College Algebra Exam I Practice Guidance

Q1. Draw the graphs of 2 different equations that are functions. Justify why they are functions.

Background

Topic: Functions and Graphs

This question tests your understanding of what makes an equation a function and how to represent it graphically.

Key Terms:

• Function: A relation where each input (x-value) has exactly one output (y-value).

• Vertical Line Test: A graphical method to determine if a graph represents a function.

Step-by-Step Guidance

1. Choose two equations that are functions, such as $y = x$ and $y = x^2$.

2. Sketch the graphs for each equation on separate axes.

3. Apply the vertical line test: For each graph, check that no vertical line crosses the graph more than once.

4. Justify why each equation is a function based on the definition and the vertical line test.

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Q2. Draw the graphs of 2 different equations that are not functions. Justify why they are not functions.

Background

Topic: Relations vs. Functions

This question asks you to distinguish between functions and non-functions using graphs.

Key Terms:

• Non-function: A relation where at least one input (x-value) corresponds to more than one output (y-value).

• Vertical Line Test: Used to check if a graph is not a function.

Step-by-Step Guidance

1. Choose two equations that are not functions, such as $x = y^2$ (a sideways parabola) and $x^2 + y^2 = 1$ (a circle).

2. Sketch the graphs for each equation.

3. Apply the vertical line test: For each graph, show that some vertical lines cross the graph more than once.

4. Justify why each equation is not a function based on the definition and the vertical line test.

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Q3. Graph the equation $6x + 2y = 4$ on one set of axes. Indicate at least two points and sketch the general behavior. Do the same for a line with slope 3 and y-intercept -1. Do the same for $y = x^2$.

Background

Topic: Graphing Linear and Quadratic Equations

This question tests your ability to graph equations, identify points, and understand slope and intercepts.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• Quadratic equation: $y = x^2$

Step-by-Step Guidance

1. For $6x + 2y = 4$, solve for $y$ to get it in slope-intercept form.

2. Choose two values for $x$ (e.g., $x = 0$ and $x = 1$), and solve for $y$ to find two points.

3. Plot these points and sketch the line.

4. For the line with slope 3 and y-intercept -1, write the equation as $y = 3x - 1$ and plot the y-intercept and another point using the slope.

5. For $y = x^2$, plot several points (e.g., $x = -2, -1, 0, 1, 2$) and sketch the parabola.

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Q4. Solve the following equations:

Background

Topic: Solving Linear Equations

This question tests your ability to manipulate and solve equations for unknown variables.

Key Terms and Formulas:

• Isolate the variable: Use algebraic operations to get the variable alone on one side.

Step-by-Step Guidance

1. For (a) $\frac{b-1}{2} = b$, multiply both sides by 2 to eliminate the denominator.

2. Rearrange the equation to collect like terms and solve for $b$.

3. For (b) $2(x-1) - 2(x-1) = 0$, expand both terms and simplify.

4. Combine like terms and solve for $x$.

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Q5. For the equation $\frac{2y}{3} + 2 = \frac{3x}{9}$:

Background

Topic: Solving Equations and Finding Intercepts

This question tests your ability to check solutions, find intercepts, and solve for variables.

Key Terms and Formulas:

• Ordered pair: $(x, y)$ values that satisfy the equation.

• x-intercept: Where $y = 0$.

• y-intercept: Where $x = 0$.

Step-by-Step Guidance

1. For (a), substitute $x = \frac{15}{2}$ and $y = 9$ into the equation and check if both sides are equal.

2. For (b), set $y = 0$ to find the x-intercept, and set $x = 0$ to find the y-intercept.

3. For (c), choose values for $x$ and $y$ (not zero) and solve the equation to find a solution.

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Q6. Calculate the following for the given functions:

Background

Topic: Function Evaluation and Composition

This question tests your ability to evaluate functions and compose them.

Key Terms and Formulas:

• Function notation: $f(x)$, $g(x)$, $h(x)$, $k(x)$

• Composition: $k(g(h(9)))$ means evaluate $h(9)$, then $g$ at that value, then $k$ at that result.

Step-by-Step Guidance

1. For (a), evaluate each function at the specified input and add the results.

2. For (b), substitute $a + b + c$ into $f(x)$ and simplify.

3. For (c), start by evaluating $h(9)$, then use that result as the input for $g$, then use that result as the input for $k$.

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Q7. Calculate the domain and range of $g(x)$ and $k(x)$.

Background

Topic: Domain and Range of Functions

This question tests your understanding of which inputs and outputs are possible for a function.

Key Terms:

• Domain: Set of all possible input values ($x$).

• Range: Set of all possible output values ($y$).

Step-by-Step Guidance

1. For $g(x) = \sqrt{2x + 1}$, determine for which $x$ values the expression under the square root is non-negative.

2. For $k(x) = \frac{1}{x}$, determine for which $x$ values the denominator is not zero.

3. Describe the range for each function based on their definitions.

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Q8. Find both zeroes of $f(x)$. Find the zeroes of $g(x)$.

Background

Topic: Finding Roots of Functions

This question tests your ability to solve equations by setting the function equal to zero.

Key Terms and Formulas:

• Zeroes: Values of $x$ where $f(x) = 0$.

• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2 + bx + c = 0$.

Step-by-Step Guidance

1. Set $f(x) = 2x^2 + 7x + 3 = 0$ and identify $a$, $b$, and $c$.

2. Apply the quadratic formula to solve for $x$.

3. For $g(x) = \sqrt{2x + 1}$, set $g(x) = 0$ and solve for $x$.

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Q9. Write the equation for the function $g(x)$ such that $g(3) = 16$ and the graph passes through $(1,5)$.

Background

Topic: Constructing Functions from Points

This question tests your ability to write a function given specific values and points.

Key Terms and Formulas:

• Linear function: $g(x) = mx + b$

• System of equations: Use two points to solve for $m$ and $b$.

Step-by-Step Guidance

1. Assume $g(x)$ is linear: $g(x) = mx + b$.

2. Plug in $x = 3$, $g(3) = 16$ to get one equation.

3. Plug in $x = 1$, $g(1) = 5$ to get a second equation.

4. Solve the system for $m$ and $b$.

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Q10. Write the equation for the line that has y-intercept 3 and x-intercept 2.

Background

Topic: Writing Linear Equations

This question tests your ability to write the equation of a line given intercepts.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• Intercepts: y-intercept ($b$), x-intercept (where $y = 0$)

Step-by-Step Guidance

1. Write the points: y-intercept $(0,3)$, x-intercept $(2,0)$.

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

3. Write the equation using the slope and y-intercept.

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Q11. Write the equation for the line that has slope 2 and x-intercept 6.

Background

Topic: Writing Linear Equations

This question tests your ability to write the equation of a line given a slope and an intercept.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• x-intercept: $(6, 0)$

Step-by-Step Guidance

1. Use the x-intercept $(6, 0)$ and the slope $m = 2$.

2. Plug these values into the point-slope form: $y - y_1 = m(x - x_1)$.

3. Simplify to get the equation in slope-intercept form.

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Q12. John rents a car for $200 plus $0.20 per mile. Write a linear function for the cost per mile. Use it to determine the cost for 350 miles.

Background

Topic: Linear Functions and Applications

This question tests your ability to model real-world situations with linear functions.

Key Terms and Formulas:

• Linear function: $C(m) = 200 + 0.20m$

• Where $m$ is the number of miles driven.

Step-by-Step Guidance

1. Write the function: $C(m) = 200 + 0.20m$.

2. Substitute $m = 350$ into the function.

3. Set up the calculation for the total cost.

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Q13. An oil company increased production from 1.3 million barrels in 2012 to 1.6 million barrels in 2015. What is the average rate of change per year?

Background

Topic: Average Rate of Change

This question tests your ability to calculate the average rate of change over an interval.

Key Terms and Formulas:

• Average rate of change: $\frac{f(b) - f(a)}{b - a}$

Step-by-Step Guidance

1. Identify the initial and final values: $f(a) = 1.3$, $f(b) = 1.6$.

2. Identify the years: $a = 2012$, $b = 2015$.

3. Plug values into the formula: $\frac{1.6 - 1.3}{2015 - 2012}$.

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Q14. Sketch the lines $x = 2$ and $y = 6$.

Background

Topic: Graphing Vertical and Horizontal Lines

This question tests your ability to graph lines defined by $x$ or $y$ equals a constant.

Key Terms:

• Vertical line: $x = 2$

• Horizontal line: $y = 6$

Step-by-Step Guidance

1. For $x = 2$, draw a vertical line through $x = 2$ on the axes.

2. For $y = 6$, draw a horizontal line through $y = 6$ on the axes.

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Q15. India purchased 32,000 industrial robots in 2016, which is 60% more than the US purchased that year. How many did the US purchase?

Background

Topic: Percent Increase and Solving for Original Value

This question tests your ability to work with percentages and solve for unknowns.

Key Terms and Formulas:

• Percent increase: $\text{India} = \text{US} + 0.6 \times \text{US}$

• Equation: $32,000 = 1.6 \times \text{US}$

Step-by-Step Guidance

1. Set up the equation: $32,000 = 1.6 \times \text{US}$.

2. Solve for the US value by dividing both sides by 1.6.

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Q16. Two trains leave a station at different times and speeds. When will they be the same distance from the station?

Background

Topic: Distance, Rate, and Time Problems

This question tests your ability to set up and solve equations involving distance and time.

Key Terms and Formulas:

• Distance formula: $d = rt$

Step-by-Step Guidance

1. Let $t$ be the time after the second train leaves.

2. First train has a 2-hour head start: its time is $t + 2$.

3. Set up equations for each train's distance: $60(t + 2)$ and $80t$.

4. Set the distances equal and solve for $t$.

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Q17. Two runners start on opposite ends of a 30 km trail and run toward each other. Alex runs at 9 km/h, Brady at 7 km/h. How far will Alex have run when they meet?

Background

Topic: Relative Motion and Meeting Point

This question tests your ability to use rates and distances to find when two moving objects meet.

Key Terms and Formulas:

• Combined rate: $9 + 7 = 16$ km/h

• Time to meet: $t = \frac{30}{16}$

• Distance Alex runs: $9t$

Step-by-Step Guidance

1. Calculate the time it takes for them to meet: $t = \frac{30}{16}$.

2. Multiply Alex's speed by the time to find the distance he runs: $9t$.

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Q18. State the domain and range of the functions pictured below (with points at (2,-4) and (-2,-4)).

Background

Topic: Domain and Range from Graphs

This question tests your ability to interpret domain and range from a graph.

Key Terms:

• Domain: All possible $x$ values.

• Range: All possible $y$ values.

Step-by-Step Guidance

1. Look at the graph and identify the $x$ values for which the function is defined.

2. Identify the $y$ values the function takes.

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Q19. Solve the inequality $4|3x - 1| \leq 16$ and write your answer in interval notation and on a number line.

Background

Topic: Solving Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values.

Key Terms and Formulas:

• Absolute value inequality: $|A| \leq B$ means $-B \leq A \leq B$

Step-by-Step Guidance

1. Divide both sides by 4: $|3x - 1| \leq 4$.

2. Rewrite as two inequalities: $-4 \leq 3x - 1 \leq 4$.

3. Solve for $x$ in both inequalities.

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Q20. Solve the inequality $\frac{|x + 1|}{3} + 2 > 305$ and write your answer in interval notation and on a number line.

Background

Topic: Solving Absolute Value Inequalities

This question tests your ability to solve inequalities involving absolute values and fractions.

Key Terms and Formulas:

• Absolute value inequality: $|A| > B$ means $A > B$ or $A < -B$

Step-by-Step Guidance

1. Subtract 2 from both sides: $\frac{|x + 1|}{3} > 303$.

2. Multiply both sides by 3: $|x + 1| > 909$.

3. Rewrite as two inequalities: $x + 1 > 909$ or $x + 1 < -909$.

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