Precalculus Function Transformations and Graphing Guidance

Q44. Graph the function $f(x) = (x + 2)^3 - 3$

Background

Topic: Transformations of Functions (Cubic Functions)

This question tests your understanding of how to graph a cubic function by applying horizontal and vertical shifts to the parent function $y = x^3$.

Key Terms and Formulas

• Parent function: $y = x^3$

• Horizontal shift: $f(x + h)$ shifts the graph left by $h$ units.

• Vertical shift: $f(x) + k$ shifts the graph up by $k$ units (or down if $k$ is negative).

Step-by-Step Guidance

1. Start with the basic cubic function $y = x^3$.

2. Apply the horizontal shift: Replace $x$ with $x + 2$ to shift the graph 2 units to the left.

3. Apply the vertical shift: Subtract 3 from the function to move the graph 3 units down.

4. Identify key points on the parent graph (such as $(-2, -3)$, $(-1, -2)$, $(0, -3)$, etc.) and apply the transformations to these points.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $f(x) = (x + 2)^3 - 3$ is the graph of $y = x^3$ shifted 2 units left and 3 units down. The domain is $(-\infty, \infty)$ and the range is $(-\infty, \infty)$.

Q50. Graph the function $f(x) = 3(x - 2)^2 + 1$

Background

Topic: Transformations of Functions (Quadratic Functions)

This question tests your ability to graph a quadratic function by applying vertical stretching, horizontal shifting, and vertical shifting to the parent function $y = x^2$.

Key Terms and Formulas

• Parent function: $y = x^2$

• Vertical stretch: $a f(x)$ stretches the graph by a factor of $a$ if $a > 1$.

• Horizontal shift: $f(x - h)$ shifts the graph right by $h$ units.

• Vertical shift: $f(x) + k$ shifts the graph up by $k$ units.

Step-by-Step Guidance

1. Start with the basic quadratic function $y = x^2$.

2. Apply the horizontal shift: Replace $x$ with $x - 2$ to shift the graph 2 units to the right.

3. Apply the vertical stretch: Multiply the function by 3 to stretch it vertically.

4. Apply the vertical shift: Add 1 to move the graph 1 unit up.

5. Identify the new vertex and a few key points after all transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $f(x) = 3(x - 2)^2 + 1$ is a parabola opening upwards, stretched vertically by a factor of 3, shifted 2 units right and 1 unit up. The vertex is at $(2, 1)$. The domain is $(-\infty, \infty)$ and the range is $[1, \infty)$.

Q52. Graph the function $g(x) = 3|x + 1| - 3$

Background

Topic: Transformations of Functions (Absolute Value Functions)

This question tests your understanding of how to graph an absolute value function by applying horizontal and vertical shifts, as well as vertical stretching.

Key Terms and Formulas

• Parent function: $y = |x|$

• Vertical stretch: $a|x|$ stretches the graph by a factor of $a$ if $a > 1$.

• Horizontal shift: $|x + h|$ shifts the graph left by $h$ units.

• Vertical shift: $|x| + k$ shifts the graph up by $k$ units (or down if $k$ is negative).

Step-by-Step Guidance

1. Start with the basic absolute value function $y = |x|$.

2. Apply the horizontal shift: Replace $x$ with $x + 1$ to shift the graph 1 unit to the left.

3. Apply the vertical stretch: Multiply the function by 3 to stretch it vertically.

4. Apply the vertical shift: Subtract 3 to move the graph 3 units down.

5. Identify the new vertex and a few key points after all transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $g(x) = 3|x + 1| - 3$ is a V-shaped graph, stretched vertically by a factor of 3, shifted 1 unit left and 3 units down. The vertex is at $(-1, -3)$. The domain is $(-\infty, \infty)$ and the range is $[-3, \infty)$.

Q54. Graph the function $h(x) = \frac{4}{x} + 2$

Background

Topic: Transformations of Rational Functions

This question tests your understanding of how to graph a rational function by applying vertical shifts to the parent function $y = \frac{1}{x}$.

Key Terms and Formulas

• Parent function: $y = \frac{1}{x}$

• Vertical stretch: $a \cdot \frac{1}{x}$ stretches the graph by a factor of $a$.

• Vertical shift: $\frac{1}{x} + k$ shifts the graph up by $k$ units.

Step-by-Step Guidance

1. Start with the basic rational function $y = \frac{1}{x}$.

2. Apply the vertical stretch: Multiply the function by 4 to stretch it vertically.

3. Apply the vertical shift: Add 2 to move the graph 2 units up.

4. Identify the new horizontal asymptote and vertical asymptote after the transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $h(x) = \frac{4}{x} + 2$ is a hyperbola with a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 2$. The domain is $(-\infty, 0) \cup (0, \infty)$ and the range is $(-\infty, 2) \cup (2, \infty)$.

Q55. Graph the function $f(x) = -(x + 1)^3 - 1$

Background

Topic: Transformations of Functions (Cubic Functions)

This question tests your understanding of how to graph a cubic function by applying horizontal and vertical shifts, as well as reflection and vertical shifting.

Key Terms and Formulas

• Parent function: $y = x^3$

• Reflection: $-f(x)$ reflects the graph over the x-axis.

• Horizontal shift: $f(x + h)$ shifts the graph left by $h$ units.

• Vertical shift: $f(x) + k$ shifts the graph up by $k$ units (or down if $k$ is negative).

Step-by-Step Guidance

1. Start with the basic cubic function $y = x^3$.

2. Apply the horizontal shift: Replace $x$ with $x + 1$ to shift the graph 1 unit to the left.

3. Apply the reflection: Multiply the function by -1 to reflect it over the x-axis.

4. Apply the vertical shift: Subtract 1 to move the graph 1 unit down.

5. Identify the new inflection point and a few key points after all transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $f(x) = -(x + 1)^3 - 1$ is the graph of $y = x^3$ shifted 1 unit left, reflected over the x-axis, and shifted 1 unit down. The domain is $(-\infty, \infty)$ and the range is $(-\infty, \infty)$.

Q58. Graph the function $g(x) = 4\sqrt{2 - x}$

Background

Topic: Transformations of Functions (Square Root Functions)

This question tests your understanding of how to graph a square root function by applying horizontal shifts, reflections, and vertical stretching.

Key Terms and Formulas

• Parent function: $y = \sqrt{x}$

• Reflection: $\sqrt{-x}$ reflects the graph over the y-axis.

• Horizontal shift: $\sqrt{a - x}$ shifts the graph and may reflect it depending on the sign of $x$.

• Vertical stretch: $a \sqrt{x}$ stretches the graph vertically by a factor of $a$.

Step-by-Step Guidance

1. Start with the basic square root function $y = \sqrt{x}$.

2. Apply the reflection and horizontal shift: Replace $x$ with $2 - x$ to reflect the graph over the y-axis and shift it horizontally.

3. Apply the vertical stretch: Multiply the function by 4 to stretch it vertically.

4. Determine the domain by setting $2 - x \geq 0$ and solve for $x$.

5. Identify the starting point and a few key points after all transformations.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $g(x) = 4\sqrt{2 - x}$ starts at $x = 2$ and extends to the left. The domain is $(-\infty, 2]$ and the range is $[0, \infty)$.

Q62. Given the graph of $f(x)$, graph $F(x) = f(x) + 3$

Background

Topic: Vertical Shifts of Functions

This question tests your understanding of how to shift a function vertically on its graph.

Key Terms and Formulas

• Vertical shift: $f(x) + k$ shifts the graph up by $k$ units.

Step-by-Step Guidance

1. Start with the given graph of $f(x)$.

2. Shift every point on the graph up by 3 units.

3. Redraw the graph with the new positions of the key points.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $F(x) = f(x) + 3$ is the graph of $f(x)$ shifted up by 3 units. The domain remains the same, but the range increases by 3 units.

Q63. Given the graph of $f(x)$, graph $G(x) = f(x + 2)$

Background

Topic: Horizontal Shifts of Functions

This question tests your understanding of how to shift a function horizontally on its graph.

Key Terms and Formulas

• Horizontal shift: $f(x + h)$ shifts the graph left by $h$ units.

Step-by-Step Guidance

1. Start with the given graph of $f(x)$.

2. Shift every point on the graph left by 2 units.

3. Redraw the graph with the new positions of the key points.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $G(x) = f(x + 2)$ is the graph of $f(x)$ shifted left by 2 units. The domain and range remain the same, but the graph is moved horizontally.

Q64. Given the graph of $f(x)$, graph $P(x) = -f(x)$

Background

Topic: Reflections of Functions

This question tests your understanding of how to reflect a function over the x-axis.

Key Terms and Formulas

• Reflection over x-axis: $-f(x)$ reflects the graph of $f(x)$ over the x-axis.

Step-by-Step Guidance

1. Start with the given graph of $f(x)$.

2. Reflect every point on the graph over the x-axis (change the sign of the y-coordinate).

3. Redraw the graph with the new positions of the key points.

Try solving on your own before revealing the answer!

Final Answer:

The graph of $P(x) = -f(x)$ is the graph of $f(x)$ reflected over the x-axis. The domain remains the same, but the range is reflected about the x-axis.