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

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Q1. Find the slope and y-intercept of

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:
Slope (): The coefficient of in the slope-intercept form.
Y-intercept (): The constant term in the slope-intercept form.

Step-by-Step Guidance

Start by isolating in the equation .
Add to both sides to get .
Divide both sides by $7yy = mx + b$.
Identify the slope () and y-intercept () from your new equation.

Try solving on your own before revealing the answer!

Q2. Find the x- and y-intercepts of

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-intercept: The point where .

Step-by-Step Guidance

To find the x-intercept, set in the equation and solve for .
To find the y-intercept, set in the equation and solve for .
Write your answers as points: for the x-intercept and for the y-intercept.

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Q3. Find the slope of the line between and .

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:

Step-by-Step Guidance

Label your points: as and as .
Substitute the values into the slope formula: .
Simplify the numerator and denominator separately before dividing.

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Q4. Write the slope-intercept form of the line that goes through and has slope .

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:
Point-slope form:

Step-by-Step Guidance

Start with the point-slope form: .
Plug in the given point and the slope .
Simplify the equation to get it into the slope-intercept form .

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Q5a. Assuming the trend is linear, write an equation that gives the number of adults living alone in year , with 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:
Slope ():

Step-by-Step Guidance

Assign the years to values: for 1960 and for 2020.
Assign the corresponding values: (1960), (2020).
Calculate the slope using the two points.
Use one point and the slope to write the equation in 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, (since for 1960).

Step-by-Step Guidance

Substitute into your equation from part (a).
Simplify to find the estimated number of adults living alone in 2025.

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Q6a. If fixed costs are $200 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:
Fixed cost (): The cost when .
Variable cost per item (): The additional cost per item produced.

Step-by-Step Guidance

Let be the total cost to produce items.
You know (fixed cost) and .
Set up two equations and solve for the variable cost per item ().
Write the cost function .

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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 in the cost function .

Step-by-Step Guidance

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

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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:

Step-by-Step Guidance

Find using your cost function from part (a).
Divide by $150$ to get the average cost per item.

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Q7a. The revenue (in millions of dollars) from the sale of units at a home supply outlet is given by . The profit (in millions of dollars) from the sale of units is given by . 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:
Cost function:

Step-by-Step Guidance

Write the equation .
Rearrange to solve for : .
Substitute the given expressions for and 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 .

Key Terms and Formulas:

Use from part (a).

Step-by-Step Guidance

Plug into your cost function .
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 and solve for .

Step-by-Step Guidance

Set using the given profit function.
Solve the resulting equation for 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: , demand: . 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 .

Step-by-Step Guidance

Set .
Solve for 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 into either the supply or demand equation to find .

Step-by-Step Guidance

Substitute the equilibrium from part (a) into or .
Simplify to find the equilibrium price.

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Q9. For , 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:
Vertex:
Direction: If , opens upward; if , opens downward.

Step-by-Step Guidance

Identify , , and from the given function.
Determine the direction the parabola opens based on the sign of .
State the vertex as .

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Q10a. A store owner finds that, at a price of dollars, units of a certain item will be sold. The cost function is linear with fixed costs of $5000 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:

Step-by-Step Guidance

Write .
Substitute into the formula.
Expand and simplify the expression for .

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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:
is the variable cost per item, is the fixed cost.

Step-by-Step Guidance

Identify the fixed cost ($5000).
Write the cost function as .

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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:

Step-by-Step Guidance

Use your expressions for and from parts (a) and (b).
Subtract from to get .
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 where profit is zero.

Key Terms and Formulas:

Set and solve for .

Step-by-Step Guidance

Set your profit function equal to zero.
Solve the resulting equation for 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 , the maximum occurs at if .

Step-by-Step Guidance

Identify and from your profit function.
Use the formula to find the 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 value found in part (e).

Key Terms and Formulas:

Plug the maximizing from part (e) into .

Step-by-Step Guidance

Substitute the value from part (e) into your profit function.
Simplify to find the maximum profit.