College Algebra Test 1 Review – Step-by-Step Guidance

Q1. Find the slope and y-intercept of $-2x + 7y = -21$

Background

Topic: Linear Equations – Slope and Intercept

This question tests your ability to rewrite a linear equation in slope-intercept form and identify the slope and y-intercept.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

• Slope ($m$): The coefficient of $x$ in the slope-intercept form.

• Y-intercept ($b$): The constant term in the slope-intercept form.

Step-by-Step Guidance

1. Start by isolating $y$ in the equation $-2x + 7y = -21$.

2. Add $2x$ to both sides to get $7y = 2x - 21$.

3. Divide both sides by $7$to solve for$y$and write the equation in the form$y = mx + b$.

4. Identify the slope ($m$) and y-intercept ($b$) from your new equation.

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Q2. Find the x- and y-intercepts of $4x - 5y + 12 = 0$

Background

Topic: Intercepts of Linear Equations

This question checks your understanding of how to find where a line crosses the x-axis and y-axis.

Key Terms and Formulas:

• X-intercept: The point where $y = 0$.

• Y-intercept: The point where $x = 0$.

Step-by-Step Guidance

1. To find the x-intercept, set $y = 0$ in the equation and solve for $x$.

2. To find the y-intercept, set $x = 0$ in the equation and solve for $y$.

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

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Q3. Find the slope of the line between $(-3, 5)$ and $(-1, -1)$.

Background

Topic: Slope Between Two Points

This question tests your ability to use the slope formula for two points on a line.

Key Terms and Formulas:

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

Step-by-Step Guidance

1. Label your points: $(-3, 5)$ as $(x_1, y_1)$ and $(-1, -1)$ as $(x_2, y_2)$.

2. Substitute the values into the slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

3. Simplify the numerator and denominator separately before dividing.

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Q4. Write the slope-intercept form of the line that goes through $(1, 6)$ and has slope $-\frac{3}{5}$.

Background

Topic: Writing Equations of Lines

This question asks you to use a point and a slope to write the equation of a line in slope-intercept form.

Key Terms and Formulas:

• Slope-intercept form: $y = mx + b$

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

Step-by-Step Guidance

1. Start with the point-slope form: $y - y_1 = m(x - x_1)$.

2. Plug in the given point $(1, 6)$ and the slope $-\frac{3}{5}$.

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

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Q5a. Assuming the trend is linear, write an equation that gives the number of adults living alone in year $x$, with $x=60$ corresponding to the year 1960.

Background

Topic: Linear Modeling

This question tests your ability to create a linear model from two data points.

Key Terms and Formulas:

• Linear equation: $y = mx + b$

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

Step-by-Step Guidance

1. Assign the years to $x$ values: $x=60$ for 1960 and $x=120$ for 2020.

2. Assign the corresponding $y$ values: $y_1 = 7.5$ (1960), $y_2 = 36$ (2020).

3. Calculate the slope $m$ using the two points.

4. Use one point and the slope to write the equation in $y = mx + b$ form.

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Q5b. Assuming the linear trend continues, estimate the number of adults living alone in the year 2025.

Background

Topic: Linear Extrapolation

This question asks you to use your linear model to predict a future value.

Key Terms and Formulas:

• Use the equation from part (a).

• For 2025, $x = 125$ (since $x=60$ for 1960).

Step-by-Step Guidance

1. Substitute $x = 125$ into your equation from part (a).

2. Simplify to find the estimated number of adults living alone in 2025.

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Q6a. If fixed costs are $200$and $50$ items cost $2000$ to produce, find the cost function.

Background

Topic: Linear Cost Functions

This question tests your ability to construct a linear cost function from fixed and variable costs.

Key Terms and Formulas:

• Cost function: $C(x) = mx + b$

• Fixed cost ($b$): The cost when $x = 0$.

• Variable cost per item ($m$): The additional cost per item produced.

Step-by-Step Guidance

1. Let $C(x)$ be the total cost to produce $x$ items.

2. You know $C(0) = 200$ (fixed cost) and $C(50) = 2000$.

3. Set up two equations and solve for the variable cost per item ($m$).

4. Write the cost function $C(x) = mx + 200$.

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Q6b. Find the marginal cost.

Background

Topic: Marginal Cost

This question asks you to identify the cost to produce one additional item, which is the slope of the cost function.

Key Terms and Formulas:

• Marginal cost: The coefficient of $x$ in the cost function $C(x) = mx + b$.

Step-by-Step Guidance

1. From your cost function in part (a), identify the value of $m$.

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Q6c. Find the average cost to produce 150 items.

Background

Topic: Average Cost

This question tests your ability to compute the average cost per item for a given production level.

Key Terms and Formulas:

• Average cost: $\text{Average cost} = \frac{C(x)}{x}$

Step-by-Step Guidance

1. Find $C(150)$ using your cost function from part (a).

2. Divide $C(150)$ by $150$ to get the average cost per item.

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Q7a. The revenue (in millions of dollars) from the sale of $x$ units at a home supply outlet is given by $R(x) = 0.21x$. The profit (in millions of dollars) from the sale of $x$ units is given by $P(x) = 0.084x - 1.5$. Find the cost equation.

Background

Topic: Revenue, Cost, and Profit Functions

This question tests your understanding of the relationship between revenue, cost, and profit.

Key Terms and Formulas:

• Profit function: $P(x) = R(x) - C(x)$

• Cost function: $C(x) = R(x) - P(x)$

Step-by-Step Guidance

1. Write the equation $P(x) = R(x) - C(x)$.

2. Rearrange to solve for $C(x)$: $C(x) = R(x) - P(x)$.

3. Substitute the given expressions for $R(x)$ and $P(x)$ and simplify.

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Q7b. What is the cost of producing seven units?

Background

Topic: Evaluating Cost Functions

This question asks you to use the cost function from part (a) to find the cost for a specific value of $x$.

Key Terms and Formulas:

• Use $C(x)$ from part (a).

Step-by-Step Guidance

1. Plug $x = 7$ into your cost function $C(x)$.

2. Simplify to find the cost for seven units.

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Q7c. What is the break-even point?

Background

Topic: Break-Even Analysis

This question tests your ability to find the production level where revenue equals cost (or profit is zero).

Key Terms and Formulas:

• Break-even point: Set $P(x) = 0$ and solve for $x$.

Step-by-Step Guidance

1. Set $P(x) = 0$ using the given profit function.

2. Solve the resulting equation for $x$ to find the break-even quantity.

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Q8a. Let the supply and demand for bananas in cents per pound be given by supply: $p = 25q$, demand: $p = 100 - 25q$. Find the equilibrium quantity.

Background

Topic: Supply and Demand – Equilibrium

This question tests your ability to find the equilibrium point where supply equals demand.

Key Terms and Formulas:

• Equilibrium: Set supply equal to demand and solve for $q$.

Step-by-Step Guidance

1. Set $25q = 100 - 25q$.

2. Solve for $q$ to find the equilibrium quantity.

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Q8b. Find the equilibrium price.

Background

Topic: Equilibrium Price

This question asks you to use the equilibrium quantity to find the corresponding price.

Key Terms and Formulas:

• Plug the equilibrium $q$ into either the supply or demand equation to find $p$.

Step-by-Step Guidance

1. Substitute the equilibrium $q$ from part (a) into $p = 25q$ or $p = 100 - 25q$.

2. Simplify to find the equilibrium price.

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Q9. For $f(x) = 3(x - 2)^2 - 3$, determine if the parabola opens upward or downward and find its vertex.

Background

Topic: Quadratic Functions – Vertex Form

This question tests your understanding of the vertex form of a quadratic and how to interpret its graph.

Key Terms and Formulas:

• Vertex form: $f(x) = a(x - h)^2 + k$

• Vertex: $(h, k)$

• Direction: If $a > 0$, opens upward; if $a < 0$, opens downward.

Step-by-Step Guidance

1. Identify $a$, $h$, and $k$ from the given function.

2. Determine the direction the parabola opens based on the sign of $a$.

3. State the vertex as $(h, k)$.

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Q10a. A store owner finds that, at a price of $p(x) = -2x + 900$ dollars, $x$ units of a certain item will be sold. The cost function is linear with fixed costs of $5000$and variable costs of $100$ per item. Find the revenue function.

Background

Topic: Revenue Functions

This question tests your ability to write a revenue function using price and quantity.

Key Terms and Formulas:

• Revenue function: $R(x) = p(x) \cdot x$

Step-by-Step Guidance

1. Write $R(x) = p(x) \cdot x$.

2. Substitute $p(x) = -2x + 900$ into the formula.

3. Expand and simplify the expression for $R(x)$.

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Q10b. Find the cost function.

Background

Topic: Cost Functions

This question asks you to write a linear cost function from fixed and variable costs.

Key Terms and Formulas:

• Cost function: $C(x) = mx + b$

• $m$ is the variable cost per item, $b$ is the fixed cost.

Step-by-Step Guidance

1. Identify the fixed cost ($5000$) and variable cost per item ($100$).

2. Write the cost function as $C(x) = 100x + 5000$.

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Q10c. Find the profit function.

Background

Topic: Profit Functions

This question tests your ability to find profit as the difference between revenue and cost.

Key Terms and Formulas:

• Profit function: $P(x) = R(x) - C(x)$

Step-by-Step Guidance

1. Use your expressions for $R(x)$ and $C(x)$ from parts (a) and (b).

2. Subtract $C(x)$ from $R(x)$ to get $P(x)$.

3. Simplify the resulting expression.

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Q10d. Find the break-even point(s).

Background

Topic: Break-Even Analysis

This question asks you to find the value(s) of $x$ where profit is zero.

Key Terms and Formulas:

• Set $P(x) = 0$ and solve for $x$.

Step-by-Step Guidance

1. Set your profit function equal to zero.

2. Solve the resulting equation for $x$ to find the break-even quantity or quantities.

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Q10e. Find the number of items that leads to the maximum profit.

Background

Topic: Maximizing Quadratic Functions

This question tests your ability to find the vertex of a downward-opening parabola to maximize profit.

Key Terms and Formulas:

• For $P(x) = ax^2 + bx + c$, the maximum occurs at $x = -\frac{b}{2a}$ if $a < 0$.

Step-by-Step Guidance

1. Identify $a$ and $b$ from your profit function.

2. Use the formula $x = -\frac{b}{2a}$ to find the $x$ value that maximizes profit.

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Q10f. Find the maximum profit.

Background

Topic: Maximum Value of a Quadratic

This question asks you to evaluate the profit function at the $x$ value found in part (e).

Key Terms and Formulas:

• Plug the maximizing $x$ from part (e) into $P(x)$.

Step-by-Step Guidance

1. Substitute the $x$ value from part (e) into your profit function.

2. Simplify to find the maximum profit.

Try solving on your own before revealing the answer!