Q1. Write the two forms of a quadratic equation.

Background

Topic: Quadratic Functions

This question tests your understanding of the two main ways to express a quadratic function: vertex form and general (standard) form.

Key Terms and Formulas

• Vertex Form:

• General Form:

Step-by-Step Guidance

1. Recall that the general form is the expanded version of a quadratic, showing all terms.

2. The vertex form highlights the vertex of the parabola and is useful for graphing.

3. Write both forms clearly, using the correct variables and structure.

Try solving on your own before revealing the answer!

Q2. Match the letters on the graph to their appropriate labels: axis of symmetry, solutions/roots/zeroes, y-intercept, vertex.

Background

Topic: Quadratic Graph Features

This question tests your ability to identify key features of a quadratic graph, such as the axis of symmetry, roots, y-intercept, and vertex.

Key Terms

• Axis of Symmetry: The vertical line that divides the parabola into two mirror images.

• Solutions/Roots/Zeroes: The points where the graph crosses the x-axis.

• Y-intercept: The point where the graph crosses the y-axis.

• Vertex: The lowest or highest point on the parabola.

Step-by-Step Guidance

1. Examine the graph and locate the points labeled a, b, c, and d.

2. Identify which label corresponds to each letter based on their position:

3. Look for the points where the graph crosses the x-axis (roots), y-axis (y-intercept), the turning point (vertex), and the dashed vertical line (axis of symmetry).

4. Assign each label to the correct letter, but stop before stating the final matches.

Try solving on your own before revealing the answer!

Q3. Identify if each quadratic has a minimum or maximum.

Background

Topic: Quadratic Function Properties

This question tests your ability to determine whether a quadratic function opens upward (minimum) or downward (maximum) based on the leading coefficient.

Key Terms

• Minimum: The lowest point of the parabola (vertex) when .

• Maximum: The highest point of the parabola (vertex) when .

Step-by-Step Guidance

1. For each quadratic, identify the leading coefficient ().

2. If , the parabola opens upward and has a minimum.

3. If , the parabola opens downward and has a maximum.

4. Apply this reasoning to each function, but stop before stating the final classification for each.

Try solving on your own before revealing the answer!

Q4. Rewrite in vertex form and state the vertex as an ordered pair.

Background

Topic: Completing the Square

This question tests your ability to convert a quadratic from general form to vertex form by completing the square.

Key Formula

• Vertex Form:

• Completing the Square:

Step-by-Step Guidance

1. Start with .

2. Group the and terms: .

3. Find the value to complete the square: Take half of , square it, and add/subtract as needed.

4. Rewrite the function in the form , but stop before stating the final vertex form and vertex.

Try solving on your own before revealing the answer!

Q5. Rewrite in vertex form and state the vertex as an ordered pair.

Background

Topic: Completing the Square with a Leading Coefficient

This question tests your ability to convert a quadratic with a leading coefficient other than 1 into vertex form.

Key Formula

• Vertex Form:

• Completing the Square: Factor from and terms before completing the square.

Step-by-Step Guidance

1. Start with .

2. Factor out $2x^2x.

3. Complete the square inside the parentheses: Take half of $4$, square it, and adjust the constant term accordingly.

4. Rewrite the function in vertex form, but stop before stating the final vertex form and vertex.

Try solving on your own before revealing the answer!

Q6. Rewrite in vertex form, sketch the function, and state the requested properties: vertex, axis of symmetry, solutions, domain, range.

Background

Topic: Quadratic Function Analysis

This question tests your ability to convert a quadratic to vertex form, graph it, and identify key properties.

Key Formula

• Vertex Form:

• Axis of Symmetry:

• Solutions: Set and solve for

• Domain: All real numbers ()

• Range: Depends on the vertex and direction of opening

Step-by-Step Guidance

1. Start with .

2. Complete the square to rewrite in vertex form.

3. Identify the vertex from the vertex form.

4. Determine the axis of symmetry using .

5. Set and solve for to find the solutions.

6. State the domain and describe how to find the range, but stop before stating the final values.

Try solving on your own before revealing the answer!

Q7. Find the solutions (roots/zeroes) of .

Background

Topic: Solving Quadratic Equations

This question tests your ability to find the roots of a quadratic equation by factoring or using the quadratic formula.

Key Formula

• Quadratic Formula:

• Factoring:

Step-by-Step Guidance

1. Set to find the roots: .

2. Try to factor the quadratic or use the quadratic formula.

3. Identify two numbers that multiply to and add to .

4. Write the factors and solve for , but stop before stating the final solutions.

Try solving on your own before revealing the answer!

Q8. Find the solutions (roots/zeroes) of .

Background

Topic: Solving Quadratic Equations

This question tests your ability to find the roots of a quadratic equation, which may be real or complex.

Key Formula

• Quadratic Formula:

Step-by-Step Guidance

1. Set to find the roots: .

2. Identify , , .

3. Calculate the discriminant: .

4. Plug values into the quadratic formula, but stop before stating the final solutions.

Try solving on your own before revealing the answer!

Q9. A baseball player throws a rock upward from a 112-foot cliff at 96 ft/s. The height is . Answer the following:

Background

Topic: Quadratic Applications – Projectile Motion

This question tests your ability to analyze a quadratic function modeling projectile motion, including finding maximum height and time to hit the ground.

Key Formula

• Vertex: for maximum/minimum

• Maximum height: Plug into

• Time to hit the ocean: Set and solve for

Step-by-Step Guidance

1. For maximum height, use with , .

2. Plug this value into to find the maximum height.

3. To find when the rock hits the ocean, set and solve for using the quadratic formula.

4. Be sure to include units in your answers, but stop before stating the final values.

Try solving on your own before revealing the answer!

Q10. You have 200 feet of fencing. What should the length (L) and width (W) of a rectangular fence be to maximize the area?

Background

Topic: Optimization with Quadratics

This question tests your ability to use quadratic functions to solve optimization problems, specifically maximizing area given a fixed perimeter.

Key Formula

• Perimeter:

• Area:

• Express in terms of and substitute into the area formula.

Step-by-Step Guidance

1. Start with the perimeter equation: .

2. Solve for in terms of : .

3. Substitute into the area formula: .

4. Write the area as a quadratic function of and find the value of that maximizes using the vertex formula, but stop before stating the final values.

Try solving on your own before revealing the answer!