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Evaluate "},{"type":"inlineMath","attrs":{"latex":"f(-1)"}},{"type":"text","marks":[{"type":"bold"}],"text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = 7x^2 - 3x + 2"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Function Evaluation"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to substitute a value into a function and simplify the result."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function: "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" is a rule that assigns each input "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to exactly one output."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitution: Replace "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" with the given value."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the function: "},{"type":"inlineMath","attrs":{"latex":"f(x) = 7x^2 - 3x + 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substitute "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" into the function: "},{"type":"inlineMath","attrs":{"latex":"f(-1) = 7(-1)^2 - 3(-1) + 2"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"(-1)^2"}},{"type":"text","text":" and multiply by 7."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply "},{"type":"inlineMath","attrs":{"latex":"-3"}},{"type":"text","text":" by "},{"type":"inlineMath","attrs":{"latex":"-1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Add all terms together to get the result."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer: "},{"type":"inlineMath","attrs":{"latex":"f(-1) = 12"}}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Substituting "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":" gives "},{"type":"inlineMath","attrs":{"latex":"f(-1) = 7(1) + 3 + 2 = 12"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Determine the domain and range of the relation. Is it a function? Is it one-to-one?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Relations and Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of domain, range, and the definition of a function and one-to-one function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: Set of all possible input values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: Set of all possible output values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function: Each input has exactly one output."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"One-to-one: Each output is paired with only one input."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the set of inputs (domain) and outputs (range) from the relation."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if each input is paired with only one output to determine if it is a function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if each output is paired with only one input to determine if it is one-to-one."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the domain and range as sets or lists."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: Countries Range: Continents Function? Yes One-to-one? No"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Each country is paired with one continent, but some continents may be paired with multiple countries."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. Determine the domain and range of the relation. Is it a function? Is it one-to-one?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Relations and Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question is similar to Q2, testing your ability to identify domain, range, and function properties."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: Set of all possible input values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: Set of all possible output values."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function: Each input has exactly one output."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"One-to-one: Each output is paired with only one input."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the countries as the domain and continents as the range."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if each country is paired with only one continent."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Check if any continent is paired with more than one country."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the domain and range."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: {Belgium, Kenya, Norway, Thailand} Range: {Asia, Africa, Europe} Function? 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Sketch the graph of the equation "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Linear Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to graph a vertical line."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical Line: All points where "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" is constant."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recognize that "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":" is a vertical line."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"On a coordinate plane, plot the line where every point has "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The line will pass through "},{"type":"inlineMath","attrs":{"latex":"(4, y)"}},{"type":"text","text":" for all "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" values."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try sketching the graph before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The graph is a vertical line passing through "},{"type":"inlineMath","attrs":{"latex":"x = 4"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Sketch the graph of the equation "},{"type":"inlineMath","attrs":{"latex":"y = -\\frac{5}{6}x + 4"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Linear Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to graph a line in slope-intercept form."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope-intercept form: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope ("},{"type":"inlineMath","attrs":{"latex":"m"}},{"type":"text","text":"): "},{"type":"inlineMath","attrs":{"latex":"-\\frac{5}{6}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Y-intercept ("},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":"): $4$"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the slope "},{"type":"inlineMath","attrs":{"latex":"m = -\\frac{5}{6}"}},{"type":"text","text":" and y-intercept "},{"type":"inlineMath","attrs":{"latex":"b = 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Plot the y-intercept at "},{"type":"inlineMath","attrs":{"latex":"(0, 4)"}},{"type":"text","text":" on the graph."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"From the y-intercept, use the slope to find another point: go down 5 units and right 6 units."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Draw a straight line through these points."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try sketching the graph before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"The graph is a straight line with slope "},{"type":"inlineMath","attrs":{"latex":"-\\frac{5}{6}"}},{"type":"text","text":" and y-intercept $4$."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. Find the slope-intercept form of the line passing through "},{"type":"inlineMath","attrs":{"latex":"(6, -3)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"(3, 2)"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Linear Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the equation of a line given two points."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope formula: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{y_2 - y_1}{x_2 - x_1}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope-intercept form: "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Label the points: "},{"type":"inlineMath","attrs":{"latex":"(x_1, y_1) = (6, -3)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"(x_2, y_2) = (3, 2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate the slope: "},{"type":"inlineMath","attrs":{"latex":"m = \\frac{2 - (-3)}{3 - 6}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the slope and one point to solve for "},{"type":"inlineMath","attrs":{"latex":"b"}},{"type":"text","text":" in "},{"type":"inlineMath","attrs":{"latex":"y = mx + b"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the final equation in slope-intercept form."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Slope "},{"type":"inlineMath","attrs":{"latex":"m = -1.67"}},{"type":"text","text":", y-intercept "},{"type":"inlineMath","attrs":{"latex":"b = 7"}},{"type":"text","text":"; equation: "},{"type":"inlineMath","attrs":{"latex":"y = -1.67x + 7"}},{"type":"text","text":"."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q7. Solve the inequality "},{"type":"inlineMath","attrs":{"latex":"3(x-3) + 1 > 5(x-2)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and write your answer in interval notation."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Linear Inequalities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve and express solutions to inequalities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Distributive property"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Combining like terms"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Interval notation"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Expand both sides: "},{"type":"inlineMath","attrs":{"latex":"3(x-3) + 1"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"5(x-2)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify each side by distributing and combining like terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Move all terms involving "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to one side and constants to the other."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the solution in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x < 3.5"}},{"type":"text","text":"; interval notation: "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, 3.5)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q8. State the open intervals over which the function is increasing, decreasing, and constant. On what open intervals is the graph continuous?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graph Analysis"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to interpret a graph and describe its behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Increasing: Function values rise as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" increases."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Decreasing: Function values fall as "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" increases."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Constant: Function values stay the same."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Continuous: No breaks or holes in the graph."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Examine the graph and identify intervals where the function rises, falls, or stays constant."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the intervals using open interval notation (e.g., "},{"type":"inlineMath","attrs":{"latex":"(a, b)"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify where the graph is continuous."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try describing the intervals before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Increasing: (intervals where function rises) Decreasing: (intervals where function falls) Constant: (intervals where function is flat) Continuous: (intervals where graph has no breaks)"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q9. Use transformations to sketch the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x) = |x-3| - 4"}},{"type":"text","marks":[{"type":"bold"}],"text":". State the domain and range."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Absolute Value Functions and Transformations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of graph transformations and absolute value functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Absolute value function: "},{"type":"inlineMath","attrs":{"latex":"f(x) = |x|"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal shift: "},{"type":"inlineMath","attrs":{"latex":"x-3"}},{"type":"text","text":" shifts right by 3 units"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical shift: "},{"type":"inlineMath","attrs":{"latex":"-4"}},{"type":"text","text":" shifts down by 4 units"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Start with the parent function "},{"type":"inlineMath","attrs":{"latex":"f(x) = |x|"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply a horizontal shift right by 3 units: "},{"type":"inlineMath","attrs":{"latex":"|x-3|"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply a vertical shift down by 4 units: "},{"type":"inlineMath","attrs":{"latex":"|x-3| - 4"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the resulting graph."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"State the domain and range."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try sketching and stating domain/range before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain: "},{"type":"inlineMath","attrs":{"latex":"(-\\infty, \\infty)"}},{"type":"text","text":" Range: "},{"type":"inlineMath","attrs":{"latex":"[-4, \\infty)"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q10. 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Use completing the square to rewrite "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2x^2 - 12x + 12"}},{"type":"text","marks":[{"type":"bold"}],"text":" for graphing. 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Circle which end behavior best describes the graph of "},{"type":"inlineMath","attrs":{"latex":"f(x) = x^4 + x^3 - x + 1"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: End Behavior of Polynomial Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how the degree and leading coefficient affect end behavior."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree: Highest power of "},{"type":"inlineMath","attrs":{"latex":"x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Leading coefficient: Coefficient of highest degree term"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the degree (4) and leading coefficient (1)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For even degree and positive leading coefficient, both ends go up."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Describe the end behavior as "},{"type":"inlineMath","attrs":{"latex":"x \\to \\pm\\infty"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try describing the end behavior before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"As "},{"type":"inlineMath","attrs":{"latex":"x \\to \\pm\\infty"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"f(x) \\to \\infty"}},{"type":"text","text":" (both ends up)."}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. List all local and absolute extrema of the function defined by the graph. Write answers as "},{"type":"inlineMath","attrs":{"latex":"f(a) = b"}},{"type":"text","marks":[{"type":"bold"}],"text":". Write N/A if none."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Extrema of Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to identify local and absolute maxima and minima from a graph."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Local maximum/minimum: Highest/lowest point in a neighborhood"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Absolute maximum/minimum: Highest/lowest point overall"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Examine the graph for peaks (maxima) and valleys (minima)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify coordinates for local and absolute extrema."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write answers in the form "},{"type":"inlineMath","attrs":{"latex":"f(a) = b"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try identifying extrema before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Local Maxima: "},{"type":"inlineMath","attrs":{"latex":"f(-0.693) = -0.397"}},{"type":"text","text":" Local Minima: "},{"type":"inlineMath","attrs":{"latex":"f(1.443) = -2.833"}},{"type":"text","text":" Absolute Maxima: "},{"type":"inlineMath","attrs":{"latex":"f(0) = 0"}},{"type":"text","text":" Absolute Minima: $f(1.443) = -2.833$"}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. Solve the polynomial equation "},{"type":"inlineMath","attrs":{"latex":"x^3 + x^2 - 4x - 4 = 0"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving Polynomial Equations"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve cubic equations, possibly by factoring or using rational root theorem."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factorization"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational Root Theorem"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Try factoring by grouping or use rational root theorem to find possible roots."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Test possible rational roots (e.g., "},{"type":"inlineMath","attrs":{"latex":"\\pm1, \\pm2, \\pm4"}},{"type":"text","text":")."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Once a root is found, use synthetic or long division to reduce the cubic to a quadratic."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Solve the resulting quadratic equation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try factoring and solving before revealing the answer!"}]},{"type":"collapsible","content":[{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Final Answer:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Roots: "},{"type":"inlineMath","attrs":{"latex":"x = -1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = -2"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x = 2"}}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. 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Find "