Q1. Find a polynomial function of degree 4 that has real zeros of -3 (multiplicity 2), 7 (multiplicity 1), and 1/3 (multiplicity 1).

Background

Topic: Constructing Polynomial Functions from Zeros

This question tests your ability to build a polynomial function when given its real zeros and their multiplicities.

Key Terms and Formulas:

• Zero (Root): A value of where the polynomial equals zero.

• Multiplicity: The number of times a particular zero occurs.

• General Form: If is a zero of multiplicity , then is a factor.

Step-by-Step Guidance

1. Write the factors for each zero, using the correct multiplicity. For example, for zero with multiplicity 2, the factor is .

2. Write the factors for the other zeros: $7(x-7) (multiplicity 1) gives .

3. Multiply all the factors together to form the general polynomial: , where is a leading coefficient (often set to 1 unless otherwise specified).

4. To avoid fractions in the polynomial, multiply by 3 to get an equivalent factor with integer coefficients: .

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Q2. Determine the zeros and their multiplicities for the polynomial .

Background

Topic: Identifying Zeros and Multiplicities of a Polynomial

This question asks you to find all real zeros of a given polynomial and state their multiplicities.

Key Terms and Formulas:

• Zero: A value of that makes .

• Multiplicity: The exponent on the factor corresponding to each zero.

Step-by-Step Guidance

1. Set each factor in the polynomial equal to zero to find the zeros.

2. For , set ; for , set ; for , set .

3. Identify the multiplicity of each zero by looking at the exponent on each factor.

4. List each zero with its corresponding multiplicity.

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Q3. Determine a potential polynomial function (with smallest degree possible) for the graph shown below. (0,6)

Background

Topic: Constructing a Polynomial from a Graph

This question tests your ability to deduce a polynomial's equation from its graph, using features like intercepts and turning points.

Key Terms and Formulas:

• Degree: The highest power of in the polynomial.

• Y-intercept: The value of .

• Turning Points: The number of local maxima and minima; a degree polynomial can have up to turning points.

Step-by-Step Guidance

1. Identify the -intercept from the graph, which is the value when (here, ).

2. Estimate the number of turning points to determine the minimum degree of the polynomial.

3. Look for -intercepts (real zeros) and their behavior (touching or crossing the axis) to infer multiplicities.

4. Write a general form for the polynomial using the zeros and multiplicities, and use the -intercept to solve for the leading coefficient.

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Q4. Analyze and graph a polynomial function. (a)

Background

Topic: Graphing Polynomial Functions

This question tests your understanding of end behavior, intercepts, and turning points for polynomial functions.

Key Terms and Formulas:

• End Behavior: Determined by the leading term ().

• Intercepts: Where the graph crosses the axes.

• Turning Points: Maximum number is degree minus one.

Step-by-Step Guidance

1. Identify the degree and leading coefficient to determine end behavior. For , as , .

2. Find the -intercept by evaluating .

3. Set to find -intercepts (real zeros).

4. Determine the maximum number of turning points (degree minus one).

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Q5. Analyze and graph a rational function. (a)

Background

Topic: Graphing Rational Functions

This question tests your ability to analyze rational functions for asymptotes, intercepts, and holes.

Key Terms and Formulas:

• Vertical Asymptotes: Set denominator equal to zero and solve.

• Horizontal Asymptotes: Compare degrees of numerator and denominator.

• Holes: Occur if a factor cancels from numerator and denominator.

• Intercepts: Set numerator or denominator to zero as appropriate.

Step-by-Step Guidance

1. Factor numerator and denominator completely.

2. Find vertical asymptotes by setting the denominator factors equal to zero.

3. Check for holes by seeing if any factors cancel.

4. Determine horizontal asymptote by comparing degrees.

5. Find - and -intercepts by setting numerator and denominator to zero, respectively.

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Q6. Solve the polynomial inequality algebraically.

Background

Topic: Solving Polynomial Inequalities

This question tests your ability to solve inequalities involving polynomials by finding critical points and testing intervals.

Key Terms and Formulas:

• Critical Points: Values where the expression equals zero.

• Sign Chart: Used to test intervals between critical points.

Step-by-Step Guidance

1. Move all terms to one side: .

2. Factor the expression completely.

3. Set each factor equal to zero to find critical points.

4. Use a sign chart to test the sign of the expression in each interval determined by the critical points.

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Q7. Solve the rational inequality algebraically:

Background

Topic: Solving Rational Inequalities

This question tests your ability to solve inequalities involving rational expressions by finding critical points and testing intervals.

Key Terms and Formulas:

• Critical Points: Values where the numerator or denominator is zero.

• Sign Chart: Used to test intervals between critical points.

Step-by-Step Guidance

1. Subtract 1 from both sides to get a single rational expression: .

2. Combine into a single fraction and simplify the numerator.

3. Factor the numerator and denominator if possible.

4. Find critical points by setting the numerator and denominator equal to zero.

5. Use a sign chart to test the sign of the expression in each interval.

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Q8. Solve this system using the method of elimination:

Background

Topic: Solving Systems of Linear Equations (Elimination Method)

This question tests your ability to solve a system of equations by eliminating one variable.

Key Terms and Formulas:

• Elimination Method: Multiply equations as needed so that adding or subtracting them eliminates one variable.

Step-by-Step Guidance

1. Multiply one or both equations so that the coefficients of or are opposites.

2. Add or subtract the equations to eliminate one variable.

3. Solve for the remaining variable.

4. Substitute back into one of the original equations to solve for the other variable.

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Q9. Solve this system using the method of substitution:

Background

Topic: Solving Systems of Linear Equations (Substitution Method)

This question tests your ability to solve a system by expressing one variable in terms of the other and substituting.

Key Terms and Formulas:

• Substitution Method: Solve one equation for one variable, then substitute into the other equation.

Step-by-Step Guidance

1. Solve the first equation for or .

2. Substitute this expression into the second equation.

3. Solve for the remaining variable.

4. Substitute back to find the other variable.

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Q10. (a) Let and . Determine , its domain, and simplify the result.

Background

Topic: Composition of Functions

This question tests your ability to compose two functions and determine the domain of the resulting function.

Key Terms and Formulas:

• Composition:

• Domain: Set of values for which the composition is defined.

Step-by-Step Guidance

1. Substitute into to get .

2. Simplify the resulting expression.

3. Determine the domain by finding values of that do not make any denominator zero.

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Q10. (b) Let and . Determine , its domain, and simplify the result.

Background

Topic: Composition of Functions with Radicals

This question tests your ability to compose functions involving square roots and polynomials, and to find the domain.

Key Terms and Formulas:

• Composition:

• Domain: Set of values for which the composition is defined (here, ).

Step-by-Step Guidance

1. Substitute into to get .

2. Simplify the resulting expression.

3. Determine the domain by setting .

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Q11. The surface area, (in ), of a spherical balloon with radius (in ) is given by . A helium tank can fill the balloon so that its radius is given by , , where is the number of seconds since the start of filling the balloon. Find the surface area of the balloon as a function of time.

Background

Topic: Applications of Function Composition

This question tests your ability to compose functions in a real-world context.

Key Terms and Formulas:

• Surface Area of a Sphere:

• Composition:

Step-by-Step Guidance

1. Substitute into to get .

2. Simplify the resulting expression to write as a function of .

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Q12. The graph of the function is given below. Determine the graph of . (Points: (0,1), (1,3))

Background

Topic: Graphs of Inverse Functions

This question tests your understanding of how to graph the inverse of a function using its original graph.

Key Terms and Formulas:

• Inverse Function: reverses the roles of and in .

• Graphical Relationship: The graph of is the reflection of across the line .

Step-by-Step Guidance

1. Identify key points on the graph of , such as and .

2. For each point on , plot the point on .

3. Reflect the entire graph of over the line to obtain .

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Q13. The function is one-to-one. Find its inverse function, .

Background

Topic: Finding Inverse Functions Algebraically

This question tests your ability to find the inverse of a rational function.

Key Terms and Formulas:

• Inverse Function: Swap and and solve for .

Step-by-Step Guidance

1. Replace with and write .

2. Swap and to get .

3. Solve for in terms of by multiplying both sides by and isolating .

4. Express as the final formula.

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Q14. Demonstrate that and are inverse functions.

Background

Topic: Verifying Inverse Functions

This question tests your ability to verify that two functions are inverses by showing and .

Key Terms and Formulas:

• Inverse Functions: and for all in the domain.

Step-by-Step Guidance

1. Compute by substituting into and simplify.

2. Compute by substituting into and simplify.

3. Show that both compositions simplify to .

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