College Algebra Worksheet Guidance: Functions, Graphs, and Linear Equations

Q1. Determine whether the relation is a function: {(−8, −9), (−8, −1), (−1, 8), (3, 9), (8, 3)}

Background

Topic: Functions and Relations

This question tests your understanding of what makes a relation a function, specifically whether each input (x-value) is associated with only one output (y-value).

Key Terms:

• Relation: A set of ordered pairs (x, y).

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

Step-by-Step Guidance

1. List all the x-values from the given pairs: −8, −8, −1, 3, 8.

2. Check if any x-value is repeated with a different y-value.

3. If an x-value is paired with more than one y-value, the relation is not a function.

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Q2. Evaluate the function $f(x) = \frac{x^2 + 6}{x^3 + 6x}$ at $x = 3$.

Background

Topic: Function Evaluation

This question asks you to substitute a specific value into a function and simplify the result.

Key Terms and Formula:

• Function Evaluation: Substitute the given value for x in the function.

Step-by-Step Guidance

1. Write the function: $f(x) = \frac{x^2 + 6}{x^3 + 6x}$.

2. Substitute $x = 3$ into the function: $f(3) = \frac{3^2 + 6}{3^3 + 6 \times 3}$.

3. Calculate the numerator and denominator separately before simplifying the fraction.

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Q3. Use the vertical line test to determine whether the graph is a function of x.

Background

Topic: Graphs of Functions

This question tests your ability to use the vertical line test to determine if a graph represents a function.

Key Terms:

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

Step-by-Step Guidance

1. Visualize or sketch the graph if not provided.

2. Imagine drawing vertical lines at various x-values.

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

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Q4. Use the graph to determine the function's domain and range.

Background

Topic: Domain and Range

This question asks you to identify all possible x-values (domain) and y-values (range) from a graph.

Key Terms:

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

Step-by-Step Guidance

1. Look at the graph and identify the leftmost and rightmost points for the domain.

2. Identify the lowest and highest points for the range.

3. Express the domain and range using interval notation.

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Q5. Identify the intervals where the function is increasing.

Background

Topic: Increasing and Decreasing Intervals

This question asks you to determine where the function's graph is rising as you move from left to right.

Key Terms:

• Increasing Interval: Where the function's y-values increase as x increases.

Step-by-Step Guidance

1. Examine the graph and note where the curve moves upward as you move right.

2. Identify the x-intervals where this occurs.

3. Write the intervals in interval notation.

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Q6. Use the graph of $f(x) = x^3 - 3x^2 + 1$ to find any relative maxima and minima.

Background

Topic: Relative Extrema

This question asks you to find the high and low points (relative maxima and minima) on the graph of a cubic function.

Key Terms:

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

Step-by-Step Guidance

1. Find the critical points by setting the derivative $f'(x)$ to zero.

2. Calculate $f'(x)$ for $f(x) = x^3 - 3x^2 + 1$.

3. Solve $f'(x) = 0$ to find the x-values of possible extrema.

4. Use the second derivative or the graph to determine if each critical point is a maximum or minimum.

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Q7. Evaluate the piecewise function $f(x) = \begin{cases} x - 2 & \text{if } x > -5 \\ -(x - 2) & \text{if } x \leq -5 \end{cases}$ at $x = -8$.

Background

Topic: Piecewise Functions

This question tests your ability to evaluate a function defined by different expressions depending on the value of x.

Key Terms:

• Piecewise Function: A function defined by different formulas for different intervals of the domain.

Step-by-Step Guidance

1. Determine which part of the piecewise function applies for $x = -8$.

2. Substitute $x = -8$ into the appropriate formula.

3. Simplify the expression to find the value of $f(-8)$.

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Q8. Find and simplify the difference quotient $\frac{f(x + h) - f(x)}{h}$, $h \neq 0$, for $f(x) = 9x^2$.

Background

Topic: Difference Quotient

This question asks you to compute the difference quotient, which is foundational for understanding derivatives.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Find $f(x + h)$ by substituting $x + h$ into $f(x)$: $f(x + h) = 9(x + h)^2$.

2. Expand $9(x + h)^2$ using the binomial theorem.

3. Subtract $f(x) = 9x^2$ from $f(x + h)$.

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

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Q9. Find and simplify the difference quotient $\frac{f(x + h) - f(x)}{h}$, $h \neq 0$, for $f(x) = x^2 + 2x - 3$.

Background

Topic: Difference Quotient

This question is similar to Q8 but with a different quadratic function.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Find $f(x + h)$ by substituting $x + h$ into $f(x)$: $f(x + h) = (x + h)^2 + 2(x + h) - 3$.

2. Expand and simplify $f(x + h)$.

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

4. Divide the result by $h$ and simplify.

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Q10. Graph the line whose equation is $y = -\frac{2}{5}x + 3$.

Background

Topic: Graphing Linear Equations

This question asks you to graph a line given in slope-intercept form.

Key Terms and Formula:

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Step-by-Step Guidance

1. Identify the slope ($m$) and y-intercept ($b$) from the equation.

2. Plot the y-intercept on the y-axis.

3. Use the slope to find another point on the line.

4. Draw the line through these points.

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Q11. Graph the equation $3x - 2 = -17$ in the rectangular coordinate system.

Background

Topic: Graphing Linear Equations

This question asks you to rearrange and graph a linear equation.

Key Terms and Formula:

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

Step-by-Step Guidance

1. Solve the equation for $x$ or $y$ to put it in a recognizable form.

2. Identify if the equation represents a vertical or horizontal line.

3. Plot the line on the coordinate plane accordingly.

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Q12. Determine the slope and the y-intercept of the graph of the equation $5x + y - 6 = 0$.

Background

Topic: Slope and Intercept

This question asks you to rewrite a linear equation in slope-intercept form and identify its slope and y-intercept.

Key Terms and Formula:

• Slope-Intercept Form: $y = mx + b$

• Slope ($m$): The coefficient of $x$.

• Y-intercept ($b$): The constant term.

Step-by-Step Guidance

1. Rewrite the equation $5x + y - 6 = 0$ in the form $y = mx + b$.

2. Identify the values of $m$ and $b$ from the new equation.

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Q13. Write the equation of the line passing through (4, 2) and perpendicular to the line $y = 9x + 7$ in point-slope form.

Background

Topic: Equations of Lines

This question tests your understanding of perpendicular lines and writing equations in point-slope form.

Key Terms and Formula:

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Perpendicular Slopes: If two lines are perpendicular, their slopes are negative reciprocals.

Step-by-Step Guidance

1. Identify the slope of the given line ($m_1 = 9$).

2. Find the perpendicular slope ($m_2 = -\frac{1}{9}$).

3. Use the point-slope form with point (4, 2) and $m_2$.

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Q14. Write the equation of the line passing through (2, 3) and parallel to the line $4x + y - 5 = 0$ in slope-intercept form.

Background

Topic: Equations of Lines

This question tests your ability to find the equation of a line parallel to a given line and passing through a specific point.

Key Terms and Formula:

• Slope-Intercept Form: $y = mx + b$

• Parallel Lines: Have the same slope.

Step-by-Step Guidance

1. Rewrite $4x + y - 5 = 0$ in slope-intercept form to find its slope.

2. Use the same slope for the new line.

3. Plug in the point (2, 3) to solve for the y-intercept $b$.

4. Write the final equation in $y = mx + b$ form.

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Q15. Find the average rate of change of $f(x) = 2x$ from $x_1 = 2$ to $x_2 = 8$.

Background

Topic: Average Rate of Change

This question asks you to calculate the average rate of change of a function over a given interval.

Key Terms and Formula:

• Average Rate of Change: $\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Step-by-Step Guidance

1. Find $f(x_1)$ and $f(x_2)$ by substituting $x_1 = 2$ and $x_2 = 8$ into $f(x)$.

2. Calculate the difference $f(x_2) - f(x_1)$.

3. Divide by $x_2 - x_1$ to find the average rate of change.

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