College Algebra Exam I Review – Step-by-Step Guidance

Q1. Find the center and radius of the following circles:

• $(x + 5)^2 + (y + 9.5)^2 = 30.25$

• $(x - 2)^2 + (y + 2)^2 = 25$

Background

Topic: Equations of Circles

This question tests your understanding of the standard form of a circle's equation and how to extract the center and radius from it.

Key Terms and Formulas

• Standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$

• Center: $(h, k)$

• Radius: $r$ (where $r^2$ is the number on the right side of the equation)

Step-by-Step Guidance

1. Compare each given equation to the standard form $(x - h)^2 + (y - k)^2 = r^2$.

2. Identify the values of $h$ and $k$ by looking at the expressions inside the parentheses for $x$ and $y$.

3. Remember that $(x + a)^2$ can be rewritten as $(x - (-a))^2$, so the center's coordinates may be negative.

4. Find the value of $r^2$ on the right side of the equation, then take the square root to get $r$.

5. Write the center as $(h, k)$ and the radius as $r$ for each circle.

Try solving on your own before revealing the answer!

Final Answer:

• For $(x + 5)^2 + (y + 9.5)^2 = 30.25$: Center: $(-5, -9.5)$ Radius: $r = \sqrt{30.25} = 5.5$

• For $(x - 2)^2 + (y + 2)^2 = 25$: Center: $(2, -2)$ Radius: $r = \sqrt{25} = 5$

We matched each equation to the standard form and identified the center and radius accordingly.

Q2. Solve each inequality:

• $-5x + 13 \geq 4$ or $x - 7 \geq 4x + 3$

• $-24 < 2x + 18 \leq 12$

• $25x - 4 \geq 7x - 40$

Background

Topic: Solving Linear Inequalities

This question tests your ability to solve single and compound linear inequalities, including those with 'or' and 'and' statements.

Key Terms and Formulas

• Linear inequality: An inequality involving a linear expression.

• "Or" means the solution is any value that satisfies at least one inequality.

• "And" (as in compound inequalities) means the solution must satisfy all parts simultaneously.

Step-by-Step Guidance

1. For each inequality, isolate $x$ by performing inverse operations (add, subtract, multiply, or divide both sides).

2. For compound inequalities (like $-24 < 2x + 18 \leq 12$), break it into two parts and solve each, then find the intersection.

3. For inequalities with "or," solve each part separately, then combine the solution sets (union).

4. Remember: If you multiply or divide both sides by a negative number, reverse the inequality sign.

5. Express your answers in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

• $-5x + 13 \geq 4$ or $x - 7 \geq 4x + 3$ $x \leq 1.8$ or $x \leq -3.33$ Solution: $x \leq 1.8$

• $-24 < 2x + 18 \leq 12$ $-21 < x \leq -3$

• $25x - 4 \geq 7x - 40$ $x \geq -2$

Each inequality was solved step by step, and the solutions are given in interval notation where appropriate.

Q3. Find the domains of the following functions:

• $f(x) = \sqrt{1 - 1 - 3x}$

• $f(x) = \frac{\sqrt{x}}{1 - x}$

• $f(x) = \sqrt[3]{x - 9}$

• $f(x) = 12x^2 - 300$

Background

Topic: Domain of Functions

This question tests your ability to determine the set of all real numbers for which a function is defined, considering roots and denominators.

Key Terms and Formulas

• Domain: The set of all real numbers $x$ for which $f(x)$ is defined.

• Square root: The radicand must be $\geq 0$.

• Denominator: Cannot be zero.

• Cube root: Defined for all real numbers.

• Polynomial: Defined for all real numbers.

Step-by-Step Guidance

1. For each function, identify any restrictions (square roots, denominators, etc.).

2. For square roots, set the radicand $\geq 0$ and solve for $x$.

3. For denominators, set the denominator $\neq 0$ and solve for $x$.

4. For cube roots and polynomials, recall that their domains are all real numbers unless otherwise restricted.

5. Write the domain in interval notation for each function.

Try solving on your own before revealing the answer!

Final Answer:

• $f(x) = \sqrt{1 - 1 - 3x}$ Domain: $x \leq 0$

• $f(x) = \frac{\sqrt{x}}{1 - x}$ Domain: $x \geq 0$, $x \neq 1$

• $f(x) = \sqrt[3]{x - 9}$ Domain: $(-\infty, \infty)$

• $f(x) = 12x^2 - 300$ Domain: $(-\infty, \infty)$

We considered the restrictions for each function and wrote the domains accordingly.

Q4. For each line, find an equation of the parallel line passing through $(-1, 3)$:

• $y = 2x - 3$

• $y = x$

Background

Topic: Parallel Lines and Point-Slope Form

This question tests your ability to write the equation of a line parallel to a given line and passing through a specific point.

Key Terms and Formulas

• Parallel lines have the same slope.

• Point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Identify the slope $m$ of the given line (from the equation).

2. Use the same slope for the parallel line.

3. Plug the point $(-1, 3)$ and the slope $m$ into the point-slope form.

4. Simplify the equation to slope-intercept form if needed.

Try solving on your own before revealing the answer!

Final Answer:

• Parallel to $y = 2x - 3$ through $(-1, 3)$: $y = 2x + 5$

• Parallel to $y = x$ through $(-1, 3)$: $y = x + 4$

We used the same slope as the given line and the point-slope formula to find the equations.

Q5. For each line, find the equation of the perpendicular line passing through $(-2, 4)$:

• $y = 2x - 3$

• $y = x$

Background

Topic: Perpendicular Lines and Point-Slope Form

This question tests your ability to find the equation of a line perpendicular to a given line and passing through a specific point.

Key Terms and Formulas

• Perpendicular lines: Slopes are negative reciprocals ($m_1 = -1/m_2$).

• Point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Find the slope $m$ of the given line.

2. Calculate the negative reciprocal of $m$ for the perpendicular line.

3. Use the point $(-2, 4)$ and the new slope in the point-slope form.

4. Simplify to slope-intercept form if needed.

Try solving on your own before revealing the answer!

Final Answer:

• Perpendicular to $y = 2x - 3$ through $(-2, 4)$: $y = -\frac{1}{2}x + 3$

• Perpendicular to $y = x$ through $(-2, 4)$: $y = -x + 2$

We found the negative reciprocal of the original slope and used the point-slope formula.

Q6. Consider the functions $f(x) = \sqrt{2 - 3x}$ and $g(x) = x^2 - 3x + 2$. Find the following:

• $(f + g)(-2)$

• $(f / g)(-4)$

• $(f \circ g)(-5)$

• $(g \circ f)(-1)$

Background

Topic: Function Operations and Composition

This question tests your ability to evaluate function operations (addition, division) and compositions at specific values.

Key Terms and Formulas

• Function addition: $(f + g)(x) = f(x) + g(x)$

• Function division: $(f / g)(x) = \frac{f(x)}{g(x)}$

• Composition: $(f \circ g)(x) = f(g(x))$

Step-by-Step Guidance

1. For each part, substitute the given value into the appropriate function(s).

2. For $(f + g)(-2)$, calculate $f(-2)$ and $g(-2)$, then add the results.

3. For $(f / g)(-4)$, calculate $f(-4)$ and $g(-4)$, then divide $f(-4)$ by $g(-4)$.

4. For compositions, first evaluate the inner function, then substitute that result into the outer function.

5. Be careful with domains: ensure the values you substitute are valid for the functions involved.

Try solving on your own before revealing the answer!

Final Answer:

• $(f + g)(-2) = 4 + 16 = 20$

• $(f / g)(-4) = 4 / 42 = 2 / 21$

• $(f \circ g)(-5) = \sqrt{2 - 3(42)} = \sqrt{2 - 126} = \sqrt{-124}$ (not a real number)

• $(g \circ f)(-1) = g(f(-1)) = g(5) = 12$

We evaluated each function or composition step by step, checking for domain issues.

Q7. Graph the following lines on the grid paper provided:

• $y - 5 = 1.5(x + 2)$

• $y = -2x + 4$

Background

Topic: Graphing Linear Equations

This question tests your ability to interpret and graph lines given in point-slope and slope-intercept forms.

Key Terms and Formulas

• Point-slope form: $y - y_1 = m(x - x_1)$

• Slope-intercept form: $y = mx + b$

• Slope ($m$): Rise over run

• Y-intercept ($b$): Where the line crosses the y-axis

Step-by-Step Guidance

1. Rewrite $y - 5 = 1.5(x + 2)$ in slope-intercept form by solving for $y$.

2. Identify the slope and y-intercept for each equation.

3. Plot the y-intercept on the graph for each line.

4. Use the slope to find another point on the line (rise over run).

5. Draw the line through the points for each equation.

Try solving on your own before revealing the answer!

Final Answer:

• $y - 5 = 1.5(x + 2)$ becomes $y = 1.5x + 8$

• $y = -2x + 4$ is already in slope-intercept form

Graph each line using the slope and y-intercept identified above.

Q8. Solve this system of equations:

• $-4x + 3y = 17$

• $5x + 2y = 27$

Background

Topic: Solving Systems of Linear Equations

This question tests your ability to solve a system of two linear equations using substitution or elimination.

Key Terms and Formulas

• System of equations: Two or more equations with the same variables.

• Elimination method: Add or subtract equations to eliminate a variable.

• Substitution method: Solve one equation for a variable and substitute into the other.

Step-by-Step Guidance

1. Decide whether to use substitution or elimination. Here, elimination is efficient.

2. Multiply one or both equations if necessary so that the coefficients of $x$ or $y$ are opposites.

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

4. Solve for the remaining variable.

5. Substitute back into one of the original equations to find the other variable.

Try solving on your own before revealing the answer!

Final Answer:

$x = 2$, $y = 5$

We used the elimination method to solve for one variable, then substituted back to find the other.