Precalculus (Unit 2)

ADD THESE FROM NOTEBOOK WORKSHEETS

Velocity equation

h(t)=1/2gt²+v(nought)t+s(nought)

v0=initial speed

s0=initial height

g= -9.8 (negative!!)

Continued Fractions process (normal)

• Set it equal to X, then take one loop and substitute X for where it restarts

• x=1/2+x

• Solve, you should get a quadratic

• ONLY CHOOSE THE POSITIVE ANSWER IF THE FRACTION IS POSITIVE! IT CANNOT BE BOTH!

Continued Fractions process (“complicated and creative way”)

• Try to split the fraction into one you can solve and the extra numbers (Via addition, subtraction, or factoring out a number)

• 3+1/2+1/… » Take out a 1 so it is 1+2/1+2/1+2/…

• 39/1+3/1+3/… »39 × 1/1+3/1+3/1+3/…

When given a vertex and a point…

Make Vertex form equation and plug in the points to solve for a!

5(x+3)(x-(4/7)) = f(x), integer coefficients

5(x+3)(7x-4) = f(x)

Make any fractions into integer factors using common sense!

Name the 4 ways to find the vertex

• Complete the square

  • x²+6x+4 » (x²+6x+9)+4-9

  • (x+3)²-5

  • Half the b-value squared, then c subtracted by a*0.5b² (in this case, a=1), then factor for vertex form

  • 2x²+12x+4 » 2(x²+6x+9)+4-18

• Average of the x-intercepts

• (-b/2a, f(-b/2a))

• Calculator (not recommended)

Find x-intercepts of an equation in vertex form

DO NOT MULTIPLY OUT, SOLVE AS NORMAL

• (x+3)²-5 » x+3= ±sqrt(5)

• x = ±sqrt(5)-3

Quadratics and word problems

• Form two linear equations based off of the two parts of the problem!

• Revenue is People * Price, and Price is 70-4x and People is 34+2x, so therefore…

• r(x) = (70-4x)(34+2x), from here you can solve

• DO NOT FORGET TO PLUG IN THE X VALUE IN THE APPROPRIATE LINEAR FACTORS WHEN COMPLETE, X IS ONLY A SCALE FACTOR IN THIS CASE AND IS NOT PRICE/PEOPLE!!!!

• And don’t forget that x scale factor may also need to be rounded before you plug it in

In problems that yield multiple solutions… (outside of cont. fractions)

Check them all! There could be extraneous solutions, especially in radicals or division!

For all problems, do not forget to…

SHOW YOUR WORK! You can skip some baby steps, but be sure to keep the general steps in place!

Sometimes PEMDAS could be issue, so at least get credit for right work (BUT USE THAT FRACTION FUNCTION)

Basically, do all the steps on paper but do the calculations to get to each step on calculator!

Example: Find x-intercepts » use average to get vertex » take the x-value from vertex » plug it in » answer

When finding the x-value of a vertex given the y…

Don’t forget to actually substitute y with the given value once you get the right equation to solve for the x!

Example: 4k²+3k+8=5 « the given value

Geo-Optimization Steps

1. Form the “given” equation (sometimes not directly, may have to use Pythagorean Theorem for example to find these)

2. Form the “unknown” equation (usually a formula)

   1. A=lw, V=lwh, D=r/t, etc.

   2. One of the values will be given for the 2nd equation (rate, volume, etc.)

3. Isolate a variable in the first equation and plug and solve for the 2nd equation

4. Find the vertex, and understand what x is, if it is only a scale factor or something else, plug and solve in previous equations!

Geo-Optimization Tips

• Don’t assume every side to a shape is equal, especially when dealing with rectangular prisms (long side areas are l*h,w*h, etc.)

• ENSURE YOU’RE TYPING EVERYTHING IN THE CALCULATOR CORRECTLY

  • DO EVERY LITTLE CALCULATION IN THE CALCULATOR, DOUBLE CHECK!

Rounding and Revenue Equations

• Applies not just for revenue, but all quadratic word problems

• IF you need an integer x value, whether you round it up or down depends on the situation. Try both the rounded down and up values!

For fencing/cubes…

be careful not to accidentally double your dimensions twice!

Use common logic and go slow!