College Algebra Quiz 2 Review – Functions, Graphs, and Linear Equations

Q1. What is the first step to determine if y is a function of x?

Background

Topic: Functions and Relations

This question is testing your understanding of how to determine if a relation is a function, specifically using the vertical line test or by checking if each input (x-value) has only one output (y-value).

Key Terms:

• Function: A relation where each input has exactly one output.

• Vertical Line Test: A graphical method to determine if a curve is a function.

Step-by-Step Guidance

1. Examine the relation or graph provided. Ask yourself: For every x-value, is there only one corresponding y-value?

2. If you are looking at a graph, imagine drawing vertical lines through the graph. Does any vertical line cross the graph more than once?

3. If a vertical line crosses the graph more than once, the relation is not a function.

Try solving on your own before revealing the answer!

Q2. How do you write the domain of a function in interval notation?

Background

Topic: Domain and Range

This question is about expressing the set of all possible input values (x-values) for a function using interval notation.

Key Terms:

• Domain: The set of all possible x-values for which the function is defined.

• Interval Notation: A way to write subsets of the real numbers using parentheses and brackets.

Step-by-Step Guidance

1. Identify the set of x-values for which the function is defined (no division by zero, no square roots of negative numbers, etc.).

2. Write the domain using interval notation. Use parentheses for values not included, and brackets for values that are included.

3. For example, if x can be any real number except 2, you would write $(-\infty, 2) \cup (2, \infty)$.

Try solving on your own before revealing the answer!

Q3. What is the distance formula, and how is it used?

Background

Topic: Coordinate Geometry

This question is about finding the distance between two points in the coordinate plane using the distance formula.

Key Formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

• $d$ is the distance between the points.

Step-by-Step Guidance

1. Identify the coordinates of the two points you want to find the distance between.

2. Subtract the x-coordinates and y-coordinates to find $(x_2 - x_1)$ and $(y_2 - y_1)$.

3. Square both differences, add them together, and then take the square root of the sum.

Try solving on your own before revealing the answer!

Q4. How do you write the midpoint formula, and what does it find?

Background

Topic: Coordinate Geometry

This question is about finding the midpoint between two points in the coordinate plane.

Key Formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points.

• $M$ is the midpoint between the points.

Step-by-Step Guidance

1. Add the x-coordinates of the two points and divide by 2 to get the x-coordinate of the midpoint.

2. Add the y-coordinates of the two points and divide by 2 to get the y-coordinate of the midpoint.

3. Write the midpoint as an ordered pair.

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Q5. What is the general form (slope-intercept form) of a line?

Background

Topic: Linear Equations

This question is about writing the equation of a line in slope-intercept form.

Key Formula:

$y = mx + b$

• $m$ is the slope of the line.

• $b$ is the y-intercept (where the line crosses the y-axis).

Step-by-Step Guidance

1. Identify the slope ($m$) and y-intercept ($b$) from the problem or graph.

2. Substitute these values into the formula $y = mx + b$.

3. Simplify the equation if needed.

Try solving on your own before revealing the answer!

Q6. What do you know about the slopes of parallel lines?

Background

Topic: Linear Equations and Geometry

This question is about understanding the relationship between the slopes of parallel lines.

Key Terms:

• Parallel Lines: Lines in the same plane that never intersect.

• Slope: The measure of steepness of a line.

Step-by-Step Guidance

1. Recall that parallel lines have the same slope.

2. If two lines are parallel, their equations will have identical $m$ values in $y = mx + b$.

3. Check the slopes in the given equations to confirm if the lines are parallel.

Try solving on your own before revealing the answer!