Q1. Find an equation of the line that contains the given pair of points: (−9, 3) and (−7, −5). Write your answer in slope-intercept form.

Background

Topic: Linear Equations

This question tests your ability to find the equation of a line given two points, and to express it in slope-intercept form ().

Key Terms and Formulas:

• Slope formula:

• Slope-intercept form:

Step-by-Step Guidance

1. Label your points: Let and .

2. Calculate the slope using the formula: .

3. Substitute the values for into the slope formula and simplify.

4. Use the point-slope form with one of the points to set up the equation.

5. Rearrange the equation into slope-intercept form (), but stop before calculating the final value of .

Try solving on your own before revealing the answer!

Q2. Find the slope-intercept equation of the line with slope 8 and y-intercept (0, 4).

Background

Topic: Linear Equations

This question checks your understanding of how to write the equation of a line when given the slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• is the slope, is the y-intercept

Step-by-Step Guidance

1. Recall the slope-intercept form: .

2. Identify the given values: , .

3. Substitute these values into the formula to write the equation.

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Q3. Graph the equation by plotting points:

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a linear equation by plotting points and understanding the slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• Slope (): rise over run

• Y-intercept (): where the line crosses the y-axis

Step-by-Step Guidance

1. Identify the slope () and y-intercept ().

2. Plot the y-intercept point on the graph.

3. From the y-intercept, use the slope to find another point: rise 4 units, run 1 unit to the right.

4. Plot at least one more point using the slope, then draw a straight line through the points.

Try solving on your own before revealing the answer!

Q4. Graph the equation by plotting points:

Background

Topic: Graphing Linear Equations

This question checks your ability to graph a line with a fractional slope and a y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• Slope (): means rise 1, run 2

• Y-intercept (): 2

Step-by-Step Guidance

1. Identify the slope () and y-intercept ().

2. Plot the y-intercept at .

3. From , use the slope: up 1, right 2 to plot another point.

4. Draw the line through the points.

Try solving on your own before revealing the answer!

Q5. Simplify by removing factors of 1:

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor polynomials and reduce rational expressions to lowest terms.

Key Terms and Formulas:

• Difference of squares:

• Perfect square trinomial:

• Reducing rational expressions: Cancel common factors in numerator and denominator

Step-by-Step Guidance

1. Factor the numerator: as a difference of squares.

2. Factor the denominator: as a perfect square trinomial.

3. Write the expression with the factored forms.

4. Identify and cancel any common factors.

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Q6. Reduce the rational expression to lowest terms:

Background

Topic: Simplifying Rational Expressions

This question tests your ability to factor quadratic expressions and reduce rational expressions.

Key Terms and Formulas:

• Factoring quadratics:

• Reducing rational expressions: Cancel common factors

Step-by-Step Guidance

1. Factor the numerator .

2. Factor the denominator .

3. Write the expression with the factored forms.

4. Identify and cancel any common factors.

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Q7. Use the graph of and transformations to sketch . Determine the domain, range, y-intercept, and equation of the horizontal asymptote.

Background

Topic: Exponential Functions and Transformations

This question tests your understanding of how to graph exponential functions and identify their key features.

Key Terms and Formulas:

• Exponential function:

• Domain: All possible values

• Range: All possible values

• Horizontal asymptote:

Step-by-Step Guidance

1. Recognize that is a horizontal shift of to the right by 6 units.

2. Recall that the domain of exponential functions is all real numbers.

3. The range is all positive real numbers (since for all ).

4. The horizontal asymptote is (no vertical shift).

5. To find the y-intercept, set and compute , but stop before calculating the final value.

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Q8. Use the graph of and transformations to sketch . Determine the domain, range, y-intercept, and equation of the horizontal asymptote.

Background

Topic: Exponential Functions and Transformations

This question tests your ability to apply transformations to exponential functions and identify their key features.

Key Terms and Formulas:

• Reciprocal of exponential:

• Vertical shift: moves the graph up by 6 units

• Horizontal asymptote:

Step-by-Step Guidance

1. Rewrite as to see the reflection and shift.

2. Domain is all real numbers.

3. Range is since is always positive.

4. Horizontal asymptote is .

5. To find the y-intercept, set and compute , but stop before calculating the final value.

Try solving on your own before revealing the answer!

Q9. Rewrite as a sum or difference of multiples of logarithms:

Background

Topic: Properties of Logarithms

This question tests your ability to expand logarithmic expressions using the product and power rules.

Key Terms and Formulas:

• Product rule:

• Power rule:

Step-by-Step Guidance

1. Apply the product rule to separate , , and .

2. Apply the power rule to and .

3. Write the expanded expression as a sum of logarithms.

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Q10. Express as a sum and/or difference of logarithms.

Background

Topic: Properties of Logarithms

This question tests your ability to expand a logarithm of a quotient and product using logarithm rules.

Key Terms and Formulas:

• Quotient rule:

• Product rule:

• Power rule:

Step-by-Step Guidance

1. Apply the quotient rule to separate numerator and denominator.

2. Apply the product rule to the numerator and denominator .

3. Apply the power rule to and .

4. Write the expanded expression as a sum and/or difference of logarithms.

Try solving on your own before revealing the answer!

Q11. Graph the function

Background

Topic: Graphing Linear Functions

This question tests your ability to graph a linear function using its slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form:

• Slope (): 6

• Y-intercept (): -9

Step-by-Step Guidance

1. Identify the slope and y-intercept from the equation.

2. Plot the y-intercept at .

3. From , use the slope: up 6, right 1 to plot another point.

4. Draw the line through the points.

Try solving on your own before revealing the answer!

Q12. Graph the function

Background

Topic: Graphing Linear Functions

This question tests your ability to graph a linear function with a fractional coefficient.

Key Terms and Formulas:

• Rewrite as

• Slope:

• Y-intercept:

Step-by-Step Guidance

1. Rewrite the function in slope-intercept form.

2. Identify the slope and y-intercept.

3. Plot the y-intercept at .

4. From the y-intercept, use the slope to plot another point.

5. Draw the line through the points.

Try solving on your own before revealing the answer!

Q13. Graph and state the domain:

Background

Topic: Rational Functions

This question tests your understanding of the graph and domain of a rational function.

Key Terms and Formulas:

• Rational function:

• Domain: All real numbers except where the denominator is zero

Step-by-Step Guidance

1. Identify values of that make the denominator zero.

2. State the domain in interval notation, excluding .

3. Recognize the graph is a hyperbola with asymptotes at and .

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Q14. Graph:

Background

Topic: Rational Functions

This question tests your ability to graph a rational function with a negative numerator.

Key Terms and Formulas:

• Rational function:

• Asymptotes: (vertical), (horizontal)

Step-by-Step Guidance

1. Recognize the function is undefined at .

2. For , is negative; for , is positive.

3. Sketch the hyperbola in the appropriate quadrants.

Try solving on your own before revealing the answer!