Algebra 1 Quadratics, Inverse Variation, COBF Test Study Guide
SOLs A.4 A.7 A.8 A.11
Name _______________________________________ Date _______________ Block _________
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Algebra 1 Quadratics, Inverse Variation, Curve of Best Fit Test STUDY GUIDE
The standard form of a parabolic quadratic function is f(x) = ax2 + bx + c
We can sketch a quadratic of the form ax2 + bx + c by finding:
o Its concavity: concave up (positive a) or concave down (negative a)
o The y-intercept (occurs at f(0), and is the same value as c)
o The axis of symmetry, the vertical line that divides the parabola into two symmetric
b
parts; the axis of symmetry is x = 
2a
b
o The vertex: the vertex always occurs on the axis of symmetry, so its x-value is 
;
2a
b
find the y-value of the vertex by plugging 
into the equation.
2a
o The number of zeros: use the discriminant b2 – 4ac:
 If b2 – 4ac is positive, there are two zeros
 If b2 – 4ac is zero, there is one and only one zero
 If b2 – 4ac is negative, there are no real zeros
o The zeros (x-intercepts), if any; find zeros using one of these methods:
 Graph using the graphing calculator; find the zeros by observation (or use the
“zeros” function to find the values of the zeros)
 Set the equation equal to 0 (all terms on one side; 0 on the other)
 Solve by factoring (factor; set each factor equal to 0 and solve)
 b  b 2  4ac
 Solve by quadratic formula: x =
2a
 Solve by completing the square: “half it, square it”
Determine if a function is a direct variation, inverse, variation, or neither:
o Direct variation is linear and always has a y-intercept through the origin, and is of
y
the form y = kx, where k is the constant of variation. We can rewrite it as k =
to
x
find the constant of variation. Direct variations are proportions.
k
o Inverse variation graphs as hyperbolas, and is of the form y =
, where k is the
x
constant of variation. We can rewrite it as k = xy to find the constant of variation
o Use these facts to determine if a table of values is a direct variation, inverse
variation, or neither.
o Solve word problems based on whether two variables vary directly or inversely.
Model data using a quadratic regression
o Use the calculator to find the quadratic curve of best fit for data, and make
predictions based on this data
o Use the coefficient of determination (R2) to determine if the curve of best fit is a
better model than the line of best fit. The closer R2 is to 1, the better.
Algebra 1 Quadratics, Inverse Variation, COBF Test Study Guide
Page 2
Study Questions
1) What are the zeros of the function shown at right?
2) Use the discriminant to find the number of real solutions:
a) x2 – 2x + 1 = 0
b) 2x2 + 3x + 5 = 0
c) 2x2 – 5x + 6 = 8
3) Sketch the graph of the following quadratics, providing the information requested.
a) f(x) = x2 – 6x + 8
b) f(x) = -x2 – 2x + 3
c) f(x) = 4x2 + 8x + 3
concave up or down:________
axis of symmetry:_________
vertex coordinate:_________
y-intercept (f(0)):_________
zeros (if any):____________
concave up or down:________
axis of symmetry:_________
vertex coordinate:_________
y-intercept (f(0)):_________
zeros (if any): ____________
concave up or down:________
axis of symmetry:_________
vertex coordinate:_________
y-intercept (f(0)):_________
zeros (if any):____________
d) f(x) = 3x2 + 12x + 8
concave up or down:________
axis of symmetry:_________
vertex coordinate:_________
y-intercept (f(0)):_________
zeros (if any):____________
4) Find the “c” value that completes the square. Then write the product as a binomial
squared.
a) x2 – 6x + c
b) x2 + 2x + c
c) x2 + 7x + c
d) x2 – 5x + c
Algebra 1 Quadratics, Inverse Variation, COBF Test Study Guide
Page 3
5) Solve by completing the square:
a) x2 + 6x = -5
b) x2 – 8x – 9 = 0
c) x2 – 6x + 3 = 0
d) 4x2 – 16x = -8
6) Determine whether the following equations represent direct variation, inverse variation,
or neither.
a) y = 5x
b) y =
10
x
c) 3x – 2y = 10
d) xy = 12
e) y =
x
5
7) Find the constant of variation. Then write the equation of the function.
a) x varies directly as y, and x = 3 when y = 9
b) x varies inversely as y, and x = 4 when y = -8
k =___________
equation______________________
k = ___________
equation______________________
c) x varies directly as y, and (4, 16) is a point on
the function.
d) x varies inversely as y, and (-3, 6) is a point
on the function.
k =___________
equation______________________
k = ___________
equation______________________
8) Determine whether the following tables represent a direct variation, inverse variation, or
neither. If it is a direct variation, write the equation. If it is neither, leave the equation
blank or write n/a.
a)
b)
x
y
1
3
3
12
5
15
7
21
x
y
direct/inverse/neither___________
equation______________________
c)
1
18
2
9
3
6
9
2
direct/inverse/neither___________
equation______________________
d)
x
y
-1
9
-3
27
5
-45
7
-63
direct/inverse/neither___________
equation______________________
x
y
1
-36
2
-18
-3
12
-4
9
direct/inverse/neither___________
equation______________________
9) Jacob works a job where the money he earns varies directly to the hours he works. In
one week, Jacob worked for 15 hours and made $146.26. How many hours did Jacob
work in his second week if he earned $117?
10) A local fast food restaurant takes in $9,000 in a 4 hour period. How many hours would
it take the restaurant to earn $20,500?
Algebra 1 Quadratics, Inverse Variation, COBF Test Study Guide
Page 4
11) The force F needed to loosen a bolt with a wrench varies inversely with the length l of
the handle. It takes 250 lbs of force to loosen a bolt with a 6 inch long handle. How
much force is needed for a 24 inch long handle?
12) The volume, V, of a gas varies inversely as the pressure, p, in a container. If the volume
of a gas is 200cc when the pressure is 1.6 liters per square centimeter, find the volume
(to the nearest tenth) when the pressure is 2.8 liters per sq centimeter.
13) On Tuesday, May 10, 2005, 17 year-old Adi Alifuddin Hussin won
the boys’ shot-putt gold medal for the fourth consecutive year. His
winning throw was 16.43 meters. A shot-putter throws a ball at an
inclination of 45° to the horizontal. The following data represent
approximate heights for a ball thrown by a shot-putter as it travels
a distance of x meters horizontally.
a) Draw a scatter plot of the data.
b) Use the calculator to find the linear regression equation (round coefficients to the
nearest hundredth). Find r2, the coefficient of determination.
c) Use the calculator to find the quadratic regression equation (round coefficients to the
nearest hundredth). Find R2, the coefficient of determination.
d) Compare the coefficient of determination (R2) of the quadratic model with the same
value (r2) for the linear model. Which model is a better fit for the data? Explain how
you came up with your conclusion.
e) Use the quadratic model to predict the height of the ball if it travels 80 meters.
f) Use the quadratic model to predict the height of the ball if it travels 100 meters.
Explain whether this prediction makes sense or not.
Algebra 1 Quadratics, Inverse Variation, COBF Test Study Guide
Page 5
STUDY QUESTION ANSWERS
1) x = -3, 2
2) a) one solution b) no solutions c) two solutions
3)
a) up; axis of sym: x=3; vertex: (3, 1), y-int = 8; zeros x = 2, 4
c) up; axis of sym: x=-1; vertex:
1), y-int=3; zeros =-1/2, -3/2
b) down; axis of sym: x=-1; vertex: (-1,
4), y-int=3; zeros =-3, -1
(-1, - d) up; axis of sym: x=-2; vertex: (-2, -4),
62 3
y-int=8; zeros =
 -3.15, -.85
3
4) a) c = 9; (x – 3)2 b) c = 1; (x + 1)2 c) c =
2
25 
7
5
;  x   d) c =
;x  
4 
4 
2
2
49 
2
5) a) x = -1, -5 b) x = -1, 9 c) x = 3± 6 ≈ 0.55, 5.45 d) x = 2± 2 ≈ 0.59, 3.41
6) a) direct b) inverse c) neither d) inverse e) direct
7) a) k = 3; y = 3x b) k = -32; xy = -32 or y = -
-18 or y= -
32
c) k = 4; y = 4x d) k = -18; xy =
x
18
x
8) a) neither (no equation) b) inverse; y =
18
36
c) direct; y = -9x d) inverse; y = x
x
9) 12 hours
10) 9.11 hours
11) 62.5 lbs
12) 114.3 cc
13) a) see right b) y=.23x+10.82; R2=.588 c) y = -.01x2+1.06x+0.24;
R2=.97 d) quadratic model is better because R 2 is closer to 1 e)
about 14 m f) about -4.8 m; no because height can’t be negative