Q1. Determine whether the given function is linear or nonlinear. If linear, determine the slope.

Background

Topic: Linear and Nonlinear Functions

This question tests your ability to identify whether a set of points represents a linear function and, if so, to find the slope.

Key Terms and Formulas:

• Linear function: A function whose graph is a straight line.

• Slope (): The rate of change between two points, calculated as .

Step-by-Step Guidance

1. List the given points and check if the change in divided by the change in is constant for all consecutive pairs.

2. Calculate the slope between each pair of points using .

3. If the slope is the same for all pairs, the function is linear; otherwise, it is nonlinear.

4. Set up the calculation for the slope using the first two points.

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Q2. A semicircle of radius is inscribed in a rectangle so that the diameter of the semicircle is the length of the rectangle.

Background

Topic: Functions and Geometry

This question tests your ability to express geometric quantities (area and perimeter) as functions of a variable.

Key Terms and Formulas:

• Area of a rectangle:

• Diameter of a semicircle:

Step-by-Step Guidance

1. Identify the length of the rectangle as the diameter of the semicircle, .

2. Express the width of the rectangle in terms of or as given.

3. Set up the area formula using the expressions found.

4. Write the area function in terms of using the given relationship for .

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Q2b. Express the perimeter of the rectangle as a function of .

Background

Topic: Functions and Geometry

This question tests your ability to express the perimeter of a rectangle as a function of a variable.

Key Terms and Formulas:

• Perimeter of a rectangle:

• Diameter of semicircle:

Step-by-Step Guidance

1. Identify the length and width of the rectangle in terms of or .

2. Set up the perimeter formula .

3. Substitute the expressions for length and width using .

4. Write the perimeter function in terms of .

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Q3. Suppose that the quantity supplied and quantity demanded of smoothies at a food festival are given by the following functions, where is the price of a smoothie:

Background

Topic: Systems of Equations and Applications

This question tests your ability to solve systems of linear equations to find equilibrium price and quantity, and analyze surplus/shortage.

Key Terms and Formulas:

• Equilibrium: Occurs when .

• Surplus: When supply exceeds demand.

• Shortage: When demand exceeds supply.

Step-by-Step Guidance

1. Set to find the equilibrium price.

2. Solve for by equating the two functions and isolating $p$.

3. Once is found, substitute back into either function to find the equilibrium quantity.

4. For surplus/shortage, plug into both and and compare the values.

5. Set up the equation for a shortage of 100 smoothies: .

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Q4. A right triangle has one vertex on the graph of , , at , another at the origin, and a third on the positive y-axis at . Express the area of the triangle as a function of .

Background

Topic: Functions and Geometry

This question tests your ability to express the area of a triangle as a function of a variable using coordinates and geometric relationships.

Key Terms and Formulas:

• Area of a triangle:

• Coordinates: , ,

Step-by-Step Guidance

1. Identify the base and height of the triangle using the coordinates.

2. Express in terms of using .

3. Set up the area formula .

4. Substitute with in the area formula.

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Q5. An island is 11 miles from the nearest point on a straight shoreline. A town is 19 miles down the shore from $P$.

Background

Topic: Functions and Applications (Distance, Rate, Time)

This question tests your ability to model travel time as a function of distance using rates and the Pythagorean theorem.

Key Terms and Formulas:

• Distance formula:

• Time formula:

Step-by-Step Guidance

1. Let be the distance from to where the person lands the boat.

2. Use the Pythagorean theorem to find the rowing distance: .

3. Rowing speed is $6 mph.

4. Set up the total time function: .

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Q5b. Find the domain of this function.

Background

Topic: Domain of Functions

This question tests your ability to determine the domain of a function based on physical constraints.

Key Terms and Formulas:

• Domain: The set of all possible input values () for which the function is defined.

Step-by-Step Guidance

1. Consider the physical meaning: must be between $0 miles (from to the town).

2. Check for any restrictions from the square root or rates.

3. Write the domain in interval notation.

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Q5c. Determine how long it will take to travel from the island to town if the person lands the boat 8 miles from .

Background

Topic: Functions and Applications (Distance, Rate, Time)

This question tests your ability to evaluate a function at a specific value.

Key Terms and Formulas:

• Time function:

Step-by-Step Guidance

1. Substitute into the time function.

2. Calculate the rowing distance: .

3. Divide the rowing distance by $6$ mph.

4. Calculate the walking distance: miles.

5. Divide the walking distance by $3$ mph.

6. Add the two times together for the total travel time.

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Q6. On a parcel-shipping service, packages up to 5 pounds ship for free, and any weight over 5 pounds is charged C(w) = 2.00(w - 5)w$ is the total weight of the package in pounds.

Background

Topic: Piecewise Functions and Applications

This question tests your ability to interpret and analyze piecewise functions in real-world contexts.

Key Terms and Formulas:

• Piecewise function: A function defined by different expressions for different intervals.

• Cost function: for

Step-by-Step Guidance

1. Identify the domain: For which values of is the function defined?

2. Write the domain in interval notation.

3. For , substitute into the cost function.

4. Calculate .

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Q7. On a separate piece of graph paper, graph the domain and range (interval notation).

Background

Topic: Domain and Range of Functions

This question tests your ability to graphically represent the domain and range of a function and express them in interval notation.

Key Terms and Formulas:

• Domain: Set of all possible input values ().

• Range: Set of all possible output values ().

• Interval notation: e.g., ,

Step-by-Step Guidance

1. Identify the domain and range for the function you are graphing.

2. Write the domain and range in interval notation.

3. Sketch the graph, marking the domain and range clearly.

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