Q1. At what time does the ball reach maximum height? What is the maximum height?

Background

Topic: Quadratic Optimization

This question tests your understanding of how to find the maximum value of a quadratic function, which models the height of a projectile.

Key Terms and Formulas:

• Vertex of a parabola: The maximum or minimum point of a quadratic function .

• Formula for the vertex:

Step-by-Step Guidance

1. Identify the coefficients in the quadratic equation: . Here, , , .

2. Recall that the maximum height occurs at the vertex of the parabola, since is negative (the parabola opens downward).

3. Use the vertex formula to find the time when the ball reaches maximum height.

4. Once you have the value of , substitute it back into to find the maximum height.

Try solving on your own before revealing the answer!

Q2. Find the maximum profit and the number of items that must be sold to achieve it.

Background

Topic: Quadratic Optimization in Business Applications

This question asks you to find the maximum value of a quadratic profit function and the input value that produces it.

Key Terms and Formulas:

• Profit function:

• Vertex formula:

Step-by-Step Guidance

1. Identify the coefficients: , , .

2. Since is negative, the parabola opens downward, so the vertex gives the maximum profit.

3. Use to find the number of items that maximizes profit.

4. Plug this value of into to find the maximum profit.

Try solving on your own before revealing the answer!

Q3. What is the maximum area that can be enclosed with 200 yards of fencing? What are the dimensions of the fence that give maximum area?

Background

Topic: Optimization with Quadratics and Geometry

This question tests your ability to use quadratic functions and geometric formulas to maximize area given a fixed perimeter.

Key Terms and Formulas:

• Perimeter of rectangle:

• Area of rectangle:

Step-by-Step Guidance

1. Set up the perimeter equation: .

2. Solve for one variable in terms of the other, e.g., .

3. Substitute into the area formula: .

4. Expand to get a quadratic in : .

5. Find the value of that maximizes using the vertex formula .

Try solving on your own before revealing the answer!

Q4. (a) At which times was the skateboarder at the top of the ramp (height = 0)? (b) At what time was he at the bottom of the ramp? (c) What is the height at the bottom of the ramp? Does this number make sense?

Background

Topic: Quadratic Equations and Vertex Analysis

This question involves finding the zeros and vertex of a quadratic function to analyze the motion of a skateboarder.

Key Terms and Formulas:

• Quadratic function:

• Zeros: Solve

• Vertex:

Step-by-Step Guidance

1. Set and solve for to find when the height is zero (top of the ramp).

2. Find the vertex using to determine the time at the bottom of the ramp.

3. Plug the vertex value into to find the height at the bottom.

4. Consider whether the height at the bottom makes sense in the context of the ramp.

Try solving on your own before revealing the answer!

Q5. For the polynomial :

Background

Topic: Higher-Degree Polynomial Analysis

This question tests your ability to analyze the end behavior, find zeros and their multiplicities, and sketch the graph of a polynomial.

Key Terms and Formulas:

• Zero: Value of where

• Multiplicity: The exponent of the factor corresponding to each zero

• End behavior: Determined by the leading term

Step-by-Step Guidance

1. Identify the zeros: (multiplicity 2), (multiplicity 3).

2. Determine the end behavior by considering the degree and leading coefficient.

3. Sketch the graph, noting how the graph behaves at each zero based on multiplicity.

Try sketching and analyzing before revealing the answer!

Q6. For the polynomial :

Background

Topic: Higher-Degree Polynomial Analysis

This question asks you to analyze zeros, multiplicities, and end behavior for a more complex polynomial.

Key Terms and Formulas:

• Zero: Value of where

• Multiplicity: The exponent of the factor corresponding to each zero

• End behavior: Determined by the degree and sign of the leading coefficient

Step-by-Step Guidance

1. Find the zeros: (multiplicity 2), (multiplicity 1), (multiplicity 3).

2. Determine the degree and leading coefficient to analyze end behavior.

3. Sketch the graph, noting how the graph behaves at each zero based on multiplicity.

Try sketching and analyzing before revealing the answer!

Q7. For the polynomial :

Background

Topic: Higher-Degree Polynomial Analysis

This question tests your ability to factor, find zeros and multiplicities, and analyze end behavior.

Key Terms and Formulas:

• Factor as

• Zero: Value of where

• Multiplicity: The exponent of the factor corresponding to each zero

Step-by-Step Guidance

1. Factor to get all zeros: (multiplicity 5), (multiplicity 1).

2. Determine the degree and leading coefficient for end behavior.

3. Sketch the graph, noting behavior at each zero.

Try sketching and analyzing before revealing the answer!

Q8. For the polynomial :

Background

Topic: Higher-Degree Polynomial Analysis

This question asks you to analyze zeros, multiplicities, and end behavior for a polynomial with multiple repeated factors.

Key Terms and Formulas:

• Zero: Value of where

• Multiplicity: The exponent of the factor corresponding to each zero

• End behavior: Determined by degree and sign of leading coefficient

Step-by-Step Guidance

1. Identify zeros: (multiplicity 1), (multiplicity 2), (multiplicity 2).

2. Determine degree and leading coefficient for end behavior.

3. Sketch the graph, noting behavior at each zero.

Try sketching and analyzing before revealing the answer!

Q9. Solve and analyze :

Background

Topic: Solving and Analyzing Cubic Polynomials

This question tests your ability to solve cubic equations, find zeros and their multiplicities, and analyze end behavior.

Key Terms and Formulas:

• Factorization techniques for cubic polynomials

• Multiplicity: How many times a zero occurs

• End behavior: Based on degree and leading coefficient

Step-by-Step Guidance

1. Set and attempt to factor or use rational root theorem to find zeros.

2. Determine multiplicities for each zero.

3. Analyze end behavior based on the degree and leading coefficient.

4. Sketch the graph, labeling zeros and their multiplicities.

Try solving and sketching before revealing the answer!

Q10. Solve and analyze :

Background

Topic: Solving and Analyzing Quartic Polynomials

This question tests your ability to factor quartic polynomials, find zeros and their multiplicities, and analyze end behavior.

Key Terms and Formulas:

• Factorization techniques for quartic polynomials

• Multiplicity: How many times a zero occurs

• End behavior: Based on degree and leading coefficient

Step-by-Step Guidance

1. Set and factor the quartic polynomial to find zeros.

2. Determine multiplicities for each zero.

3. Analyze end behavior based on the degree and leading coefficient.

4. Sketch the graph, labeling zeros and their multiplicities.

Try solving and sketching before revealing the answer!