College Algebra Study Guide: Graphs, Equations, and Algebraic Concepts

Q1. Find the equation of the line passing through the points (2, 3) and (4, 7).

Background

Topic: Linear Equations

This question tests your ability to find the equation of a line given two points. This is a fundamental skill in college algebra, involving slope calculation and using the point-slope or slope-intercept form.

Key Terms and Formulas:

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

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

• Slope-intercept form: $y = mx + b$

Step-by-Step Guidance

1. Identify the coordinates: $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (4, 7)$.

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

3. Plug the slope and one point into the point-slope form: $y - y_1 = m(x - x_1)$.

4. Simplify the equation to get it into slope-intercept form ($y = mx + b$).

Try solving on your own before revealing the answer!

Q2. Solve the equation $2x - 5 = 9$ for $x$.

Background

Topic: Solving Linear Equations

This question tests your ability to solve a basic linear equation for the variable $x$.

Key Terms and Formulas:

• Linear equation: An equation of the form $ax + b = c$.

• Solving for $x$: Isolate $x$ by performing inverse operations.

Step-by-Step Guidance

1. Add 5 to both sides to isolate the term with $x$.

2. Divide both sides by 2 to solve for $x$.

Try solving on your own before revealing the answer!

Q3. Graph the equation $y = -2x + 4$.

Background

Topic: Graphing Linear Equations

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

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• Slope ($m$): Indicates the steepness and direction of the line.

• Y-intercept ($b$): The point where the line crosses the y-axis.

Step-by-Step Guidance

1. Identify the slope ($m = -2$) and y-intercept ($b = 4$).

2. Plot the y-intercept (0, 4) on the graph.

3. Use the slope to find another point: from (0, 4), move down 2 units and right 1 unit.

4. Draw the line through these points.

Try graphing on your own before revealing the answer!

Q4. Find the coordinates of point H on the graph.

Background

Topic: Coordinate Geometry

This question tests your ability to read and interpret coordinates from a graph.

Key Terms and Formulas:

• Coordinates: $(x, y)$ values that specify a point's location on the plane.

Step-by-Step Guidance

1. Locate point H on the graph.

2. Read the x-coordinate (horizontal axis) and y-coordinate (vertical axis).

Try identifying the coordinates before revealing the answer!

Q5. Simplify the expression $3x^2 - 2x + 4x^2 + x$.

Background

Topic: Combining Like Terms

This question tests your ability to simplify algebraic expressions by combining like terms.

Key Terms and Formulas:

• Like terms: Terms with the same variable and exponent.

• Combine like terms by adding their coefficients.

Step-by-Step Guidance

1. Identify like terms: $3x^2$ and $4x^2$; $-2x$ and $x$.

2. Add the coefficients of like terms together.

Try simplifying on your own before revealing the answer!

Q6. Solve the quadratic equation $x^2 - 5x + 6 = 0$.

Background

Topic: Solving Quadratic Equations

This question tests your ability to solve quadratic equations by factoring.

Key Terms and Formulas:

• Quadratic equation: $ax^2 + bx + c = 0$

• Factoring: Expressing the equation as a product of binomials.

Step-by-Step Guidance

1. Identify $a = 1$, $b = -5$, $c = 6$.

2. Find two numbers that multiply to $c$ and add to $b$.

3. Write the equation in factored form.

Try factoring on your own before revealing the answer!

Q7. Find the distance between the points (1, 2) and (4, 6).

Background

Topic: Distance Formula

This question tests your ability to use the distance formula to find the length between two points in the coordinate plane.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Identify the coordinates: $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$.

2. Calculate the differences: $x_2 - x_1$ and $y_2 - y_1$.

3. Square the differences and add them together.

4. Take the square root of the sum.

Try calculating the distance before revealing the answer!