Polynomial Functions
Find the slope of the following equation:
f(x) = 4x
Answer: The Slope is 4
x/x2-9
Vertical Asymptotes:
3 and -3
Rewrite the following equation in logarithmic format.
52 = 25
log525 = 2
a^x = b --> logab = b
Find the center of the following equation:
(y-2)2/4 + (x-1)2/9
Answer: (1,2)
Work: (x-1)2 shows the x-value of the center (1), and (y-2)2 shows the y-value of the center (2).
Complete the square of the following equation:
x2 + 4x + 12
Answer:
(x+2)2 +8
Work/Explanation: x2 + 4x + 12 is not a perfect square, but x2 + 4x + 4 is.
(x+2)2, subtract the 4 from the original equation from 12 of x2 + 4x + 12, and you get 8.
Find the equation with a horizontal asymptote of 6 and vertical asymptote of 5 and -5.
x-6/x2-25
logx81 = 4
x4 = 81, fourth root on each side to get x by itself.
x= 811/4, x = 3
Find the center and variables a & b of the following equation:
(x-7)2/64 + (y+6)2/16
Answer:
Center: (7,-6), a = 8 , b = 4
Work: (x-7)2 indicates the x-value of the center, and (y+6)2 shows the y-value.
Square root 64 and 16, which are the a and b values, and you get b and a.
Find the highest degree, y-intercept, and x-intercept(s) of the following equation.
f(x)= -4x2 + 2x + 1 + x3
Degree: 3
y-intercept: 1, x-intercepts: 1, 3.303
Work:
Degree: x3 has a degree of 3, which is the highest.
y-intercept: f(0) = -4(0)2 + 2(0) + 1 + (0)3 = 1
x-intercepts: -4x2 + 2x + 1 + x3 = 0,factor out, 1 and 3.303.
Determine the x-intercept(s), horizontal and vertical asymptote(s) of the following equation:
(2x - 1)/(x+3)
Answer:
X-int = (1/2,0),
horizontal asymptote = y=2
Vertical Asymptote = x=-3
Work:
X-intercept: (2x-1) = 0, 2x =1, x = 1/2
Horizontal Asymp: The numerator and denominator both have a degree of one, so the coefficient of the numerator is the asymptote's value, y=2
Vertical Asymp: Focus on the denominator. (x+3) = 0, x = -3
An exponential function contains the points (0,3) (3,40). What is the exponential function?
Answer: y= 3*2.881x
Work: y = a*bx
Use (0,3) first: 3 = a*b^0 --> 3 = a * 1, divide both sides by 1. a=3
Then use (3,40): 40 = 3 * b^3, divide both sides by 3, 40/3 = b^3, cube root both sides. b = 2.881
Plug in a and b, y = 3 * 2.881x
Graph the following ellipse, and find its center, vertices, and foci.
(y-2)2/49 + (x-2)2/4
Center: (2,2)
Vertices: (2,9), (2,-5)
Foci: (2, 5.5) (2,-1.5)
Consult Desmos for graph.
For the following function, solve for the x-intercepts.
54x2 + 27x - 60
Answer: x = 5/6, -4/3
Check with the group for work.
If the y-intercept of the following function is y=3, what is the x-intercept?
f(x)= 5x+3/1-2x
x=-3/5
In order to find the x-intercept, you have to make the numerator equal to 0. 5(-⅗) + 3 → -3 + 3 = 0
You invest $12,500 of savings into a bank that provides 3% interest per year for ten years, how much will you make by the end of the three years?
[Round up to the nearest cent if necessary.]
Answer: $13,659.09
Work; P = po (1+r/n)n*t
P = 12,500 (1 + 0.03/1)1*3
P = 12,500 (1.03)3
P = $13,659.09
Find the limits of the following equations:
(x→10) x-4/x-2
Answer: 3/4.
Consult group for work.
Graph and Simplify the following polynomial:
x6 - 15x4 + 10x3 + 60x2 - 72x
Answer: (x)(x+3)2(x-2)2
Check-in with the group about the graph.
Find the slant asymptote for the following equation:
x2-x+4/2x+2
Answer: 1/2x - 1
Check with the group for work/explanation.
Condense the following logarithmic function:
6log3x - 1/3log3y - 5log3z
Answer: log3(x6 /3√y * z5)
Explanation: Since they all have a base of 3, the equation is under log 3. 6log3x is indicated in the numerator with x6 in accordance to the condensing properties. Since y's coefficient is -1/3, it means it will be on the denominator and cube rooted. Since 5log3z is negative, it will also be on the denominator but multiplied.
Find the limit(s) of the following functions.
lim x-> (√x - 3 / x-9)Answer: 1/6
Consult group for work.