Precalculus Practice: Graph Transformations and Function Operations

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: $y = f(x - h)$ shifts the graph of $y = f(x)$ horizontally.

• If $h > 0$, the shift is to the right; if $h < 0$, the shift is to the left.

Step-by-Step Guidance

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

2. Recall that $f(x - h)$ shifts the graph of $f(x)$ horizontally by $h$ 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 $h$.

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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: $y = -f(x)$

• Reflection about the y-axis: $y = f(-x)$

Step-by-Step Guidance

1. Identify the transformation: $y = -f(x)$ means every output value of $f(x)$ is multiplied by -1.

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

3. Compare with $y = f(-x)$, 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: $y = a f(x)$ stretches the graph vertically by a factor of $|a|$.

• Horizontal Compression: $y = f(bx)$ compresses the graph horizontally by a factor of $1/|b|$.

Step-by-Step Guidance

1. Identify the transformation: $y = f(6x)$ 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 $y = 6f(x)$, 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: $y = -f(x)$

• Reflection about the y-axis: $y = f(-x)$

Step-by-Step Guidance

1. Recall that $y = -f(x)$ 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 $f(-x)$.

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: $y = f(x) + k$ shifts the graph up by $k$ units if $k > 0$, down if $k < 0$.

Step-by-Step Guidance

1. Identify the base function: $y = \sqrt{x}$.

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

3. Look for the function in the answer choices that matches $y = \sqrt{x} + 1$.

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: $y = f(x - h)$ shifts the graph right by $h$ units if $h > 0$.

• For $y = |x|$, a shift right by $h$ units is $y = |x - h|$.

Step-by-Step Guidance

1. Start with the base function: $y = |x|$.

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

3. Write the new function in terms of $x$.

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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: $y = f(x) + k$ shifts the graph up by $k$ units if $k > 0$.

• For $y = x^2$, a shift up by $k$ units is $y = x^2 + k$.

Step-by-Step Guidance

1. Start with the base function: $y = x^2$.

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

3. Write the new function in terms of $x$.

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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 $x$ with $-x$ inside the function.

Key Terms and Formulas:

• Reflection about the y-axis: $y = f(-x)$

• For $y = \sqrt[3]{x}$, the reflection is $y = \sqrt[3]{-x}$.

Step-by-Step Guidance

1. Start with the base function: $y = \sqrt[3]{x}$.

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

3. Write the new function in terms of $x$.

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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: $y = a f(x)$ stretches the graph vertically by a factor of $|a|$.

• For $y = \sqrt{x}$, a vertical stretch by 8 is $y = 8\sqrt{x}$.

Step-by-Step Guidance

1. Start with the base function: $y = \sqrt{x}$.

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

3. Write the new function in terms of $x$.

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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: $y = f(x) + k$

• Reflection about the y-axis: $y = f(-x)$

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with $y = \sqrt{x}$.

2. First, shift up 7 units: $y = \sqrt{x} + 7$.

3. Next, reflect about the y-axis: replace $x$ with $-x$ to get $y = \sqrt{-x} + 7$.

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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: $y = -f(x)$

• Vertical Shift: $y = f(x) + k$

• Horizontal Shift: $y = f(x - h)$

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with $y = \sqrt{x}$.

2. First, reflect about the x-axis: $y = -\sqrt{x}$.

3. Next, shift down 4 units: $y = -\sqrt{x} - 4$.

4. Finally, shift right 6 units: replace $x$ with $x - 6$ to get $y = -\sqrt{x - 6} - 4$.

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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: $y = a f(x)$

• Vertical Shift: $y = f(x) + k$

• Horizontal Shift: $y = f(x - h)$

• Order matters: Apply transformations in the given sequence.

Step-by-Step Guidance

1. Start with $y = \sqrt{x}$.

2. First, apply the vertical stretch: $y = 3\sqrt{x}$.

3. Next, shift up 4 units: $y = 3\sqrt{x} + 4$.

4. Finally, shift right 5 units: replace $x$ with $x - 5$ to get $y = 3\sqrt{x - 5} + 4$.

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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: $y = f(x + h)$ shifts the graph left by $h$ units if $h > 0$.

Step-by-Step Guidance

1. Start with the base function: $y = x^4$.

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: $y = f(x) - k$ shifts the graph down by $k$ units if $k > 0$.

Step-by-Step Guidance

1. Start with the base function: $y = x^2$.

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: $y = a f(x)$ stretches the graph vertically by a factor of $|a|$.

Step-by-Step Guidance

1. Start with the base function: $y = \frac{1}{x}$.

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.

Try solving on your own before revealing the answer!

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: $y = -f(x)$

Step-by-Step Guidance

1. Start with the base function: $y = x^3$.

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: $y = f(x) - k$ shifts the graph down by $k$ units if $k > 0$.

Step-by-Step Guidance

1. Start with the base function: $y = |x|$.

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

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

Try solving on your own before revealing the answer!

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: $y = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Y-intercept: $c$

• X-intercepts: Solve $ax^2 + bx + c = 0$

Step-by-Step Guidance

1. Identify the vertex using $x = -\frac{b}{2a}$.

2. Find the y-intercept by evaluating $f(0)$.

3. Find the x-intercepts by solving $x^2 - 4x + 3 = 0$.

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: $y = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Y-intercept: $c$

• X-intercepts: Solve $ax^2 + bx + c = 0$

Step-by-Step Guidance

1. Identify the vertex using $x = -\frac{b}{2a}$.

2. Find the y-intercept by evaluating $f(0)$.

3. Find the x-intercepts by solving $x^2 - 6x = 0$.

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

Try solving on your own before revealing the answer!