Q1. Write the two forms of a quadratic equation.

Background

Topic: Quadratic Functions

This question is testing 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 of a quadratic function is written as , where , , and are constants.

2. The vertex form is written as , where is the vertex of the parabola.

3. Think about how each form is useful: the general form is good for expanding and factoring, while the vertex form is helpful for graphing and identifying the vertex.

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

Background

Topic: Graphing Quadratic Functions

This question is testing 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: A vertical line that divides the parabola into two mirror images.

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

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

• Vertex: The highest or lowest 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 point is at the lowest part of the parabola (vertex).

3. Find the points where the graph crosses the x-axis (roots/zeroes).

4. Locate the point where the graph crosses the y-axis (y-intercept).

5. Determine which line represents the axis of symmetry.

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

Background

Topic: Properties of Quadratic Functions

This question is testing 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: , , , .

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

Background

Topic: Completing the Square

This question is testing 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: .

4. Add and subtract 36 inside the function to rewrite it as a perfect square trinomial.

5. Express the function in the form and identify the vertex .

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

Background

Topic: Completing the Square (with a leading coefficient)

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

Key Formula

• Vertex Form:

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

Step-by-Step Guidance

1. Start with .

2. Factor out 2 from the and terms: .

3. Find the value to complete the square inside the parentheses: .

4. Add and subtract 4 inside the parentheses, then distribute the 2 and combine with the constant term.

5. Express the function in vertex form and identify the vertex .

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

Background

Topic: Quadratic Properties and Graphing

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

Key Formula

• Vertex Form:

• Axis of Symmetry:

• Solutions: Set and solve for .

• Domain: All real numbers ()

• Range: Depends on the direction the parabola opens.

Step-by-Step Guidance

1. Start with .

2. Complete the square: .

3. Write the function in vertex form: .

4. Identify the vertex and axis of symmetry .

5. Set to find the solutions (roots).

6. State the domain and range based on the graph.

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

Background

Topic: Solving Quadratic Equations

This question is testing 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: Look for two numbers that multiply to and add to .

3. If factoring is difficult, use the quadratic formula with , , .

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

Background

Topic: Solving Quadratic Equations

This question is testing your ability to find the roots of a quadratic equation, which may be complex if the discriminant is negative.

Key Formula

• Quadratic Formula:

Step-by-Step Guidance

1. Set to find the roots: .

2. Use the quadratic formula with , , .

3. Calculate the discriminant: .

4. Since the discriminant is negative, the solutions will be complex numbers.

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

• a. When does the rock reach its maximum height above the ocean?

• b. What is the maximum height of the rock above the ocean?

• c. When does the rock hit the ocean?

Background

Topic: Quadratic Applications (Projectile Motion)

This question is testing your ability to analyze a quadratic function modeling projectile motion, including finding the vertex and solving for when the height is zero.

Key Formula

• Vertex (maximum height):

• Maximum height: Plug into

• Rock hits the ocean: Solve

Step-by-Step Guidance

1. Identify , , in .

2. Find the time of maximum height using .

3. Plug this value of into to find the maximum height.

4. Set and solve for to find when the rock hits the ocean.

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

Background

Topic: Optimization with Quadratics

This question is testing your ability to use quadratic functions to solve an optimization problem involving perimeter and area.

Key Formula

• Perimeter:

• Area:

Step-by-Step Guidance

1. Write the perimeter equation: .

2. Solve for one variable in terms of the other: .

3. Write the area function: .

4. Express as a quadratic function and find its maximum by completing the square or using the vertex formula.

5. Find the values of and that maximize the area.

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