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College Algebra Chapter 4 Review: Quadratic and Polynomial Functions

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Q1. Given the quadratic function in vertex form: , answer the following:

Background

Topic: Quadratic Functions (Vertex Form)

This question tests your understanding of the vertex form of a quadratic function, how to identify the vertex, axis of symmetry, direction of opening, intercepts, and domain/range.

Key Terms and Formulas:

  • Vertex form:

  • Vertex:

  • Axis of symmetry:

  • Direction: If , opens down; if , opens up

  • x-intercepts: Set and solve for

  • y-intercept: Set and solve for

Step-by-Step Guidance

  1. Identify the values of , , and in the given function.

  2. Write the coordinates of the vertex using .

  3. Determine if the graph opens up or down by checking the sign of .

  4. Write the equation for the axis of symmetry.

  5. Set and solve for to find x-intercepts (stop before solving).

  6. Set and solve for to find the y-intercept (stop before calculating).

Try solving on your own before revealing the answer!

Q2. Rewrite in vertex form by completing the square, then answer parts a–g.

Background

Topic: Completing the Square & Vertex Form

This question tests your ability to convert a quadratic from standard to vertex form, and then analyze its properties.

Key Terms and Formulas:

  • Standard form:

  • Vertex form:

  • Completing the square: Rearranging to express as

Step-by-Step Guidance

  1. Factor from the quadratic terms:

  2. Complete the square inside the brackets: Add and subtract

  3. Rewrite the function in vertex form:

  4. Identify the vertex from the vertex form.

  5. Determine the axis of symmetry and direction of opening.

  6. Set and to find intercepts (stop before solving).

Try solving on your own before revealing the answer!

Q3. Given , does the function have a maximum or minimum value? What is that value?

Background

Topic: Quadratic Functions (Maximum/Minimum)

This question tests your ability to determine whether a quadratic has a maximum or minimum, and how to find that value.

Key Terms and Formulas:

  • If , minimum; if , maximum

  • Vertex:

  • Minimum/maximum value: at the vertex

Step-by-Step Guidance

  1. Identify and in the quadratic function.

  2. Determine if the parabola opens up or down based on .

  3. Calculate the vertex -coordinate: (stop before plugging in values).

  4. Substitute the vertex -coordinate into to find the minimum/maximum value (stop before calculating).

Try solving on your own before revealing the answer!

Q4. For , find the x-intercepts and y-intercept.

Background

Topic: Quadratic Intercepts

This question tests your ability to find the points where the graph crosses the axes.

Key Terms and Formulas:

  • x-intercepts: Solve

  • y-intercept:

  • Quadratic formula:

Step-by-Step Guidance

  1. Set and identify , , .

  2. Apply the quadratic formula to solve for (stop before calculating roots).

  3. Set and compute for the y-intercept (stop before calculating).

Try solving on your own before revealing the answer!

Q5. The height of a baseball is given by . How long does it take to reach maximum height? What is the maximum height?

Background

Topic: Quadratic Applications (Projectile Motion)

This question tests your ability to find the vertex of a quadratic function in a real-world context.

Key Terms and Formulas:

  • Vertex -coordinate:

  • Maximum height: at the vertex

Step-by-Step Guidance

  1. Identify and in the function .

  2. Calculate (stop before plugging in values).

  3. Substitute into to find the maximum height (stop before calculating).

Try solving on your own before revealing the answer!

Q6. Use the end behavior of a polynomial graph to determine if the degree is even or odd, and if the leading coefficient is positive or negative.

Background

Topic: Polynomial End Behavior

This question tests your understanding of how the degree and leading coefficient affect the graph's end behavior.

Key Terms and Formulas:

  • Even degree: Both ends go up or down

  • Odd degree: Ends go in opposite directions

  • Positive leading coefficient: Right end goes up

  • Negative leading coefficient: Right end goes down

Step-by-Step Guidance

  1. Observe the graph's left and right end behavior.

  2. Determine if the ends match (even) or differ (odd).

  3. Check which direction the right end points to decide the sign of the leading coefficient.

Try solving on your own before revealing the answer!

Q7. Find the intercepts of .

Background

Topic: Polynomial Intercepts

This question tests your ability to find where a cubic polynomial crosses the axes.

Key Terms and Formulas:

  • y-intercept:

  • x-intercepts: Solve

Step-by-Step Guidance

  1. Set to find the y-intercept.

  2. Set and factor or use rational root theorem to find x-intercepts (stop before solving).

Try solving on your own before revealing the answer!

Q8. Sketch and analyze its end behavior, y-intercept, real zeros, and multiplicities.

Background

Topic: Polynomial Graphs (Multiplicity & End Behavior)

This question tests your ability to analyze and sketch a polynomial based on its factors.

Key Terms and Formulas:

  • End behavior: Based on degree and leading coefficient

  • Multiplicity: Number of times a zero repeats

  • y-intercept:

Step-by-Step Guidance

  1. Expand the function to determine degree and leading coefficient.

  2. Identify zeros and their multiplicities from the factors.

  3. Set to find the y-intercept.

  4. Choose a test point between zeros and evaluate (stop before calculating).

Try solving on your own before revealing the answer!

Q9. Select a possible function that could be represented by a given graph.

Background

Topic: Matching Polynomial Functions to Graphs

This question tests your ability to interpret graphs and match them to possible polynomial equations based on zeros and end behavior.

Key Terms and Formulas:

  • Zeros: Where the graph crosses the x-axis

  • End behavior: Based on degree and leading coefficient

Step-by-Step Guidance

  1. Identify the zeros from the graph.

  2. Determine the degree and sign of the leading coefficient.

  3. Match the graph to the correct function from the list (stop before choosing).

Try solving on your own before revealing the answer!

Q10. Use synthetic division to divide by and write in the form .

Background

Topic: Synthetic Division & Remainder Theorem

This question tests your ability to perform synthetic division and express a polynomial in the division algorithm form.

Key Terms and Formulas:

  • Synthetic division: Shortcut for dividing by linear factors

  • Division algorithm:

Step-by-Step Guidance

  1. Set up synthetic division using .

  2. Write the coefficients of in order.

  3. Perform synthetic division steps (stop before completing).

  4. Express in the division algorithm form (stop before writing full expression).

Try solving on your own before revealing the answer!

Q11. Use synthetic division and the remainder theorem to find the remainder when is divided by .

Background

Topic: Synthetic Division & Remainder Theorem

This question tests your ability to use synthetic division to find the remainder when dividing by a linear factor.

Key Terms and Formulas:

  • Remainder theorem: is the remainder when divided by

  • Synthetic division steps

Step-by-Step Guidance

  1. Set up synthetic division with .

  2. List the coefficients of .

  3. Perform synthetic division steps (stop before calculating remainder).

Try solving on your own before revealing the answer!

Q12. Use synthetic division and the factor theorem to determine whether is a factor of .

Background

Topic: Synthetic Division & Factor Theorem

This question tests your ability to use synthetic division to check if a linear factor divides a polynomial evenly.

Key Terms and Formulas:

  • Factor theorem: is a factor if

  • Synthetic division steps

Step-by-Step Guidance

  1. Set up synthetic division with .

  2. List the coefficients of .

  3. Perform synthetic division steps (stop before checking remainder).

Try solving on your own before revealing the answer!

Q13. Given and is a zero of multiplicity 2, answer parts a–c.

Background

Topic: Polynomial Zeros & Factoring

This question tests your ability to find all zeros, write the polynomial in factored form, and sketch the graph.

Key Terms and Formulas:

  • Multiplicity: Number of times a zero repeats

  • Factored form: Express as product of linear factors

Step-by-Step Guidance

  1. Given is a zero of multiplicity 2, write as a factor.

  2. Divide by to find remaining zeros (stop before solving).

  3. Write in completely factored form (stop before completing).

Try solving on your own before revealing the answer!

Q14. Use the rational zeros theorem to list all potential rational zeros of .

Background

Topic: Rational Zeros Theorem

This question tests your ability to list possible rational zeros based on the theorem.

Key Terms and Formulas:

  • Rational zeros theorem: Possible zeros are , where divides the constant term and divides the leading coefficient.

Step-by-Step Guidance

  1. Identify the constant term () and leading coefficient ($11$).

  2. List all integer divisors of and $11$.

  3. Form all possible fractions (stop before listing all).

Try solving on your own before revealing the answer!

Q15. Solve in the complex numbers.

Background

Topic: Solving Cubic Equations (Complex Roots)

This question tests your ability to solve cubic equations and find all real and complex roots.

Key Terms and Formulas:

  • Factorization:

  • Quadratic formula for complex roots

Step-by-Step Guidance

  1. Factor as a difference of cubes.

  2. Set each factor equal to zero and solve for (stop before solving quadratic).

Try solving on your own before revealing the answer!

Q16. Form a third-degree polynomial with real coefficients and leading coefficient 1, given zeros and .

Background

Topic: Constructing Polynomials from Zeros

This question tests your ability to form a polynomial given complex and real zeros.

Key Terms and Formulas:

  • Complex zeros occur in conjugate pairs: and

  • Polynomial:

Step-by-Step Guidance

  1. Write factors for each zero, including the conjugate for the complex zero.

  2. Multiply the complex factors to get a quadratic with real coefficients (stop before expanding).

  3. Multiply by the linear factor for the real zero (stop before expanding fully).

Try solving on your own before revealing the answer!

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