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College Algebra Midterm Review Guidance

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Q1. Plot the function:

Background

Topic: Piecewise Functions

This question tests your understanding of piecewise-defined functions and how to graph them by considering different expressions for different intervals of .

Key Terms and Formulas:

  • Piecewise function: A function defined by different expressions depending on the input value.

  • Graphing: Plot each part of the function for its respective domain.

Step-by-Step Guidance

  1. Identify the two cases: For , use . For , use .

  2. For , plot the linear function . Choose a few values less than 2 (e.g., ) and calculate for each.

  3. For , plot the quadratic function . Choose values such as and calculate for each.

  4. Check the value at for both expressions to see if the function is continuous at that point.

Try solving on your own before revealing the answer!

Q2. Let . (a) Write in terms of , where the graph of is the graph of shifted left 3 units and down 2 units. (b) Graph the function.

Background

Topic: Function Transformations

This question tests your ability to apply horizontal and vertical shifts to a function.

Key Terms and Formulas:

  • Horizontal shift: shifts left by units if .

  • Vertical shift: shifts up by units if , down if .

Step-by-Step Guidance

  1. Start with .

  2. To shift left by 3 units, replace with .

  3. To shift down by 2 units, subtract 2 from the function.

  4. Combine these transformations to write in terms of .

Try solving on your own before revealing the answer!

Q3. Consider the function .

  • (a) Find and give the vertex.

  • (b) Find and give the equation of the Axis of Symmetry.

  • (c) Write the function in vertex form.

  • (d) Find the domain and range of . Use interval notation.

  • (e) Find and give the intervals where is increasing and decreasing. Use interval notation.

  • (f) Graph this function.

Background

Topic: Quadratic Functions

This question tests your understanding of quadratic functions, their graphs, and properties such as vertex, axis of symmetry, domain, range, and intervals of increase/decrease.

Key Terms and Formulas:

  • Vertex formula: for .

  • Axis of symmetry: .

  • Vertex form: where is the vertex.

  • Domain: All real numbers for quadratic functions.

  • Range: Depends on the direction the parabola opens.

Step-by-Step Guidance

  1. Identify , , .

  2. Find the vertex using .

  3. Plug back into to find the -coordinate of the vertex.

  4. Write the axis of symmetry as .

  5. Rewrite in vertex form using the vertex found.

Try solving on your own before revealing the answer!

Q4. Consider the circle given by the equation .

  • (a) Write the equation of the circle in center-radius form.

  • (b) Find and give the center and the radius of this circle.

  • (c) Sketch a graph of this circle.

Background

Topic: Circles and Completing the Square

This question tests your ability to rewrite a circle equation in standard form and identify its center and radius.

Key Terms and Formulas:

  • Standard form: where is the center and is the radius.

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

Step-by-Step Guidance

  1. Group and terms: and .

  2. Complete the square for both and terms.

  3. Rewrite the equation in the form .

  4. Identify the center and radius from the equation.

Try solving on your own before revealing the answer!

Q5. Consider the rational function .

  • (a) Find and give the and intercept(s).

  • (b) Find and give the equation of the vertical and horizontal asymptotes.

  • (c) Give the domain and range of .

  • (d) Graph this function.

Background

Topic: Rational Functions

This question tests your understanding of rational functions, including finding intercepts, asymptotes, domain, and range.

Key Terms and Formulas:

  • Vertical asymptote: Set denominator equal to zero.

  • Horizontal asymptote: Compare degrees of numerator and denominator.

  • -intercept: Set numerator equal to zero.

  • -intercept: Evaluate .

Step-by-Step Guidance

  1. Set to find -intercepts.

  2. Plug into to find the -intercept.

  3. Set to find the vertical asymptote.

  4. Compare degrees of numerator and denominator to find the horizontal asymptote.

Try solving on your own before revealing the answer!

Q6. Solve the inequality and write your answer in interval notation.

Background

Topic: Linear Inequalities

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

Key Terms and Formulas:

  • Linear inequality: An inequality involving a linear expression.

  • Interval notation: Used to express the solution set.

Step-by-Step Guidance

  1. Subtract from both sides to isolate terms with .

  2. Simplify the inequality to get .

  3. Add 3 to both sides to further isolate .

  4. Divide both sides by 2 to solve for .

Try solving on your own before revealing the answer!

Q7. (a) Solve the equation using the quadratic formula. (b) Solve the inequality and write the solution set in interval notation.

Background

Topic: Quadratic Equations and Inequalities

This question tests your ability to solve quadratic equations using the quadratic formula and to solve quadratic inequalities.

Key Terms and Formulas:

  • Quadratic formula:

  • Quadratic inequality: Solve as an equation, then test intervals.

Step-by-Step Guidance

  1. For (a): Identify , , in .

  2. Plug values into the quadratic formula.

  3. For (b): Rewrite the inequality as .

  4. Subtract 3 from both sides to get .

  5. Solve the corresponding equation to find critical points.

Try solving on your own before revealing the answer!

Q8. Let and .

  • (a) Find and give .

  • (b) Find and give .

  • (c) Give the domain of .

Background

Topic: Function Operations and Composition

This question tests your ability to add functions, compose functions, and determine domains.

Key Terms and Formulas:

  • Function addition:

  • Function composition:

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

Step-by-Step Guidance

  1. For (a): Calculate and , then add them.

  2. For (b): Substitute into to get .

  3. For (c): Determine for which values and are defined.

Try solving on your own before revealing the answer!

Q9. Given the function :

  • (a) List all possible rational zeros of .

  • (b) Find all the zeros of – use synthetic division, factoring, the quadratic formula, etc.

  • (c) Sketch a graph of versus .

Background

Topic: Polynomial Functions and Rational Root Theorem

This question tests your ability to find rational zeros, use synthetic division, and graph cubic polynomials.

Key Terms and Formulas:

  • Rational Root Theorem: Possible rational zeros are where divides the constant term and divides the leading coefficient.

  • Synthetic division: Used to test possible zeros.

Step-by-Step Guidance

  1. List all possible rational zeros using the Rational Root Theorem.

  2. Test each possible zero using synthetic division or direct substitution.

  3. Once a zero is found, factor the polynomial or use quadratic formula for remaining factors.

  4. Sketch the graph using zeros and end behavior.

Try solving on your own before revealing the answer!

Q10. Given that a function has zeros of $5, and $2, write as a polynomial of smallest degree having only real coefficients. (Leave the expression in factored form.)

Background

Topic: Polynomial Construction and Complex Roots

This question tests your ability to construct a polynomial given its zeros, including handling complex roots and multiplicities.

Key Terms and Formulas:

  • Complex roots: If is a root, so is for polynomials with real coefficients.

  • Multiplicity: The number of times a root appears.

Step-by-Step Guidance

  1. Write factors for each zero: , .

  2. For , include its conjugate and write the corresponding quadratic factor.

  3. Multiply all factors to write the polynomial in factored form.

Try solving on your own before revealing the answer!

Q11. The cost in dollars of manufacturing cellphones is given by .

  • (a) How many cellphones should be manufactured to minimize the cost?

  • (b) What will the minimum cost be?

Background

Topic: Quadratic Optimization

This question tests your ability to find the minimum of a quadratic function, which is relevant for optimization problems.

Key Terms and Formulas:

  • Vertex formula: for .

  • Minimum cost: Value of at .

Step-by-Step Guidance

  1. Identify , , .

  2. Find to determine the number of cellphones for minimum cost.

  3. Plug into to find the minimum cost.

Try solving on your own before revealing the answer!

Q12. The volume of a gas varies inversely as pressure and directly as the temperature in degrees K. If a certain gas occupies a volume of 1.6 L at 300 K and a pressure of 15 Pa, find its volume at a temperature of 340 K and a pressure of 20 Pa.

Background

Topic: Variation (Direct and Inverse)

This question tests your understanding of direct and inverse variation relationships and how to use them to solve for unknowns.

Key Terms and Formulas:

  • General variation formula: where is volume, is temperature, is pressure, and is the constant of variation.

Step-by-Step Guidance

  1. Write the general formula: .

  2. Plug in the known values (, , ) to solve for .

  3. Use the value of to set up the equation for the new conditions (, ).

  4. Set up the calculation for the new volume.

Try solving on your own before revealing the answer!

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