exam-1-review Exam 1 recovery and review

The big idea Choose the problem family first. Then use one clean setup, keep the calculator entry faithful to that setup, and check whether the answer makes sense in the problem's domain.

Start with the recovery signal

Exam 1 is a 100-point assessment due September 3. It covers Modules 1–4: the work from Basics through Regression and Quadratics.

The August 26 WebAssign snapshot records 8/69 on 1330 W - Quad, with only 2 of 21 questions answered. That makes completion and setup the first recovery target. It does not prove that every quadratic skill is weak, because most of the set was left unanswered.

Use this order: review the formula families below, work the examples without looking at the last line, take the Practice quiz, and finish with the short Explain-it response. Return to the sections behind any missed practice item before trying it again.

What Exam 1 can ask

Module 1 — Basics: order of operations, exponent rules, radicals, expanding, factoring, and solving factored equations.
Module 2 — Functions: solving equations, evaluating a function, reading tables and graphs, piecewise rules, intercepts, and the vertical-line test.
Module 3 — Business and Lines: slope, line equations, fixed and marginal cost, demand, revenue, profit, and break-even.
Module 4 — Regression and Quadratics: regression equations, residuals, correlation, quadratic graphs, vertices, zeros, quadratic regression, and model limits.

Those task families come from 1330 W - Basic, Solve Equ, Fcns, Bus Models, Linear, Lin Regress, and Quad. The review gives Module 4 extra space because that is where the incomplete 8/69 result occurred.

Formula sheet

Exponent product and quotient

x^m x^n=x^{m+n},\qquad \frac{x^m}{x^n}=x^{m-n}

The quotient rule subtracts the denominator exponent. Rewrite a negative final exponent as a reciprocal.

Zero-product rule

uv=0\quad\Longrightarrow\quad u=0\;\text{or}\;v=0

This works only after one side of the equation is zero.

Slope

m=\frac{y_2-y_1}{x_2-x_1}

Use the same point order on top and bottom. A vertical line has undefined slope.

Line forms

y=mx+b,\qquad y-y_1=m(x-x_1)

Use point-slope form when a point and slope are given; convert only if the answer box asks for another form.

Revenue and profit

R(x)=p(x)x,\qquad P(x)=R(x)-C(x)

If price changes with quantity, multiply the whole demand/price function by x. Subtract every term of cost.

Regression line and residual

\widehat y=mx+b,\qquad e=y-\widehat y

A negative residual means the observed value lies below the prediction.

Quadratic form and axis

y=ax^2+bx+c,\qquad x_{\text{vertex}}=-\frac{b}{2a}

Plug the vertex x-coordinate back into the original function to get the y-coordinate.

Quadratic formula

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Identify a, b, and c with their signs before substituting. The discriminant is the expression under the square root.

A method for every problem

1. Name the task family

Before calculating, write a two- or three-word label: simplify, solve, evaluate, line, business model, regression, or quadratic. That label chooses the formula and the calculator mode.

2. Copy the structure and units

Keep parentheses around a function being subtracted. Record whether x means units, thousands of square feet, years since a base year, or price. For a quadratic, copy a, b, and c with their signs.

P(x)=R(x)-\bigl(C(x)\bigr)

3. Solve or fit

Do algebra for exact forms. Use the regression mode requested by the problem for data sets. Keep extra calculator digits until the last line, then round exactly as directed.

4. Check the answer

Substitute a root when practical. Check that a point satisfies a line. Compare an observed value with a regression prediction. Reject a business answer outside the stated domain or with impossible units.

Traps to stop before the exam

A score is not a diagnosis when most items are blank

The W - Quad snapshot shows a completion problem first. Do not skip a problem because it looks long: identify its family, write the setup, and take the next justified step.

Dropping a sign or parenthesis

In the quadratic formula, −b and 2a each need their own parentheses in a calculator. In profit, the minus applies to the entire cost function.

Using the wrong regression

A best-fit line and a quadratic regression are different calculator commands. Read whether the problem asks for a line or a quadratic curve before entering the data.

Rounding too early

WebAssign problems in these modules often specify four decimal places. Store or copy the full calculator result, and round only the reported coefficient or final answer.

Ignoring the model's domain

A root can be algebraically correct and still be unusable. Reject negative production, a quantity outside the stated interval, or a long-term quadratic prediction that eventually becomes impossible.

Worked examples

These are review examples. The numbers either come from the captured WebAssign task or are labeled as a fresh sample; they are not a copy of Exam 1.

Module 1 · factor, then solve

Captured Basic task. $6y^2-11y-2=0$

Factor the polynomial and find both solutions.

Choose factors whose product gives −2 and whose outer-plus-inner terms give −11y.
$6y^2-11y-2=(6y+1)(y-2)$
Set each factor equal to zero.
$6y+1=0\quad\text{or}\quad y-2=0$

The solutions are y = −1/6 and y = 2. Multiplying the factors back out checks the signs.

Module 2 · read a linear function table

Fresh sample. $(-6,-1),\ (0,-3),\ (6,-5),\ (12,-7)$

Find the rule for the linear function.

Each time x rises by 6, y falls by 2, so the slope is −2/6.
$m=\frac{-2}{6}=-\frac13$
The table includes x = 0, so its y-value is the intercept.
$b=-3$

The function is f(x) = −(1/3)x − 3. Check it with x = 6: the rule gives −5.

Module 3 · build a linear cost model

Captured Linear task. $C(130)=10700,\qquad C(150)=11600$

Find the marginal cost and fixed cost.

The marginal cost is the slope between the two cost points.
$m=\frac{11600-10700}{150-130}=45$
Use either point in C(x) = 45x + b to find the fixed cost.
$10700=45(130)+b\quad\Longrightarrow\quad b=4850$

C(x) = 45x + 4,850. The marginal cost is $45 per bicycle, and the fixed cost is $4,850.

Module 4 · quadratic regression and vertex

Captured Quad data. $(1,3),\ (3,5),\ (4,4),\ (5,2)$

Fit a quadratic, round its coefficients to four decimal places, and locate the model's vertex.

Enter x-values and y-values in paired calculator lists, then run quadratic regression rather than linear regression.
$\widehat y=-0.6136x^2+3.4227x+0.2000$
Use the unrounded coefficients for the vertex x-coordinate; the rounded model gives a close check.
$x_{\text{vertex}}=-\frac{b}{2a}\approx2.789$
Evaluate the regression model at that x-coordinate.
$\widehat y(2.789)\approx4.973$

The fitted curve peaks near (2.789, 4.973). Keep full calculator precision until the requested rounding step.

Active exam practice

Open Practice quiz above and work all fourteen items without notes first. The set deliberately mixes task families the way an exam can: simplifying, solving, functions, lines, business models, regression, and quadratics.

After each miss, say the task family out loud before reading the explanation. Then redo the setup on paper. A correct second attempt matters more than memorizing the answer choice.

Final checklist

Tick these off as you go. Saved in this browser.

I can identify the task family before choosing a formula or calculator mode.
I can factor and solve a quadratic, and I can use the quadratic formula when it does not factor cleanly.
I can evaluate a function and build a line from points, a table, or a point and slope.
I can write revenue and profit without dropping the parentheses around cost.
I can run the requested regression, compute a residual, and interpret the sign of correlation.
I finished the Practice quiz and the short Explain-it response.