Q1. Graph the function g(x) = x^2 - 2 using transformations of the standard quadratic function f(x) = x^2.

Background

Topic: Transformations of Quadratic Functions

This question tests your understanding of how to graph quadratic functions using vertical shifts.

Key Terms and Formulas:

• Quadratic function:

• Transformation: is a vertical shift downward by 2 units.

Step-by-Step Guidance

1. Start by graphing the parent function .

2. Recognize that subtracting 2 from shifts the graph downward by 2 units.

3. For each point on , the corresponding point on is .

4. Plot the new vertex at instead of .

Try solving on your own before revealing the answer!

Q2. Graph the function g(x) = \sqrt{x - 1} using transformations of the standard square root function f(x) = \sqrt{x}.

Background

Topic: Transformations of Radical Functions

This question tests your ability to apply horizontal shifts to the square root function.

Key Terms and Formulas:

• Square root function:

• Transformation: is a horizontal shift right by 1 unit.

Step-by-Step Guidance

1. Graph the parent function , which starts at .

2. Identify the transformation: is replaced by , so the graph shifts right by 1 unit.

3. The new starting point is .

4. For each , is defined only when , so .

Try solving on your own before revealing the answer!

Q10. Find the domain of the function f(x) = -8x - 2.

Background

Topic: Domain of Linear Functions

This question tests your understanding of the domain for linear functions.

Key Terms and Formulas:

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

• Linear function:

Step-by-Step Guidance

1. Recognize that linear functions are defined for all real numbers.

2. There are no restrictions such as division by zero or square roots of negative numbers.

Try solving on your own before revealing the answer!

Q14. Find the coordinates of the vertex for the parabola defined by f(x) = (x - 3)^2 - 3.

Background

Topic: Vertex Form of Quadratic Functions

This question tests your ability to identify the vertex from the vertex form of a quadratic function.

Key Terms and Formulas:

• Vertex form:

• Vertex:

Step-by-Step Guidance

1. Compare to the vertex form .

2. Identify and .

3. The vertex is at .

Try solving on your own before revealing the answer!

Q25. You have 208 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area.

Background

Topic: Optimization with Quadratic Functions

This question tests your ability to use quadratic functions to solve optimization problems.

Key Terms and Formulas:

• Perimeter of rectangle:

• Area of rectangle:

• Maximum area occurs when the rectangle is a square.

Step-by-Step Guidance

1. Let and be the length and width. The perimeter constraint is .

2. Solve for one variable: .

3. Express area in terms of : .

4. Write and find the value of that maximizes .

Try solving on your own before revealing the answer!

Q32. Find the x-intercepts of the polynomial function x^5 - 31x^3 + 150x = 0. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.

Background

Topic: Polynomial Roots and Multiplicity

This question tests your ability to find roots of polynomials and interpret their multiplicity.

Key Terms and Formulas:

• x-intercept: Where

• Multiplicity: The number of times a root occurs

• Factoring:

Step-by-Step Guidance

1. Factor out :

2. Set for one intercept.

3. Factor as a quadratic in .

4. Set each factor equal to zero and solve for .

Try solving on your own before revealing the answer!

Q59. Solve the polynomial inequality (x - 2)(x + 9) > 0 and graph the solution set on a number line. Express the solution set in interval notation.

Background

Topic: Solving Polynomial Inequalities

This question tests your ability to solve inequalities involving polynomials and express the solution in interval notation.

Key Terms and Formulas:

• Critical points: Where each factor equals zero (, )

• Test intervals: Regions between and outside the critical points

Step-by-Step Guidance

1. Find the zeros: , .

2. Divide the real line into intervals: , , .

3. Test a value from each interval in the inequality to determine where it is true.

4. Write the solution set in interval notation, including only intervals where the product is positive.

Try solving on your own before revealing the answer!