Q1. Rewrite the equation in standard form and graph it:

Background

Topic: Conic Sections (Circles)

This question tests your ability to rewrite a quadratic equation in standard form and recognize the graph of a circle.

Key Terms and Formulas:

• Standard form of a circle:

• Completing the square: Used to rewrite quadratic expressions in standard form.

Step-by-Step Guidance

1. Group the and terms: .

2. Complete the square for the terms: .

3. Complete the square for the terms: .

4. Add and subtract the necessary constants to both sides to balance the equation.

5. Rewrite the equation in the form .

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Q2. Rewrite the equation in standard form and graph it:

Background

Topic: Conic Sections (Hyperbolas)

This question tests your ability to rewrite a general quadratic equation in standard form and recognize the graph of a hyperbola.

Key Terms and Formulas:

• Standard form of a hyperbola:

• Completing the square for both and terms.

Step-by-Step Guidance

1. Group and terms: .

2. Complete the square for and .

3. Adjust the equation by adding/subtracting constants to both sides.

4. Rewrite the equation in the standard form for a hyperbola.

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Q3. Find the coordinates of the foci for the curve in question #2

Background

Topic: Hyperbolas – Foci

This question tests your understanding of how to find the foci of a hyperbola given its standard form.

Key Terms and Formulas:

• For , foci are at where .

Step-by-Step Guidance

1. Identify , , , and from the standard form found in Q2.

2. Calculate using .

3. Write the coordinates of the foci: and .

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Q4. Rewrite the equation in standard form and graph it:

Background

Topic: Conic Sections (Ellipses)

This question tests your ability to rewrite a general quadratic equation in standard form and recognize the graph of an ellipse.

Key Terms and Formulas:

• Standard form of an ellipse:

• Completing the square for and terms.

Step-by-Step Guidance

1. Group and terms: .

2. Complete the square for and .

3. Factor and adjust the equation to get the standard form of an ellipse.

4. Divide both sides to get $1$ on the right side.

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Q5. Rewrite the equation in standard form and graph it:

Background

Topic: Conic Sections (Parabolas)

This question tests your ability to rewrite a quadratic equation in standard form and recognize the graph of a parabola.

Key Terms and Formulas:

• Standard form of a parabola (horizontal):

• Completing the square for terms.

Step-by-Step Guidance

1. Group terms: .

2. Factor out from terms and complete the square.

3. Rewrite the equation in the form .

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Q6. Find the coordinates of the focus and an equation of the directrix for the curve in question #5

Background

Topic: Parabolas – Focus and Directrix

This question tests your understanding of the geometric properties of a parabola in standard form.

Key Terms and Formulas:

• For , the vertex is , focus is , directrix is .

Step-by-Step Guidance

1. Identify , , and from the standard form found in Q5.

2. Calculate the focus: .

3. Write the equation of the directrix: .

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Q7. For the given parametric equations , , find its rectangular form, graph it, and indicate the orientation of the line.

Background

Topic: Parametric Equations

This question tests your ability to convert parametric equations to rectangular form and analyze the resulting graph.

Key Terms and Formulas:

• Parametric equations: ,

• Rectangular form: Eliminate to get a relationship between and .

Step-by-Step Guidance

1. Solve for in terms of .

2. Substitute this expression for into to get in terms of .

3. Sketch the graph of the resulting equation and indicate the direction as increases.

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Q8. Find two different angles, one positive and one negative, coterminal with

Background

Topic: Coterminal Angles

This question tests your understanding of how to find angles coterminal with a given angle by adding or subtracting multiples of .

Key Terms and Formulas:

• Coterminal angles: , where is any integer.

Step-by-Step Guidance

1. Add to until you get a positive angle.

2. Subtract from to get a negative coterminal angle.

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Q9. Find two different angles, one positive and one negative, coterminal with

Background

Topic: Coterminal Angles (Radians)

This question tests your ability to find coterminal angles in radians by adding or subtracting multiples of .

Key Terms and Formulas:

• Coterminal angles: , where is any integer.

Step-by-Step Guidance

1. Subtract from to get a positive coterminal angle less than $2\pi$.

2. Subtract again to get a negative coterminal angle.

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Q10. Convert degree measure to radians:

Background

Topic: Angle Conversion (Degrees to Radians)

This question tests your ability to convert degrees to radians using the conversion factor .

Key Terms and Formulas:

• Degrees to radians:

Step-by-Step Guidance

1. Multiply by .

2. Simplify the fraction to lowest terms.

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Q11. Convert radian measure to degrees:

Background

Topic: Angle Conversion (Radians to Degrees)

This question tests your ability to convert radians to degrees using the conversion factor .

Key Terms and Formulas:

• Radians to degrees:

Step-by-Step Guidance

1. Multiply by .

2. Simplify the expression to get the degree measure.

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Q12. Find the complement and supplement of angle

Background

Topic: Complementary and Supplementary Angles

This question tests your understanding of how to find the complement and supplement of a given angle in degrees.

Key Terms and Formulas:

• Complement:

• Supplement:

Step-by-Step Guidance

1. Subtract from to find the complement.

2. Subtract from to find the supplement.

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Q13. Find the complement and supplement of angle

Background

Topic: Complementary and Supplementary Angles (Radians)

This question tests your ability to find the complement and supplement of an angle given in radians.

Key Terms and Formulas:

• Complement:

• Supplement:

Step-by-Step Guidance

1. Subtract from for the complement.

2. Subtract from for the supplement.

3. Simplify the expressions to a single fraction if possible.

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Q14. Subtract , express your answer in degrees/minutes/seconds

Background

Topic: Angle Arithmetic (DMS Format)

This question tests your ability to subtract angles given in decimal degrees and degrees/minutes/seconds (DMS) format, and express the result in DMS.

Key Terms and Formulas:

• 1 degree = 60 minutes, 1 minute = 60 seconds

• Convert decimal degrees to DMS if needed.

Step-by-Step Guidance

1. Convert to DMS format.

2. Subtract from (from the conversion).

3. If necessary, borrow minutes or seconds to perform the subtraction.

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Q15. The terminal point corresponding to angle has coordinates . Find the values of all six circular functions of angle $s$

Background

Topic: Circular (Trigonometric) Functions

This question tests your ability to find sine, cosine, tangent, cosecant, secant, and cotangent given a point on the unit circle.

Key Terms and Formulas:

• , ,

• , ,

• For a point on the unit circle, ; otherwise,

Step-by-Step Guidance

1. Identify , , (since ).

2. Compute , , and using the definitions above.

3. Find , , and as reciprocals of the primary functions.

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Q16. Identify the amplitude, frequency, period, and the shifts of the equation

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to analyze the properties of a cosine function with transformations.

Key Terms and Formulas:

• General form:

• Amplitude:

• Period:

• Frequency:

• Phase shift:

• Vertical shift:

Step-by-Step Guidance

1. Identify , , , and from the equation.

2. Calculate the amplitude as .

3. Calculate the period as .

4. Find the phase shift as .

5. Identify the vertical shift .

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Q17. Graph the basic period of

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to graph a transformed cosine function over one period.

Key Terms and Formulas:

• Use amplitude, period, phase shift, and vertical shift from Q16.

• Plot key points: maximum, minimum, and midline crossings.

Step-by-Step Guidance

1. Determine the starting point of the period using the phase shift.

2. Mark the interval of one period ( units) on the -axis.

3. Plot the maximum, minimum, and midline values using amplitude and vertical shift.

4. Sketch the cosine curve over this interval.

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Q18. Identify the amplitude, frequency, period, and the shifts of the equation

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to analyze the properties of a sine function with transformations.

Key Terms and Formulas:

• General form:

• Amplitude:

• Period:

• Frequency:

• Phase shift:

• Vertical shift:

Step-by-Step Guidance

1. Identify , , , and from the equation.

2. Calculate the amplitude as .

3. Calculate the period as .

4. Find the phase shift as .

5. Identify the vertical shift .

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Q19. Graph the basic period of

Background

Topic: Graphs of Trigonometric Functions

This question tests your ability to graph a transformed sine function over one period.

Key Terms and Formulas:

• Use amplitude, period, phase shift, and vertical shift from Q18.

• Plot key points: maximum, minimum, and midline crossings.

Step-by-Step Guidance

1. Determine the starting point of the period using the phase shift ().

2. Mark the interval of one period ( units) on the -axis.

3. Plot the maximum, minimum, and midline values using amplitude and vertical shift.

4. Sketch the sine curve over this interval, noting the reflection due to the negative amplitude.

Try solving on your own before revealing the answer!