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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
Note
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