Q1. Solve Linear and Absolute Value Inequalities

Background

Topic: Linear and Absolute Value Inequalities

This question tests your ability to solve inequalities involving linear expressions and absolute values, and to express the solution set using interval notation.

Key Terms and Formulas

• Linear Inequality: An inequality involving a linear expression, such as .

• Absolute Value Inequality: An inequality involving , such as or .

For , the solution is . For , the solution is or .

Step-by-Step Guidance

1. Isolate the absolute value or variable term on one side of the inequality.

2. If dealing with an absolute value, split the inequality into two cases (for or ) and write the corresponding compound inequalities.

3. Solve each resulting linear inequality separately.

4. Express the solution set in interval notation, making sure to check for extraneous solutions if necessary.

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Q2. Find the Distance and Midpoint Between Two Points

Background

Topic: Distance and Midpoint Formulas

This question tests your ability to use the distance and midpoint formulas to find the distance between two points and the coordinates of their midpoint.

Key Terms and Formulas

• Distance Formula:

• Midpoint Formula:

Step-by-Step Guidance

1. Identify the coordinates of the two points: and .

2. Plug the coordinates into the distance formula to set up the calculation for the distance.

3. Plug the coordinates into the midpoint formula to set up the calculation for the midpoint.

4. Simplify the expressions as much as possible before performing the final calculations.

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Q3. Write the Equation of a Circle

Background

Topic: Equation of a Circle

This question tests your understanding of the standard form of the equation of a circle and how to write it given the center and radius.

Key Terms and Formulas

• Standard Form:

• is the center, is the radius.

Step-by-Step Guidance

1. Identify the center and the radius from the problem statement.

2. Substitute , , and into the standard form equation.

3. Simplify the equation as needed, but do not expand unless instructed.

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Q4. Identify and Work with Functions

Background

Topic: Functions and Their Properties

This question tests your understanding of what a function is, how to evaluate functions, and how to determine if a relation is a function.

Key Terms and Formulas

• Function: A relation where each input has exactly one output.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Function Notation:

Step-by-Step Guidance

1. Check if each input value corresponds to only one output value.

2. If given a graph, use the vertical line test to determine if it is a function.

3. To evaluate a function, substitute the given value into the function's formula.

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Q5. Graph and Write Linear Equations

Background

Topic: Linear Equations and Their Graphs

This question tests your ability to write the equation of a line in various forms and to graph it using slope and intercepts.

Key Terms and Formulas

• Slope-Intercept Form:

• Point-Slope Form:

• Slope:

Step-by-Step Guidance

1. Identify the slope and a point on the line (or two points to find the slope).

2. Use the point-slope form to write the equation if you have a point and the slope.

3. Convert to slope-intercept form if needed by solving for .

4. To graph, plot the -intercept and use the slope to find another point.

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Q6. Parent Functions, Piecewise Functions, and Transformations

Background

Topic: Parent Functions, Piecewise Functions, and Transformations

This question tests your knowledge of basic parent functions, how to interpret and graph piecewise functions, and how transformations affect graphs.

Key Terms and Formulas

• Parent Function: The simplest form of a function, e.g., , , .

• Piecewise Function: A function defined by different expressions over different intervals.

• Transformation: Shifts, stretches, compressions, and reflections of graphs.

Step-by-Step Guidance

1. Identify the parent function and its basic graph.

2. For piecewise functions, determine the expression and domain for each piece.

3. Apply transformations (shifts, stretches, reflections) as indicated by the function's formula.

4. Graph each piece on its specified interval, applying the correct transformations.

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Q7. Composition of Functions

Background

Topic: Function Composition

This question tests your ability to compose two functions, or , and to evaluate the result for a given .

Key Terms and Formulas

• Composition:

Step-by-Step Guidance

1. Identify the inner and outer functions in the composition.

2. Substitute the inner function into the outer function wherever the variable appears.

3. Simplify the resulting expression as much as possible.

4. If asked to evaluate at a specific , substitute that value into your composed function.

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Q8. Solving Quadratic, Absolute Value, Rational, Radical, and Quadratic Form Equations

Background

Topic: Solving Various Types of Equations

This question tests your ability to solve equations of different types, including quadratic, absolute value, rational, radical, and equations in quadratic form.

Key Terms and Formulas

• Quadratic Equation:

• Quadratic Formula:

• Absolute Value Equation: has solutions and

• Rational Equation: Involves fractions with variables in the denominator

• Radical Equation: Involves variables under a root

• Quadratic Form: An equation that can be rewritten as a quadratic in terms of another variable

Step-by-Step Guidance

1. Identify the type of equation you are solving.

2. For quadratic equations, use factoring, completing the square, or the quadratic formula as appropriate.

3. For absolute value equations, split into two cases and solve each.

4. For rational equations, find a common denominator and multiply both sides to clear fractions.

5. For radical equations, isolate the radical and then square both sides (be sure to check for extraneous solutions).

6. For equations in quadratic form, make a substitution to rewrite as a quadratic, solve, then substitute back.

Try solving on your own before revealing the answer!