4-Ratic
Determine the axis of symmetry and vertex of the quadratic function
f(x) = 3(x-9)^2-2
.
Axis of symmetry is x = 9. Vertex is (9, -2).
Identify the degree, leading term, constant term, and end behavior of the polynomial
f(x)=6x^3+4x^6-10
Degree: 6, Leading Term: 4, Constant Term: -10, End Behavior: up to the left and right.
Use long division to divide the number 2020 by 15. Write your answer as n=dq+r, where n = 2020, d = 15, q is the quotient and r is the remainder.
2019 = 15*134+10
List all the possible rational zeros of the polynomial
j(x)=-5x^4-17x^3-8x^2+5x+10
The possible rational zeros are (plus minus all of these): 1, 2, 5, 10, 1/5, 2/5, 1/2
What is the domain of this function?
t(x)=\frac{-8x^2}{(x-9)(x+2)}
The domain is (-inf, -2)U(-2, 9)U(9, inf)
Solve the inequality and write the answer in interval notation.
-4<-2/3x-2\leq 8
[-15, 3)
Determine the axis of symmetry and vertex of the quadratic function
f(x)=2x^2-12x+4
The axis of symmetry is x = 3. The vertex is (3, -14).
Determine the zeros and their multiplicity for the polynomial.
g(x)=4(x-7)(x+5)^2(x^2-4)
zeros: 7, -5, 2, -2
multiplicities: 1, 2, 1, 1
Divide the polynomials. State the quotient and remainder.
x^2+7x+10\divx+2
x+5 with no remainder
List all the possible rational zeros of the polynomial. Then determine the zeros.
f(x)=x^3-2x^2-9x+18
Possible zeros are (plus and minus): 1, 2, 3, 6, 9, 18.
Actual zeros are -3, 2, 3
Find the vertical asymptotes and holes in the graph of the function
f(x)=\frac{x^2-3x-18}{x^2-36}
VA: x = -6
Hole at x = 6
Solve the inequality and write the answer in interval notation.
(x-5)(x+7)>0
(-inf, -7)U(5, inf)
Determine the x- and y-intercepts of the quadratic function
f(x)=x^2+2x-8
The x-intercepts are (-4, 0) and (2, 0). The y-intercept is (0, -8).
Determine the zeros and their multiplicity for the polynomial
h(x)=x^3+4x^2-4x-16
zeros: -2, 2, -4
multiplicity: 1, 1, 1
Divide the polynomial by x - 2 to determine the quotient and remainder.
3x^5-5x^3+6x^2-7x+2
The quotient is 3x^4+6x^3+7x^2+20x+33 and the remainder is 68/(x-2)
Find a 3rd degree polynomial with 4 and 4i as zeros of the polynomial such that f(-1) = 255.
f(x) = -3x^3 +12x^2 - 48x + 192
Determine the horizontal and vertical asymptotes of the rational function.
e(x)=\frac{32x^2}{8x^2-16}
HA: y = 4
VA: x = sqrt(2), x = -sqrt(2)
Solve the inequality and write the answer in interval notation.
\frac{x-4}{x+7}\leq 0
(-7, 4]
Among all pairs of numbers whose difference is 10, find a pair whose product is as small as possible. What is the minimum product?
Determine the zeros and y-intercept of the polynomial
c(x)=-4(x-3)^2(x^2-1)
Zeros: 3 mult 2, 1 mult 1, -1 mult 1
y-intercept: (0, 36)
Evaluate the polynomial for b(-9).
b(x)=3x^3-10x^2+7x-2
-3085
Determine the zeros of the polynomial.
\alpha(x)=7x^3-5x^2-63x+45
The zeros are -3, 3, and 5/7
Determine all intercepts and asymptotes of the function.
v(x)=\frac{-x}{x+5}
x-intercept and y-intercept: (0, 0)
VA: x = -5
HA: y = -1
For what values does the inequality hold? Write the solution in interval notation.
\frac{(x+5)(x-7)}{x+3}\leq 0
(-inf, -5]U(-3, 7]
Determine the minimum value for the quadratic equation and state the domain and range of the polynomial.
d(x)=3x^2-30x-3
The minimum value is at the vertex (5, -78). The domain of the polynomial is (-inf, inf) and the range is [-78, inf).
Find the zeros and y-intercept of the polynomial
v(x)=x^3+5x^2-9x-45
Zeros: -5 mult 1, -3 mult 1, 3 mult 1
y-intercept: (0, -45)
Use synthetic division to divide the polynomial by x - 4. Then find all the zeros of w.
w(x)=x^3-2x^2-23x+60
The zeros of w(x) are 4, -5, and 3.
In our study of zeros of polynomials, which two mathematicians did we talk about?
For +$500, what countries were they from?
We talked about Carl Freidrich Gauss from Germany and Rene Descartes from France.
Describe how to graph the rational function.
u(x) = \frac{x-2}{x^2-16}
Draw a VA at x = -4 and x = 4. HA is y = 0. The graph has an x-intercept at (2, 0) and a y-intercept at (0, 1/8).
On the left interval of the domain, the graph is negative and below the HA. In the middle interval it is a negative cubic. In the last interval it is positive and is above the HA.
What is the domain of the function?
m(x)=\sqrt{6x^2-31x+5}
(-inf, 1/6]U[5, inf)