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 ($y = mx + b$).

Key Terms and Formulas:

• Slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

1. Label your points: Let $(x_1, y_1) = (−9, 3)$ and $(x_2, y_2) = (−7, −5)$.

2. Calculate the slope $m$ using the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

3. Plug in the values for $x_1, y_1, x_2, y_2$ to find $m$.

4. Use the point-slope form: $y - y_1 = m(x - x_1)$, and substitute one of the points and the slope.

5. Rearrange the equation to slope-intercept form ($y = mx + b$), but stop before calculating the final value of $b$.

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

Background

Topic: Linear Equations

This question tests 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: $y = mx + b$

• Slope ($m$): 8

• Y-intercept ($b$): 4

Step-by-Step Guidance

1. Recall the slope-intercept form: $y = mx + b$.

2. Substitute the given slope ($m = 8$) and y-intercept ($b = 4$) into the formula.

3. Write the equation, but pause before simplifying or stating the final equation.

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Q3. Graph the equation by plotting points: $y = 4x - 3$

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a linear equation by finding and plotting points.

Key Terms and Formulas:

• Linear equation: $y = mx + b$

• Slope ($m$): 4

• Y-intercept ($b$): -3

Step-by-Step Guidance

1. Identify the y-intercept: The point where $x = 0$.

2. Find another point by choosing a value for $x$ (e.g., $x = 1$) and solving for $y$.

3. Plot at least two points on the coordinate plane.

4. Draw a straight line through the points, but do not complete the graph yet.

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Q4. Graph the equation by plotting points: $y = \frac{1}{2}x + 2$

Background

Topic: Graphing Linear Equations

This question tests your ability to graph a linear equation with a fractional slope.

Key Terms and Formulas:

• Linear equation: $y = mx + b$

• Slope ($m$): $\frac{1}{2}$

• Y-intercept ($b$): 2

Step-by-Step Guidance

1. Identify the y-intercept: The point where $x = 0$.

2. Choose another value for $x$ (e.g., $x = 2$) and solve for $y$.

3. Plot at least two points on the graph.

4. Draw a straight line through the points, but do not finish the graph yet.

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Q5. Simplify by removing factors of 1: $\frac{q^2 - 4}{q^2 - 4q + 4}$

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: $a^2 - b^2 = (a - b)(a + b)$

• Perfect square trinomial: $a^2 - 2ab + b^2 = (a - b)^2$

Step-by-Step Guidance

1. Factor the numerator: $q^2 - 4$.

2. Factor the denominator: $q^2 - 4q + 4$.

3. Write the expression as a product of its factors.

4. Identify and remove any common factors from numerator and denominator, but do not write the final simplified form yet.

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Q6. Reduce the rational expression to lowest terms: $\frac{3t^2 + 21t + 30}{9t^2 - 18t - 72}$

Background

Topic: Simplifying Rational Expressions

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

Key Terms and Formulas:

• Factoring trinomials

• Reducing rational expressions

Step-by-Step Guidance

1. Factor the numerator $3t^2 + 21t + 30$.

2. Factor the denominator $9t^2 - 18t - 72$.

3. Write the expression as a product of its factors.

4. Identify and remove any common factors, but do not write the final reduced form yet.

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Q7. Use the graph of $y = 4^x$ and transformations to sketch $f(x) = 4^{x - 6}$. Determine the domain, range, y-intercept, and the equation of the horizontal asymptote.

Background

Topic: Exponential Functions and Transformations

This question tests your understanding of how exponential functions are transformed and how to find their domain, range, y-intercept, and asymptotes.

Key Terms and Formulas:

• Exponential function: $f(x) = a^{x - h} + k$

• Domain: All possible $x$ values

• Range: All possible $y$ values

• Horizontal asymptote: $y = k$

Step-by-Step Guidance

1. Recognize that $f(x) = 4^{x - 6}$ is a horizontal shift of $y = 4^x$ to the right by 6 units.

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

3. Recall that the range is all positive real numbers for $a > 0$.

4. Find the y-intercept by evaluating $f(0)$, but do not compute the final value yet.

5. State the equation of the horizontal asymptote, but do not write the final answer yet.

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Q8. Use the graph of $y = 3^x$ and transformations to sketch $f(x) = \frac{1}{3^x} + 6$. Determine the domain, range, y-intercept, and the equation of the horizontal asymptote.

Background

Topic: Exponential Functions and Transformations

This question tests your understanding of how to graph exponential functions with transformations, and how to find their domain, range, y-intercept, and asymptotes.

Key Terms and Formulas:

• Exponential function: $f(x) = a^{x} + k$

• Domain: All real numbers

• Range: Depends on vertical shift

• Horizontal asymptote: $y = k$

Step-by-Step Guidance

1. Rewrite $f(x)$ as $f(x) = 3^{-x} + 6$ to see the reflection.

2. Recognize the reflection over the y-axis and the vertical shift up by 6 units.

3. Recall the domain is all real numbers.

4. Find the range by considering the minimum value of $3^{-x}$.

5. Find the y-intercept by evaluating $f(0)$, but do not compute the final value yet.

6. State the equation of the horizontal asymptote, but do not write the final answer yet.

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Q9. Rewrite as a sum or difference of multiples of logarithms: $\log_b(x^4 y^5 z)$

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: $\log_b(MN) = \log_b M + \log_b N$

• Power rule: $\log_b(M^k) = k \log_b M$

Step-by-Step Guidance

1. Apply the product rule to separate $x^4$, $y^5$, and $z$.

2. Apply the power rule to $x^4$ and $y^5$.

3. Write the expanded expression, but do not combine or simplify further yet.

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Q10. Express $\log_b\left(\frac{x y^3}{w^3 z}\right)$ as a sum and/or difference of logarithms.

Background

Topic: Properties of Logarithms

This question tests your ability to expand logarithmic expressions involving quotients and powers.

Key Terms and Formulas:

• Quotient rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Product rule: $\log_b(MN) = \log_b M + \log_b N$

• Power rule: $\log_b(M^k) = k \log_b M$

Step-by-Step Guidance

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

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

3. Apply the power rule to $y^3$ and $w^3$.

4. Write the expanded expression, but do not combine or simplify further yet.

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Q11. Graph the function $f(x) = 6x - 9$

Background

Topic: Graphing Linear Functions

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

Key Terms and Formulas:

• Linear function: $y = mx + b$

• Slope ($m$): 6

• Y-intercept ($b$): -9

Step-by-Step Guidance

1. Identify the y-intercept: The point where $x = 0$.

2. Find another point by choosing a value for $x$ (e.g., $x = 1$) and solving for $y$.

3. Plot at least two points on the coordinate plane.

4. Draw a straight line through the points, but do not complete the graph yet.

Try solving on your own before revealing the answer!

Q12. Graph the function $f(x) = \frac{x - 7}{2}$

Background

Topic: Graphing Linear Functions

This question tests your ability to graph a linear function with a fractional slope and a horizontal shift.

Key Terms and Formulas:

• Linear function: $y = mx + b$

• Slope ($m$): $\frac{1}{2}$

• Y-intercept: Find by setting $x = 0$

Step-by-Step Guidance

1. Rewrite the function as $f(x) = \frac{1}{2}x - \frac{7}{2}$ to identify slope and y-intercept.

2. Identify the y-intercept: The point where $x = 0$.

3. Find another point by choosing a value for $x$ (e.g., $x = 2$) and solving for $y$.

4. Plot at least two points on the graph.

5. Draw a straight line through the points, but do not finish the graph yet.

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Q13. Graph and state the domain: $y = \frac{12}{x}$

Background

Topic: Rational Functions

This question tests your ability to graph a rational function and determine its domain.

Key Terms and Formulas:

• Rational function: $y = \frac{k}{x}$

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

Step-by-Step Guidance

1. Identify values of $x$ that make the denominator zero.

2. State the domain in interval notation, excluding $x = 0$.

3. Recognize the general shape of the graph (a hyperbola).

4. Plot a few points for positive and negative $x$ values, but do not complete the graph yet.

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Q14. Graph: $y = -\frac{4}{x}$

Background

Topic: Rational Functions

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

Key Terms and Formulas:

• Rational function: $y = \frac{k}{x}$

• Vertical asymptote: $x = 0$

• Horizontal asymptote: $y = 0$

Step-by-Step Guidance

1. Identify values of $x$ that make the denominator zero.

2. Recognize the general shape of the graph (a hyperbola, reflected due to the negative sign).

3. Plot a few points for positive and negative $x$ values, but do not complete the graph yet.

4. Identify the asymptotes, but do not finish the graph yet.

Try solving on your own before revealing the answer!