College Algebra Final Exam Study Guide: Step-by-Step Guidance

Q1. Find the equation of a line that passes through the points (-1,3) and (2,1). Put your final answer in slope-intercept form. State the slope and y-intercept and sketch the line.

Background

Topic: Linear Equations and Slope-Intercept Form

This question tests your ability to find the equation of a line given two points, and to express it in slope-intercept form ($y = mx + b$). You also need to identify the slope and y-intercept, and sketch the line.

Key Terms and Formulas:

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

• Slope-intercept form: $y = mx + b$

• y-intercept ($b$): The value of $y$ when $x = 0$

Step-by-Step Guidance

1. Label the points: Let $(-1,3)$ be $(x_1, y_1)$ and $(2,1)$ be $(x_2, y_2)$.

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

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

4. Rearrange the equation to slope-intercept form ($y = mx + b$).

Try solving on your own before revealing the answer!

Final Answer: $y = -\frac{2}{3}x + \frac{7}{3}$

Slope: $m = -\frac{2}{3}$

y-intercept: $b = \frac{7}{3}$

The line passes through the given points and is sketched on the provided graph.

Q2. Solve the equation: $2(x+2) - 3x = 12$

Background

Topic: Solving Linear Equations

This question tests your ability to solve a linear equation for $x$ by applying algebraic operations.

Key Terms and Formulas:

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

• Distributive property: $a(b + c) = ab + ac$

Step-by-Step Guidance

1. Expand $2(x+2)$ using the distributive property: $2x + 4$.

2. Rewrite the equation: $2x + 4 - 3x = 12$.

3. Combine like terms: $2x - 3x$.

4. Isolate $x$ by moving constants to the other side.

Try solving on your own before revealing the answer!

Final Answer: $x = -8$

After combining like terms and isolating $x$, you find $x = -8$.

Q3. Sketch the graph of $g(x) = (x+2)^2 - 1$. Include at least one point on either side of the vertex. Identify the transformations that took place from $g(x) = x^2$. State the domain and range.

Background

Topic: Quadratic Functions and Transformations

This question tests your understanding of how to graph a quadratic function, identify its vertex, and describe transformations from the parent function $y = x^2$.

Key Terms and Formulas:

• Vertex form: $y = a(x-h)^2 + k$

• Transformation: Horizontal shift ($h$), vertical shift ($k$)

• Domain: All possible $x$ values

• Range: All possible $y$ values

Step-by-Step Guidance

1. Identify the vertex: The function is in vertex form, so the vertex is $(-2, -1)$.

2. Describe the transformations: The graph is shifted left by 2 units and down by 1 unit from $y = x^2$.

3. Choose points on either side of the vertex (e.g., $x = -3$ and $x = -1$) and calculate $g(x)$ for those values.

4. State the domain and range: For a quadratic, domain is all real numbers; range starts at the vertex's $y$ value.

Try solving on your own before revealing the answer!

Final Answer:

Vertex: $(-2, -1)$

Domain: $(-\infty, \infty)$

Range: $[-1, \infty)$

Points: $(-3, 0)$ and $(-1, 0)$

The graph is a parabola opening upwards, shifted left 2 units and down 1 unit.