MS 100 College Algebra test five • Name:

1. Parabola What is the name of the shape seen on the graph below?

Data table for the above graph

2. Find a, b, and c in the y = ax² + bx + c form of the equation for the graph above. The coordinates for both x-intercepts and the vertex are depicted on the graph.

(y − 45) = a(x − (−4))²
 y − 45 = a(x + 4)²
 0 − 45 = a(−40 + 4)²
 0 − 45 = a(−36)²
 −45 = 1296a
 a = −45/1296 = −5/144 = −0.0347
 y − 45 = −5/144(x + 4)²
 y − 45 = −5/144(x² + 8x + 16)
 y − 45 = −5/144)x² + (−5/18)x + (−5/9)
 y = (−5/144)x² + (−5/18)x + (−5/9) + 45
 y = (−5/144)x² + (−5/18)x + (400/9)

This is really a relative of the marble problem:
 y − k = a(x − h)²
 0 − k = a(r − h)²
 where (±r,0) are the roots for the parabola
 −k = a(r² −2rh + h²)
 a = (−k)/(r² −2rh + h²)
 Note that if h = 0, as in the marble problem, then a = (−k)/(r²)
 

a = −5/144 = −0.03472

b = −5/18 = −0.2778

c = 400/9 = 44.44

3. y = 400/9 = 44.44 Find the y-intercept for the graph depicted above.

4. Use a Qalculate! to help you sketch a graph of the following function over the domain −5.1 ≤ x ≤ 5.1. Sketch the graph on your paper.
$\text{f(x)}=x^6-34x^4+225x^2$

5. Yes Is the equation in question four a function?

6. 6 What is the degree of the equation in question four?

7. 5 (technically six with a repeated zero at x = 0) How many zeros are there for the equation in question four?

8. Use polynomial division to divide the equation in question four by (x+3)

$\text{f(x)/(x+3)}=x^5-3x^4-25x^3+75x^2$

Gnuplot commands for plot above:

reset
set border
set xtics 1
set ytics 100
set xzeroaxis lt 9 lw 5
set yzeroaxis lt 9 lw 5
set style line 1 lt 1 lw 4
set style line 2 lt 3 lw 5
set style line 3 lt 10 lw 6
set style line 4 lt 12 lw 3
set style line 5 lt 9 lw 1
set grid ls 5
show grid
set xrange [-6:6]
set yrange [-1200:500]
set samples 1000
set key off
f(x)=x**6-34*x**4+225*x**2
plot f(x) ls 1

Maxima output

(%i1) expand((x-3)*(x+3)*(x-5)*(x+5)*x*x);
((%o1) x^6-34*x^4+225*x^2