Precalculus (Unit 5)

When graphing, always check…

• MLD

• Multiplicity

• Limits

• Discontinuities (open circle)

What can and cannot be done when drawing the graph?

• You can cross the horizontal asymptote

• You cannot create x-ints, not every line has an intercept

explain what each letter signifies

(a)²/(b)²

a is an x-intercept with a mult. of 2 and therefore it will bounce

b is a vertical asymptote with an even exponent, meaning the limits on either side will be the same

(if they were odd, they would be opposites; this rule applies to each individual factor)

When writing down aymptotes…

Include x/y=!

x = 5

y = 0

Name all three cases for horizontal asymptotes

(n = numerator exponent, d = denom. exponent)

n < d: 0

n = d + 1: Slant

n = d: The quotient of the coefficients

n > d: none, +-infinity

Five Core Tips

• Be Confident (and don’t overthink)

• When starting a problem, write what you know (what is the significance of all given numbers)

• Move in and out of problems if you get stuck

• Make new problems when studying

  • Variations of study problems, how could this problem change?

• Study all board problems, write down the concepts they reveal

  • Ask questions

When is there a slant asymptote, conceptually why is it a slant and how do you find it?

When the numerator's leading degree is exactly one more than the denominator’s leading degree

It is a slant because dividing the two leading degrees leaves just x, which is a linear equation

To find it: (Synthetically) Divide the numerator by all of the factors in the denom., discard the remainder, and the final linear equation is the slant!

Can lines pass through slant asymptotes?

Yes!

Infinite Discontinuity

When the line goes to +-oo on both sides of and asymptote

Finding points of discontinuity

Use the domain of the OG, but get the y from the reduced!

If you divide out a factor and it disappears from the denom., it still isn’t included, so just plug it in to find its y-value (from reduced) and draw an open circle on that point

Other than this, always use the reduced equation!

Long Polynomial Divison

Divide the leading terms until you can’t (aka it produces a fraction)

(and distribute all terms of the divisor to each step’s result)

What do you have to do before plotting slants on a graph?

Find the x and y intercept!

Inequalities: things to consider

• ALWAYS check the sign given in the problem, circle it! (f(x)>0, >=0, etc.)

• Use number line when needed, use graph when you have it

How do you set up inequality # line?

• Insert all x-intercepts AND vertical asymptotes (all factors present)

• All asymptotes are open circle regardless, whereas x-ints are open circles depending on the problem

Inequalities: What do you do if there are equations on both sides?

Add and subtract! Do NOT cross multiply to make one side zero, as you are potentially multiplying by (-) numbers which mess up inequality

Inequality interval simple mistake

Don’t accidentally solve for a number that isn’t in between the two # line points!

between -7 and -6 is -6.5!