College Algebra Final Exam Study Guide – Key Concepts and Practice Problems

Q1. Identify the x- and y-intercepts of the given graph.

Background

Topic: Intercepts of a Function

This question tests your ability to read a graph and determine where the function crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Key Terms and Formulas:

• x-intercept: The point(s) where the graph crosses the x-axis. Set $y = 0$ and solve for $x$.

• y-intercept: The point where the graph crosses the y-axis. Set $x = 0$ and solve for $y$.

Step-by-Step Guidance

1. Look at the graph and identify all points where the curve crosses the x-axis. These are your x-intercepts.

2. Identify the point where the curve crosses the y-axis. This is your y-intercept.

3. Write the intercepts as coordinate points: $(x, 0)$ for x-intercepts and $(0, y)$ for the y-intercept.

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Q2. Determine if the given graph represents a function.

Background

Topic: Functions and the Vertical Line Test

This question tests your understanding of what makes a relation a function, specifically using the vertical line test.

Key Terms and Formulas:

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

• Vertical Line Test: If any vertical line crosses the graph more than once, the graph is not a function.

Step-by-Step Guidance

1. Imagine drawing vertical lines through different parts of the graph.

2. Check if any vertical line crosses the graph at more than one point.

3. If all vertical lines cross at most once, the graph is a function. Otherwise, it is not.

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Q3. Describe the transformation: Shift 3 units left and 2 units down.

Background

Topic: Transformations of Functions

This question tests your understanding of how shifting a function horizontally and vertically affects its graph.

Key Terms and Formulas:

• Horizontal shift: $f(x + h)$ shifts the graph $h$ units left if $h > 0$.

• Vertical shift: $f(x) + k$ shifts the graph $k$ units up if $k > 0$, down if $k < 0$.

Step-by-Step Guidance

1. Identify the direction and magnitude of each shift from the description.

2. Write the new function in terms of $f(x)$, applying the shifts: $f(x + 3) - 2$.

3. Understand that the graph moves left by 3 units and down by 2 units.

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Q4. Identify intervals where the function is increasing, decreasing, or constant from the graph.

Background

Topic: Increasing/Decreasing/Constant Intervals

This question tests your ability to analyze a graph and describe where the function rises, falls, or stays flat.

Key Terms and Formulas:

• Increasing: The function rises as $x$ increases.

• Decreasing: The function falls as $x$ increases.

• Constant: The function remains flat as $x$ increases.

Step-by-Step Guidance

1. Look at the graph and identify intervals where the function goes up, down, or stays flat as you move from left to right.

2. Write the intervals using interval notation, based on the $x$-values where the behavior changes.

3. Label each interval as increasing, decreasing, or constant.

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Q5. Given $f(x)$ and $g(x)$, perform the indicated operations: $f(x) + g(x)$, $f(x) - g(x)$, $f(x)g(x)$, $\frac{f(x)}{g(x)}$.

Background

Topic: Operations with Functions

This question tests your ability to add, subtract, multiply, and divide functions.

Key Terms and Formulas:

• Addition: $(f + g)(x) = f(x) + g(x)$

• Subtraction: $(f - g)(x) = f(x) - g(x)$

• Multiplication: $(fg)(x) = f(x)g(x)$

• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Step-by-Step Guidance

1. Write out $f(x)$ and $g(x)$ as given in the problem.

2. For each operation, substitute the expressions for $f(x)$ and $g(x)$ and perform the indicated operation (combine like terms, multiply, or divide as appropriate).

3. Simplify each result as much as possible.

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Q6. Find and simplify the difference quotient for the given function.

Background

Topic: Difference Quotient

This question tests your ability to compute and simplify the difference quotient, which is foundational for calculus.

Key Terms and Formulas:

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$

Step-by-Step Guidance

1. Find $f(x+h)$ by substituting $x+h$ into the function $f(x)$.

2. Subtract $f(x)$ from $f(x+h)$.

3. Divide the result by $h$ and simplify the expression as much as possible.

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Q7. Determine whether the given function is even, odd, or neither.

Background

Topic: Even and Odd Functions

This question tests your ability to classify functions based on their symmetry.

Key Terms and Formulas:

• Even function: $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the y-axis).

• Odd function: $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

Step-by-Step Guidance

1. Substitute $-x$ into the function and simplify $f(-x)$.

2. Compare $f(-x)$ to $f(x)$ and $-f(x)$ to determine if the function is even, odd, or neither.

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Q8. Find the vertex (maximum or minimum) of the quadratic function $f(x) = ax^2 + bx + c$.

Background

Topic: Vertex of a Parabola

This question tests your ability to find the vertex of a quadratic function, which is the maximum or minimum point of the parabola.

Key Terms and Formulas:

• Vertex formula: $x = -\frac{b}{2a}$

• Plug this $x$ value into $f(x)$ to find the $y$ coordinate of the vertex.

Step-by-Step Guidance

1. Identify the coefficients $a$ and $b$ from the quadratic function.

2. Calculate $x = -\frac{b}{2a}$.

3. Substitute this $x$ value back into $f(x)$ to find the $y$ coordinate of the vertex.

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Q9. Find the inverse of the one-to-one function.

Background

Topic: Inverse Functions

This question tests your ability to find the inverse of a function, which "undoes" the original function.

Key Terms and Formulas:

• Inverse function: Swap $x$ and $y$ in the equation, then solve for $y$.

Step-by-Step Guidance

1. Replace $f(x)$ with $y$ in the function.

2. Swap $x$ and $y$ in the equation.

3. Solve for $y$ to get the inverse function $f^{-1}(x)$.

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Q10. Find all the zeros of the polynomial function.

Background

Topic: Zeros of Polynomial Functions

This question tests your ability to find the roots (zeros) of a polynomial, which are the $x$-values where $f(x) = 0$.

Key Terms and Formulas:

• Zero of a function: A value of $x$ for which $f(x) = 0$.

• Set the polynomial equal to zero and solve for $x$ (factoring, quadratic formula, or synthetic division).

Step-by-Step Guidance

1. Set the polynomial equal to zero: $f(x) = 0$.

2. Factor the polynomial if possible, or use the quadratic formula if it is quadratic.

3. Solve for all possible $x$ values that make $f(x) = 0$.

Try solving on your own before revealing the answer!