Study Guide: Transformations of Graphs in College Algebra

Q1. Sketch graphs of basic functions discussed in lecture. Make sure you plot at least 3 accurate coordinates in each graph and that you have the correct shape.

Background

Topic: Parent Functions and Their Graphs

This question is testing your understanding of the basic shapes and key points of common parent functions in algebra, such as linear, quadratic, absolute value, cubic, square root, and reciprocal functions.

Key Terms and Formulas

• Parent function: The simplest form of a function type (e.g., $f(x) = x^2$ for quadratics).

• Linear function: $f(x) = x$

• Quadratic function: $f(x) = x^2$

• Absolute value function: $f(x) = |x|$

• Cubic function: $f(x) = x^3$

• Square root function: $f(x) = \sqrt{x}$

• Reciprocal function: $f(x) = \frac{1}{x}$

Step-by-Step Guidance

1. Identify each parent function you are asked to graph (e.g., linear, quadratic, etc.).

2. For each function, choose at least three $x$-values (for example, $x = -1, 0, 1$) and calculate the corresponding $y$-values using the function's formula.

3. Plot these points on the coordinate plane. Make sure to use the correct scale and label your axes.

4. Draw the general shape of the graph that fits through these points, matching the known shape of the parent function (e.g., a parabola for $x^2$, a V-shape for $|x|$).

5. Check that your graph matches the expected behavior for large positive and negative $x$-values (end behavior).

Try solving on your own before revealing the answer!

Final Answer:

Each parent function should be graphed as follows (with at least three accurate points):

• Linear ($f(x) = x$): Points: $(-1, -1)$, $(0, 0)$, $(1, 1)$. Straight line through the origin.

• Quadratic ($f(x) = x^2$): Points: $(-1, 1)$, $(0, 0)$, $(1, 1)$. Parabola opening upwards.

• Absolute Value ($f(x) = |x|$): Points: $(-1, 1)$, $(0, 0)$, $(1, 1)$. V-shape with vertex at the origin.

• Cubic ($f(x) = x^3$): Points: $(-1, -1)$, $(0, 0)$, $(1, 1)$. S-shaped curve through the origin.

• Square Root ($f(x) = \sqrt{x}$): Points: $(0, 0)$, $(1, 1)$, $(4, 2)$. Starts at the origin and curves upward to the right.

• Reciprocal ($f(x) = \frac{1}{x}$): Points: $(-1, -1)$, $(1, 1)$, $(2, 0.5)$. Two branches, one in the first quadrant and one in the third, with asymptotes at $x=0$ and $y=0$.

Be sure to plot these on the provided coordinate grids for accuracy.

Q2. For each function, name the parent function and describe the transformation in words.

Background

Topic: Identifying Parent Functions and Describing Transformations

This question tests your ability to recognize the basic function underlying a more complex function and to describe how it has been shifted, stretched, reflected, or otherwise transformed.

Key Terms and Formulas

• Parent function: The simplest form of a function type.

• Transformation: Any change to the graph, such as shifting, stretching, compressing, or reflecting.

• Common transformations include:

  • Vertical shift: $f(x) + c$ or $f(x) - c$

  • Horizontal shift: $f(x + c)$ or $f(x - c)$

  • Reflection: $-f(x)$ (over $x$-axis), $f(-x)$ (over $y$-axis)

  • Vertical stretch/shrink: $a f(x)$ where $|a| > 1$ (stretch), $0 < |a| < 1$ (shrink)

Step-by-Step Guidance

1. For each function, identify the parent function by removing all constants and coefficients that represent transformations.

2. Analyze the function to determine what transformations have been applied (e.g., shifts, stretches, reflections).

3. Describe each transformation in words, specifying the direction and magnitude (e.g., "shifted up 5 units," "reflected over the $x$-axis").

4. Write your answer in a clear format: Parent function: __; Transformation(s): __.

Try solving on your own before revealing the answer!

Final Answer:

• $f(x) = x^2 + 5$ Parent function: $x^2$ Transformation: Shifted up 5 units.

• $T(x) = (x - 3)^2$ Parent function: $x^2$ Transformation: Shifted right 3 units.

• $h(x) = |x + 4|$ Parent function: $|x|$ Transformation: Shifted left 4 units.

• $c(x) = -2\sqrt{x}$ Parent function: $\sqrt{x}$ Transformation: Reflected over $x$-axis and vertically stretched by 2.

• $J(x) = \frac{1}{3}x^3$ Parent function: $x^3$ Transformation: Vertically shrunk by a factor of $\frac{1}{3}$.

• $H(x) = (x + 1)^2 - 6$ Parent function: $x^2$ Transformation: Shifted left 1 unit and down 6 units.

• $\mu(x) = -|x - 2|$ Parent function: $|x|$ Transformation: Shifted right 2 units and reflected over $x$-axis.

• $t(x) = 3x + 2$ Parent function: $x$ Transformation: Vertically stretched by 3 and shifted up 2 units.

Q3. For the given functions, state the parent function and the transformations applied, then sketch the graph.

Background

Topic: Graphing Transformed Functions

This question tests your ability to identify the parent function, describe the transformations, and sketch the resulting graph.

Key Terms and Formulas

• Parent function: The basic function before any transformations.

• Transformation: Shifts, stretches, compressions, and reflections applied to the parent function.

Step-by-Step Guidance

1. For each function, identify the parent function (e.g., $\sqrt{x}$, $x^2$, $|x|$).

2. Determine the transformations (shifts, stretches, reflections) by comparing the given function to its parent.

3. Describe the transformations in words.

4. Sketch the graph by applying the transformations to the parent function's graph. Plot at least three key points.

5. Label the axes and key points for clarity.

Try solving on your own before revealing the answer!

Final Answer:

• $T(x) = \sqrt{x} + 5$ Parent function: $\sqrt{x}$ Transformation: Shifted up 5 units.

• $g(x) = (x + 3)^2 - 1$ Parent function: $x^2$ Transformation: Shifted left 3 units and down 1 unit.

• $J(x) = -|x - 5| + 6$ Parent function: $|x|$ Transformation: Shifted right 5 units, reflected over $x$-axis, and shifted up 6 units.

• $h(x) = 2x^3$ Parent function: $x^3$ Transformation: Vertically stretched by 2.

Sketch each graph on a coordinate grid, applying the transformations to the parent function's shape and key points.

Q4. A city’s water consumption $W(t)$ (in million gallons per day) peaks at 180 in summer and drops to 90 in winter. The function $W(t)$ models an average year. Due to a multi-year drought, authorities estimate that next year’s consumption will run 25 million gallons per day higher than average at every point in the year.

Background

Topic: Function Transformations in Real-World Contexts

This question tests your ability to apply vertical shifts to a function in a real-world scenario.

Key Terms and Formulas

• Vertical shift: Adding or subtracting a constant to a function shifts its graph up or down.

• Transformed function: $f(x) + c$ shifts the graph up by $c$ units.

Step-by-Step Guidance

1. Recognize that the new function should be $W(t)$ shifted up by 25 units to represent the increased consumption.

2. Write the new function as $R(t) = W(t) + 25$.

3. To answer if the city will have enough water, determine the new peak consumption by adding 25 to the original peak (180).

4. Compare the new peak to the reservoir capacity (210 million gallons).

Try solving on your own before revealing the answer!

Final Answer:

a. The function for next year’s water usage is $R(t) = W(t) + 25$.

b. The new peak is $180 + 25 = 205$ million gallons per day, which is less than the reservoir capacity of 210. So, the city will have enough water next summer.

Q5. An audiologist records the hearing threshold curve $A(f)$ for a healthy young adult, measuring the softest sound (in decibels) audible at each frequency $f$ (in Hz). The curve $B(f)$ is used for a middle-aged patient with noise-induced hearing loss. At every frequency, the patient needs sounds to be louder to hear.

Background

Topic: Interpreting Function Transformations in Context

This question tests your understanding of horizontal shifts and how they affect real-world data.

Key Terms and Formulas

• Horizontal shift: $f(x - c)$ shifts the graph right by $c$ units; $f(x + c)$ shifts left by $c$ units.

Step-by-Step Guidance

1. Interpret $B(f) = A(f - 500)$ as a horizontal shift of the hearing threshold curve.

2. Describe in words what this means for the patient's hearing at each frequency.

3. For the second part, use the transformation to find $B(1000) = A(500)$.

4. Given that $A(1000) = 10$ dB, determine what $A(500)$ represents and how it relates to the patient's threshold at 1000 Hz.

Try solving on your own before revealing the answer!

Final Answer:

a. $B(f) = A(f - 500)$ means the patient's hearing threshold at frequency $f$ is the same as a healthy adult's threshold at a frequency 500 Hz lower. This suggests the patient hears as well at $f$ as a healthy adult does at $f - 500$ Hz, indicating a loss of sensitivity at higher frequencies.

b. The patient can hear a 1000 Hz tone at the decibel level required for a healthy adult to hear a 500 Hz tone. If $A(500)$ is, for example, 15 dB, then the patient needs 15 dB at 1000 Hz.

Q6. A physiologist models the force $F(v)$ a muscle can exert as a function of its contraction speed $v$. For a trained athlete, all muscle forces are roughly 1.8 times those of an untrained person at the same speed.

Background

Topic: Vertical Stretch Transformations in Real-World Contexts

This question tests your ability to apply vertical stretches and shrinks to functions in a real-world scenario.

Key Terms and Formulas

• Vertical stretch: $a f(x)$ multiplies all output values by $a$.

• Vertical shrink: $0 < a < 1$.

Step-by-Step Guidance

1. Write the athlete's force function as $g(v) = 1.8 F(v)$.

2. To find the athlete's force at a given speed, multiply the untrained person's force at that speed by 1.8.

3. For the injured athlete, write $c(v) = 0.6 g(v)$ to represent 60% of the normal force.

Try solving on your own before revealing the answer!

Final Answer:

a. $g(v) = 1.8 F(v)$

b. If $F(2) = 200$ N, then $g(2) = 1.8 \times 200 = 360$ N.

c. $c(v) = 0.6 g(v) = 0.6 \times 1.8 F(v) = 1.08 F(v)$

Q7. The price per unit of a product is $P$, and the number of units is $Q$. The supply function is given by $P = \frac{2}{3}Q$ and the demand function is given by $P = \frac{4000}{Q} - 40$.

Background

Topic: Identifying and Transforming Rational and Linear Functions

This question tests your ability to identify parent functions and describe how they are transformed in the context of supply and demand.

Key Terms and Formulas

• Linear function: $f(x) = x$

• Rational function: $f(x) = \frac{1}{x}$

• Transformation: Multiplying, adding, or subtracting constants changes the slope, vertical stretch, or shifts the graph.

Step-by-Step Guidance

1. For the supply function, identify the parent function ($P = Q$) and describe the effect of multiplying by $\frac{2}{3}$.

2. For the demand function, identify the parent function ($P = \frac{1}{Q}$) and describe the effect of multiplying by 4000 and subtracting 40.

3. Describe each transformation in words (e.g., vertical stretch, vertical shift).

Try solving on your own before revealing the answer!

Final Answer:

a. Parent function: $P = Q$. Transformation: Vertically shrunk by a factor of $\frac{2}{3}$.

b. Parent function: $P = \frac{1}{Q}$. Transformation: Vertically stretched by 4000 and shifted down by 40 units.