Math 130 · Fall 2026 · Final Lecture

Final Exam Guided Review

A 45-minute review based on the current practice blueprint. Canvas lists the final for Dec 14 at 1:00 PM; confirm the official content coverage in Canvas or in class.

1

Build the formula map.

2

Practice the problem types.

3

Leave with an attack plan.

Priority Map: Final Review and Earlier Quiz Practice

Skill	Final practice	Earlier Quiz 13 practice	Student priority
Arc length / sector area	Yes	Yes	High: angle conversion required
Linear velocity	Yes	Yes	High: radius and radians
Coordinate geometry	Yes	Yes	High: multi-part work
Logs / exponents	Yes	Yes	High: quick algebra points
Rational equations	Yes	Yes	High: restrictions matter
Vectors, intervals, bearings	Yes	No	Final-only focus
Compound interest / projectile maximum	Yes	No	Formula-pattern focus

Review plan: start with overlapping skills, then close with the remaining cumulative topics.

What Students Must Be Ready To Do

Current final-review emphasis

Trigonometry and right triangles
Radians, degrees, arc length, and sector area
Polygon area and angle relationships
Coordinate geometry: distance, midpoint, slope, equations
Logs, exponents, rational equations, compound interest
Projectile maximum height, bearing, and vectors

Earlier Quiz 13 practice emphasis

Arc length and sector area with missing values
Linear velocity from rotations per minute
Quadratic minimum/maximum
Exponent simplification
Distance, line equations, perpendicular lines
Rational equations with restrictions

Instructor move: frame the session as pattern recognition, not as a list of formulas.

Exam-Day Formula Sheet: Use It Correctly

Provided

$s = r\theta$

$A_{sector}=\dfrac{1}{2}r^2\theta$

$A=\dfrac{1}{2}bc\sin(A)$

$C=2\pi r$

Common trap

For $s=r\theta$ and $A=\frac12r^2\theta$, the angle must be in radians.

Degrees go in only after conversion.

$\theta_{rad}=\theta^\circ\cdot\dfrac{\pi}{180}$

What Is Provided vs. What Must Be Known

Provided or directly referenced

$s=r\theta$
$A=\frac12r^2\theta$
$A=\frac12bc\sin(A)$
$C=2\pi r$

Must know how to use

Degree/radian conversion
$v=r\omega$ and unit conversion
Distance, midpoint, slope, and line equations
Exponent rules and log/exponential conversion
Rational equation restrictions
Quadrant correction for inverse tangent

Calculator Mode Rules

Degree mode
ordinary trig with degree angles

Radian mode
arc length and sector area formulas

Inverse tangent
check the quadrant manually

Common wrong move

$s=7.6(140)=1064$

Wrong because 140 is degrees, not radians.

Correct setup

$s=7.6\left(140\cdot\frac{\pi}{180}\right)$

Convert the angle first, then calculate.

Three Angle Conversions Students Need Fast

Degrees to radians

$\theta^\circ\cdot\dfrac{\pi}{180}$

Example: $63.4^\circ\approx1.106$ rad

Radians to degrees

$\theta\cdot\dfrac{180}{\pi}$

Example: $2.8\approx160.428^\circ$

Decimal to DMS

Keep whole degrees.
Multiply decimal by 60 for minutes.
Multiply remaining decimal by 60 for seconds.

Quick check: 85° is not 85 radians. It is $85\cdot\pi/180\approx1.484$ radians.

Your Turn: Arc Length

Find the arc length of a circle with radius 7.6 units and central angle 140°.

Step 1

Identify the formula.

$s=r\theta$

Step 2

Convert $140^\circ$ to radians.

Step 3

Multiply $r\theta$ and round.

Instructor workspace

Solution Reveal: Arc Length

Work

$140^\circ=140\cdot\frac{\pi}{180}=\frac{7\pi}{9}\approx2.443$ radians

$s=r\theta=7.6\left(\frac{7\pi}{9}\right)\approx18.581$ units

What to say aloud

The formula is short, but the setup is the problem. The angle must become radians before it enters the formula.

Arc Length and Sector Area: Missing Quantity Map

Question form	Equation	Most common mistake
Find $s$	$s=r\theta$	Using degrees without converting.
Find $r$	$r=s/\theta$	Dividing by a degree measure.
Find $\theta$	$\theta=s/r$	Giving radians when degrees are requested.
Find sector area	$A=\frac12r^2\theta$	Forgetting to square the radius.
Find $r$ from sector area	$r=\sqrt{\frac{2A}{\theta}}$	Forgetting the square root.

Instructor line: “The formula sheet gives the formula, but it does not convert your angle for you.”

Linear vs. Angular Velocity

The relationship

$v=r\omega$

$r$ = radius from center to moving object
$\omega$ = angular velocity in radians per unit time
$v$ = linear velocity in distance per unit time

Two setup traps

Wingspan is diameter. Radius is half the wingspan.
Rotations and degrees are not radians. Convert before multiplying.

$1\text{ rotation}=2\pi\text{ radians}$

Unit conversion: $1\text{ ft/sec}=\dfrac{3600}{5280}\text{ mi/hr}\approx0.6818\text{ mi/hr}$.

Roll Rate Diagram: Radius Is Half the Wingspan

Visual setup

The center of the airplane is the center of rotation.
The wingtip travels around a circle.
The radius is the distance from center to wingtip.
Therefore, radius is half the wingspan.

Your Turn: Roll Rate

A plane has a wingspan of 30 feet. It rolls at 500° per second.

Find the wingtip linear velocity in ft/sec and mph.

Radius

$r=15$ ft

Angular velocity

$500^\circ/sec=\frac{500\pi}{180}$ rad/sec

Linear velocity

$v=r\omega$

Instructor workspace

Solution Reveal: Roll Rate

Work

$r=30/2=15$ ft

$\omega=500\cdot\frac{\pi}{180}\approx8.727$ rad/sec

$v=r\omega=15(8.727)\approx130.900$ ft/sec

$130.900\cdot\frac{3600}{5280}\approx89.250$ mph

Scoring emphasis

Half the wingspan.
Convert degrees to radians.
Report both requested units.

Right Triangle and Exact Trig Review

Right triangle setup

Label hypotenuse first.
Relative to angle $A$: identify opposite and adjacent.
Use sine, cosine, or tangent based on what is given.

$\sin A=\frac{opp}{hyp},\quad \cos A=\frac{adj}{hyp},\quad \tan A=\frac{opp}{adj}$

Six trig functions

$\sin\theta$	$y/r$
$\cos\theta$	$x/r$
$\tan\theta$	$y/x$
$\csc,\sec,\cot$	reciprocals

For coordinate trig, calculate $r=\sqrt{x^2+y^2}$, then apply signs by quadrant.

Coordinate Trig: Point to Functions

Example point: $P(-4,8)$

Process

Find $r=\sqrt{x^2+y^2}$.
Use $\sin\theta=y/r$ and $\cos\theta=x/r$.
Use $\tan\theta=y/x$.
Take reciprocals for $\csc$, $\sec$, and $\cot$.

Sign rule: Quadrant II has negative cosine and tangent, positive sine.

Area: Polygons and Triangle Formula

Trapezoid

$A=\dfrac{b_1+b_2}{2}h$

Use vertical height, not slant side.

Regular polygon

$A=\dfrac12aP$

$a$ = apothem; $P$ = perimeter.

Triangle with included angle

$A=\dfrac12bc\sin(A)$

Angle $A$ must be between sides $b$ and $c$.

Rounding convention: polygon and triangle dimensions are commonly rounded to one decimal place when requested.

Your Turn: Polygon Area

A regular hexagon has side length 10 and apothem 8.7.

Find its area to one decimal place.

Step 1

Find perimeter: $P=6s$.

Step 2

Use $A=\frac12aP$.

Step 3

Round to one decimal place.

Instructor workspace

Solution Reveal: Polygon Area

Work

$P=6(10)=60$

$A=\frac12aP=\frac12(8.7)(60)$

$A=261.0$ square units

Coordinate Geometry: Four-Part Problem

These problems are high-value practice in the current final review. They reward an organized setup.

Given two points $P(x_1,y_1)$ and $Q(x_2,y_2)$

Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Midpoint: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
Slope: $m=\frac{y_2-y_1}{x_2-x_1}$
Line: use $y-y_1=m(x-x_1)$

Perpendicular line

Use the opposite reciprocal slope.

$m_{\perp}=-\dfrac{1}{m}$

If it goes through the origin, the equation is often simple: $y=m_{\perp}x$.

Your Turn: Coordinate Geometry

For $P=(-4,8)$ and $Q=(6,-2)$, find:

Distance

Midpoint

Equation of $PQ$

Perpendicular line through $(0,0)$

Instructor workspace

Solution Reveal: Coordinate Geometry

Distance	$d=\sqrt{(6-(-4))^2+(-2-8)^2}=\sqrt{200}=10\sqrt2\approx14.142$
Midpoint	$M=\left(\frac{-4+6}{2},\frac{8+(-2)}{2}\right)=(1,3)$
Slope	$m=\frac{-2-8}{6-(-4)}=-1$
Line through $P,Q$	$y-8=-1(x+4)$, so $y=-x+4$
Perpendicular through origin	$m_\perp=1$, so $y=x$

Logs and Exponents: Convert Forms Immediately

Logarithmic form

$\log_b(y)=x$

Read as: “The exponent on $b$ that gives $y$ is $x$.”

Exponential form

$b^x=y$

Same relationship, different format.

Mini-drill: $\log_3(81)=x \Rightarrow 3^x=81 \Rightarrow x=4$.

Reverse drill: $2^x=30 \Rightarrow x=\log_2(30)=\frac{\log(30)}{\log(2)}$.

Exponent Simplification Checklist

Rules

Multiply same base: add exponents.
Divide same base: subtract exponents.
Power of a power: multiply exponents.
Negative exponent: move across the fraction bar.
Fraction exponent: root/power meaning.

Examples

$x^{1/2}=\sqrt{x}$

$x^{-3}=\dfrac{1}{x^3}$

$x^{1/2}x^{5/2}=x^3$

Required finish: positive exponents only when the problem asks for it.

Quadratics: Solve vs. Find the Vertex

Solve the equation

Example: $x^2-7x+10=0$

Factor if possible.
Set each factor equal to zero.
List all solutions.

Find minimum or maximum

Example: $h(x)=4x^2-4x+21$

Use $x=-b/(2a)$.
Plug the $x$-value back in.
If $a>0$, it is a minimum. If $a<0$, it is a maximum.

Quick Reveal: Quadratic Minimum

For $h(x)=4x^2-4x+21$, use $x=-\frac{b}{2a}$.

$x=-\frac{-4}{2(4)}=\frac12$

$h\left(\frac12\right)=4\left(\frac14\right)-4\left(\frac12\right)+21=20$

Conclusion

Because $a=4>0$, the parabola opens upward.

Minimum value: $20$

Projectile Motion: Maximum Height

Pattern

Height is given by a quadratic:

$h(t)=-16t^2+bt+c$

The maximum height occurs at the vertex.

Example

$h(t)=-16t^2+88t$

Find $t=-b/(2a)$.
Plug $t$ into $h(t)$.
State the maximum height with units.

Do not stop at time. The problem asks for the maximum height.

Compound Interest Word Problems

$A=P\left(1+\dfrac{r}{n}\right)^{nt}$

$P$	principal / starting amount
$r$	annual rate as a decimal
$n$	number of compounds per year
$t$	time in years

Common mistakes

Using 4.8 instead of 0.048.
Forgetting $nt$ in the exponent.
Using $n=12$ for quarterly compounding.

quarterly: $n=4$

Rational Equations: Clear Fractions Safely

Process

State restrictions from denominators.
Find the LCD.
Multiply every term by the LCD.
Solve the resulting equation.
Reject any restricted solution.

Rational-equation practice pattern

$\dfrac{x+2}{6x}+1=\dfrac{x-7}{x}$

Restriction: $x\ne0$.

$\dfrac{2}{x-5}-\dfrac{1}{x+5}=\dfrac{11}{x^2-25}$

Restrictions: $x\ne5,-5$.

Intervals: Union and Intersection

Union

Everything in either set.

$A\cup B$

Think: “combine coverage.”

Intersection

Only what the sets share.

$A\cap B$

Think: “overlap only.”

If $A=[2,9)$ and $B=(-1,6]$, find $A\cup B$ and $A\cap B$.

Bearings and Descent Angle

Direct bearing to an airport

Draw an east-north coordinate sketch.
Use right-triangle trig to find the angle.
State direction carefully.
Distance uses the Pythagorean theorem.

Descent angle

Convert units first: $1\text{ NM}=6076\text{ ft}$.

$\tan\theta=\dfrac{\text{altitude loss}}{\text{forward travel}}$

Vectors: Component Method

Add horizontal with horizontal; vertical with vertical.

Given $u=\langle a,b\rangle$ and $v=\langle c,d\rangle$

$r=u+v=\langle a+c,b+d\rangle$
$|r|=\sqrt{x^2+y^2}$
$\theta=\tan^{-1}(y/x)$, then adjust for quadrant.

Calculator warning: inverse tangent alone may give a reference angle in the wrong quadrant.

Your Turn: Vectors

Find $r=u+v$, $|r|$, and $\theta_r$ to one decimal place.

$u=\langle 9,12\rangle$, $v=\langle -15,4\rangle$

Resultant

$r=\langle 9-15,12+4\rangle$

Magnitude

$|r|=\sqrt{x^2+y^2}$

Direction

Use quadrant for $(-,+)$.

Instructor workspace

Solution Reveal: Vectors

$r=\langle9+(-15),12+4\rangle=\langle-6,16\rangle$

$|r|=\sqrt{(-6)^2+16^2}=\sqrt{292}\approx17.1$

Reference angle: $\tan^{-1}\left|\frac{16}{-6}\right|\approx69.4^\circ$

Quadrant II direction: $180^\circ-69.4^\circ=110.6^\circ$

High-Value Error Check Before Submitting

Check these every time

Did I convert degrees to radians for arc/sector formulas?
Did I use radius rather than diameter/wingspan?
Did I square the radius for sector area?
Did I round to the requested place?
Did I include units for word problems?

Algebra checks

Are exponents positive if required?
Did I reject restricted values in rational equations?
Did I answer the actual question: time, height, amount, angle, or distance?
Did I adjust inverse tangent for the correct quadrant?

Mixed Review Attack Strategy

Start with algebra. Simplify exponents and solve rational equations while fresh.
Do arc and sector problems as a table. Write $r$, $\theta$, and units before calculating.
Draw every word problem. Linear velocity, bearings, descent angle, and coordinate geometry all improve with a sketch.
Use formula sheet intentionally. Verify whether the angle must be radians.
Check rounding last. Do not round too early inside multistep problems.

Take-home expectation: show enough work that another person can follow the setup.

Final 8-Minute Mixed Drill

Set A

Convert $125^\circ$ to radians.
Find $s$ if $r=9.2$ and $\theta=125^\circ$.
Simplify $x^{1/2}y^{3/5}\cdot x^{5/2}y^{4/3}$.
Find the vertex value of $h(x)=4x^2-4x+21$.

Set B

Find the midpoint of $(-6,5)$ and $(8,-3)$.
Find $r$ if $s=36$ and $\theta=1.2$.
For $u=\langle10,7\rangle$ and $v=\langle-22,-2\rangle$, find $u+v$.
Write $\log_5(125)=x$ in exponential form and solve.

Use as a quick individual check, then ask students to explain one setup to a neighbor.

Solution Reveal: Mixed Drill

Set A

$125^\circ=\frac{25\pi}{36}\approx2.182$ rad
$s=9.2(2.182)\approx20.076$
$x^{1/2+5/2}y^{3/5+4/3}=x^3y^{29/15}$
Minimum value: $20$

Set B

Midpoint: $(1,1)$
$r=36/1.2=30$
$u+v=\langle-12,5\rangle$
$5^x=125$, so $x=3$

Printable Final Checklist

Setup checks

I identified what the problem is asking for.
I wrote the correct formula before substituting.
I converted degrees to radians when required.
I used radius, not diameter or wingspan.
I drew a diagram for word, vector, and bearing problems.

Finish checks

I did not round too early.
I used the requested rounding rule.
I included units when appropriate.
I rejected restricted values in rational equations.
I checked the quadrant for inverse tangent.

Closing

Prepared means organized.

The strongest test strategy is not memorizing every problem. It is recognizing the type, writing the setup, checking the units, and finishing the requested answer.

Formula

Choose and convert.

Setup

Draw and label.

Check

Round and answer.