using .

2. Find the y-intercept by evaluating .

3. Find the x-intercepts by solving .

4. Match these features to the correct graph among the choices.

Try solving on your own before revealing the answer!