Summer ll Pre-Calculus- Chapter 3

panola summer ll precalculus chapter 3 exam w chatcpt curated questions and answers and notes

Determine the long-run behavior of the polynomial function f(x)=3×4−2x3+5x2−x+7.

The polynomial f(x)=3x4−2x3+5x2−x+7

has a degree of 4 (even degree) and a positive leading coefficient (3). Therefore, as x→∞, f(x)→∞ and as x→−∞, f(x)→∞

For the polynomial g(x)=2x3−4x2+6x−8

• Identify the degree.

• Determine the leading coefficient.

• Predict the number of real solutions using the degree and leading coefficient.

• For g(x)=2x3−4x2+6x−8

  • Degree: 3

  • Leading Coefficient: 2

  • Number of Real Solutions: Since the degree is odd and the leading coefficient is positive, there is at least one real solution.

Express the equation y=2(x−3)2+4 in the standard form y=ax2+bx+c

y=2(x−3)2+4

Find the equation of a quadratic function given the vertex (2,−3) and the leading coefficient 5.

• Given vertex (2,−3) and leading coefficient 5: y=5(x−2)2−3

Use synthetic division to find all zeros of f(x)=x3+2x2−5x−6f(x)=x3+2x2−5x−6.

The zeros are x=−2,1,−3

Rewrite −3 as a complex number: −3+0i

Multiply the complex numbers (2+3i) and (1−2i)

Multiply (2+3i) and (1−2i):

(2+3i)(1−2i)=2−4i+3i−6i2

2−i−6(−1)=

2−i+6=

8−i

Determine the holes in the rational function f(x)=x/2−9/x+3

Hole at x=−3

Find the intercepts, vertical asymptotes, and horizontal asymptote of g(x)=x/2−4x−2

• Intercepts: x-intercepts at (2,0) and (−2,0)

• y-intercept at (0,2)

• Vertical Asymptote: x=2

• Horizontal Asymptote: y=x+2

if the degree of the numerator is greater than the degree of the denominator

horizontal asymptote is y=0

if the degree of the denominator is greater than the degree of the numerator

there is no horizontal asymptote

if the degree of the numerator is equal to the degree of the denominator

the horizontal asymptote is the ratio of the leading coefficients

when trying to find the x intercept of a rational function

factor if possible, set top equal to zero, if none cancel out

when trying to find the y intercept of a rational function

look at ratio of constants if possible, if not set equal to zero and get x/2

to find vertical asymptote

set factored, if possible, bottom equal to zero

when the degree of the numerator is one degree greater than the denominator

it is a slant asymptote

when trying to find slant asymptote

use long division, top divided by bottom

the slant asymptote is

the quotient after the long division

to write an equation of a rational function given the x intercepts, vertical asymptote, and y intercept

make intercepts and asymptotes factors (ex: x=0)(intercepts on top, vertical asymptote on bottom) and set it up to solve for y intercept by setting the equation equal to the y intercept and the equation equals a()()/()()

to write an equation of a rational function given the x intercepts, vertical asymptotes, and horizontal asymptotes

set up x intercepts and vertical asymptotes as factors with intercepts on top and vertical asymptotes on bottom, horizontal asymptote is the a value

where do x intercepts go in a rational equations

on the top

where do vertical asymptotes go in a rational equations

on the bottom

when writing a rational equation from a graph

use the y intercept to find a value (watch for ends going the same direction, if so, square that vertical asymptote)

if a x-intercept bounces off of the x-axis

x is squared

if a x-intercept crosses the x-axis

x is not squared

To find the vertex of a polynomial

h=-b/2a, k=f(h)

vertex form

y=a(x-h)/2+k

to solve the rocket equation

height of rocket launched= y-intercept

splash down time= positive x value when doing quadratic formula

rocket reaching peak= k value (f(x)=a(x-h)/2-k)

when adding complex numbers

add the real, then imaginary

when subtracting complex numbers

change the sign to adding and make the other side opposite, then add as normal

i/2 =

-1

when dividing complex numbers

multiply by the conjugate (3x +4) * (3x-4)

to find holes in a polynomial

factor top and bottom, if any cancel out there’s the hole

vertical asymptote

factor bottom, x values are asymptotes

discontinuities are

holes