Precalculus Study Guide: Absolute Value Equations & Library of Functions

Q1. Solve the equation: $|3x| = 5$

Background

Topic: Absolute Value Equations

This question tests your understanding of how to solve equations involving absolute values, which often have two possible solutions.

Key Terms and Formulas:

• Absolute value: $|a|$ is the distance of $a$ from zero, always non-negative.

• General rule: $|A| = B$ means $A = B$ or $A = -B$ (if $B \geq 0$).

Step-by-Step Guidance

1. Recognize that $|3x| = 5$ means $3x = 5$ or $3x = -5$.

2. Set up two separate equations: $3x = 5$ and $3x = -5$.

3. To solve for $x$, divide both sides of each equation by $3$.

Try solving on your own before revealing the answer!

Final Answer: $x = \frac{5}{3}$ or $x = -\frac{5}{3}$

Both values satisfy the original absolute value equation.

Q2. Solve the equation: $|x + 2| = 10$

Background

Topic: Absolute Value Equations

This question tests your ability to solve for $x$ when the absolute value is set equal to a positive number.

Key Terms and Formulas:

• Absolute value property: $|A| = B$ implies $A = B$ or $A = -B$.

Step-by-Step Guidance

1. Rewrite $|x + 2| = 10$ as two equations: $x + 2 = 10$ and $x + 2 = -10$.

2. Subtract $2$from both sides of each equation to isolate$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 8$ or $x = -12$

Both solutions are valid for the absolute value equation.

Q3. Solve the equation: $| -x + 1| = 11$

Background

Topic: Absolute Value Equations

This question tests your ability to solve for $x$ when the absolute value contains a linear expression.

Key Terms and Formulas:

• Absolute value property: $|A| = B$ means $A = B$ or $A = -B$.

Step-by-Step Guidance

1. Set up two equations: $-x + 1 = 11$ and $-x + 1 = -11$.

2. Subtract $1$ from both sides of each equation.

3. Multiply both sides by $-1$ to solve for $x$.

Try solving on your own before revealing the answer!

Final Answer: $x = -10$ or $x = 12$

Both values satisfy the original equation.

Q4. Solve the equation: $|x - 5| = 3$

Background

Topic: Absolute Value Equations

This question tests your ability to solve for $x$ when the absolute value is set equal to a positive number.

Key Terms and Formulas:

• Absolute value property: $|A| = B$ means $A = B$ or $A = -B$.

Step-by-Step Guidance

1. Set up two equations: $x - 5 = 3$ and $x - 5 = -3$.

2. Add $5$to both sides of each equation to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 8$ or $x = 2$

Both values are solutions to the absolute value equation.

Q5. Solve the equation: $|2x + 5| = 14$

Background

Topic: Absolute Value Equations

This question tests your ability to solve for $x$ when the absolute value is set equal to a positive number.

Key Terms and Formulas:

• Absolute value property: $|A| = B$ means $A = B$ or $A = -B$.

Step-by-Step Guidance

1. Set up two equations: $2x + 5 = 14$ and $2x + 5 = -14$.

2. Subtract $5$ from both sides of each equation.

3. Divide both sides by $2$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 4.5$ or $x = -9.5$

Both values satisfy the original equation.

Q6. Solve the equation: $|8x - 2| = 7$

Background

Topic: Absolute Value Equations

This question tests your ability to solve for $x$ when the absolute value is set equal to a positive number.

Key Terms and Formulas:

• Absolute value property: $|A| = B$ means $A = B$ or $A = -B$.

Step-by-Step Guidance

1. Set up two equations: $8x - 2 = 7$ and $8x - 2 = -7$.

2. Add $2$ to both sides of each equation.

3. Divide both sides by $8$to solve for$x$.

Try solving on your own before revealing the answer!

Final Answer: $x = 1.125$ or $x = -0.625$

Both values are solutions to the absolute value equation.

Q7. For $f(x) = 3(x-1)^2 + 1$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to recognize parent functions and describe how the function is transformed (stretched, shifted, etc.).

Key Terms and Formulas:

• Parent function: The basic form, e.g., $f(x) = x^2$.

• Transformations: Vertical/horizontal shifts, stretches, compressions.

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x^2$.

2. Notice the coefficient $3$in front: vertical stretch by a factor of $3$.

3. $(x-1)$ inside the squared term: horizontal shift $1$ unit right.

4. $+1$ outside: vertical shift $1$ unit up.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x^2$

• Vertical stretch by $3$

• Horizontal shift $1$ unit right

• Vertical shift $1$ unit up

Q8. For $f(x) = \frac{1}{2}x^2 - 10$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify the parent function and describe vertical stretches/compressions and shifts.

Key Terms and Formulas:

• Parent function: $f(x) = x^2$

• Vertical compression: coefficient less than $1$

• Vertical shift: constant added/subtracted outside

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x^2$.

2. Coefficient $\frac{1}{2}$: vertical compression (parabola is wider).

3. $-10$ outside: vertical shift $10$ units down.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x^2$

• Vertical compression by $\frac{1}{2}$

• Vertical shift $10$ units down

Q9. For $f(x) = \frac{1}{2}(x-10)^2$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify horizontal shifts and vertical compressions.

Key Terms and Formulas:

• Parent function: $f(x) = x^2$

• Horizontal shift: $(x-h)$ moves graph $h$ units right

• Vertical compression: coefficient less than $1$

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x^2$.

2. Coefficient $\frac{1}{2}$: vertical compression.

3. $(x-10)$: horizontal shift $10$ units right.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x^2$

• Vertical compression by $\frac{1}{2}$

• Horizontal shift $10$ units right

Q10. For $f(x) = e^{x-1}$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify exponential parent functions and horizontal shifts.

Key Terms and Formulas:

• Parent function: $f(x) = e^x$

• Horizontal shift: $(x-h)$ moves graph $h$ units right

Step-by-Step Guidance

1. Identify the parent function: $f(x) = e^x$.

2. $(x-1)$ in exponent: horizontal shift $1$ unit right.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = e^x$

• Horizontal shift $1$ unit right

Q11. For $f(x) = \frac{1}{x-1}$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify rational parent functions and horizontal shifts.

Key Terms and Formulas:

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

• Horizontal shift: $(x-h)$ moves graph $h$ units right

Step-by-Step Guidance

1. Identify the parent function: $f(x) = \frac{1}{x}$.

2. $(x-1)$ in denominator: horizontal shift $1$ unit right.

Try describing the transformations before revealing the answer!

Final Answer:

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

• Horizontal shift $1$ unit right

Q12. For $f(x) = \frac{1}{x-1}$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify rational parent functions and horizontal shifts.

Key Terms and Formulas:

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

• Horizontal shift: $(x-h)$ moves graph $h$ units right

Step-by-Step Guidance

1. Identify the parent function: $f(x) = \frac{1}{x}$.

2. $(x-1)$ in denominator: horizontal shift $1$ unit right.

Try describing the transformations before revealing the answer!

Final Answer:

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

• Horizontal shift $1$ unit right

Q13. For $f(x) = 2x - 3$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify linear parent functions and describe vertical stretches and shifts.

Key Terms and Formulas:

• Parent function: $f(x) = x$

• Vertical stretch: coefficient greater than $1$

• Vertical shift: constant added/subtracted outside

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x$.

2. Coefficient $2$: vertical stretch by $2$.

3. $-3$ outside: vertical shift $3$ units down.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x$

• Vertical stretch by $2$

• Vertical shift $3$ units down

Q14. For $f(x) = 2x - 3$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify linear parent functions and describe vertical stretches and shifts.

Key Terms and Formulas:

• Parent function: $f(x) = x$

• Vertical stretch: coefficient greater than $1$

• Vertical shift: constant added/subtracted outside

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x$.

2. Coefficient $2$: vertical stretch by $2$.

3. $-3$ outside: vertical shift $3$ units down.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x$

• Vertical stretch by $2$

• Vertical shift $3$ units down

Q15. For $f(x) = x^3 + 6$, identify the parent function and all transformations.

Background

Topic: Library of Functions & Transformations

This question tests your ability to identify cubic parent functions and vertical shifts.

Key Terms and Formulas:

• Parent function: $f(x) = x^3$

• Vertical shift: constant added/subtracted outside

Step-by-Step Guidance

1. Identify the parent function: $f(x) = x^3$.

2. $+6$ outside: vertical shift $6$ units up.

Try describing the transformations before revealing the answer!

Final Answer:

Parent function: $f(x) = x^3$

• Vertical shift $6$ units up

Q16. Analyze the function $f(x) = x^2 + 6x$ in terms of shifts relative to $f(x) = x^2$.

Background

Topic: Quadratic Functions & Transformations

This question tests your understanding of how adding terms to a quadratic function affects its graph, and how to correctly describe vertical and horizontal shifts.

Key Terms and Formulas:

• Parent function: $f(x) = x^2$

• Vertical shift: $f(x) = x^2 + k$ shifts up/down by $k$

• Horizontal shift: $f(x) = (x + h)^2$ shifts left/right by $h$

• Completing the square: $x^2 + 6x = (x + 3)^2 - 9$

Step-by-Step Guidance

1. Compare $f(x) = x^2 + 6x$ to $f(x) = x^2$.

2. Notice that $6x$ is not a simple vertical or horizontal shift; it changes the shape and position.

3. Complete the square: rewrite $x^2 + 6x$ as $(x + 3)^2 - 9$.

4. Now, identify the transformations: horizontal shift $3$units left, vertical shift $9$ units down.

Try identifying which student is correct and write the shifted equations before revealing the answer!

Final Answer:

Neither student is fully correct. $f(x) = x^2 + 6x$ is not a simple vertical or horizontal shift of $x^2$.

After completing the square: $f(x) = (x + 3)^2 - 9$

• To shift $f(x) = x^2$ up by $6$:$f(x) = x^2 + 6$

• To shift $f(x) = x^2$ left by $6$:$f(x) = (x + 6)^2$

Graphing $f(x) = x^2 + 6x$ shows the vertex at $(-3, -9)$, not simply shifted up or left by $6$.

Bonus: Create a study sheet or resource with pictures of each function in the library of functions.

Background

Topic: Library of Functions & Key Features

This task encourages you to visually organize the parent functions and their key features, such as intercepts and asymptotes.

Key Terms and Formulas:

• Parent functions: $f(x) = x$, $f(x) = x^2$, $f(x) = x^3$, $f(x) = \sqrt{x}$, $f(x) = |x|$, $f(x) = \frac{1}{x}$, $f(x) = e^x$

• Key features: intercepts, asymptotes, domain, range

Step-by-Step Guidance

1. List each parent function and sketch its graph (on paper or digitally).

2. Label important features: x-intercepts, y-intercepts, asymptotes, domain, and range.

3. Use color or arrows to highlight transformations if you add them.

Try making your own study sheet before revealing the answer!

Final Answer:

For your study sheet, include:

• $f(x) = x$ (linear): y-intercept at $0$, domain and range are all real numbers.

• $f(x) = x^2$ (quadratic): vertex at $(0,0)$, domain all real, range $y \geq 0$.

• $f(x) = x^3$ (cubic): inflection at $(0,0)$, domain and range all real.

• $f(x) = \sqrt{x}$ (square root): starts at $(0,0)$, domain $x \geq 0$, range $y \geq 0$.

• $f(x) = |x|$ (absolute value): vertex at $(0,0)$, domain all real, range $y \geq 0$.

• $f(x) = \frac{1}{x}$ (rational): vertical and horizontal asymptotes at $x=0$ and $y=0$.

• $f(x) = e^x$ (exponential): y-intercept at $1$, horizontal asymptote at$y=0$.

Label all features and transformations for each function. Use this as a reference for future problems!