Algebra III: 4.6 Residuals
Solutions
Name _______________________
Period ____ Date ______
Bell Work: Write an equation for each function described.
a. A quadratic function 𝒇 such that 𝑓(−1) = 3, 𝑓(1) = 9, and 𝑓(4) = 48 𝑓(𝑥) = 2𝑥 2 + 3𝑥 + 4
b. A linear function 𝒈 such that 𝑔(−1) = 3 and 𝑔(1) = 9 𝑔(𝑥) = 6𝑥 + 3
c. An exponential function 𝒉 such that ℎ(1) = 9 and ℎ(2) = 27 ℎ(𝑥) = 3(3)𝑥
6-10. Battle Creek Cereal is trying a variety of packaging
sizes for their Crispy Puffs cereal. Below is a list of six
current packages. (Label Axes)
a.
Create a Scatterplot in the calculator and above.
Describe the association
b.
Generate a regression equation to predict the net weight of
cereal based on the amount of cardboard used for the package.
c.
If 260 in2 of cardboard is used, then predict the amount of cereal it will hold.
6-11. A residual is a measure of how far a prediction is from what is actually observed.
residual = actual – predicted. The 260 in2 box from problem 6-10 will actually hold 355 g of cereal.
a.
b.
c.
Indicate a residual as a distance on your scatterplot from problem 6-10. What is the residual for the
260 in2 box?
What is the difference between a positive and a negative residual in the context of this problem?
How could graphing the actual and predicted data points help?
6-12. What is the residual for the 471 in2 box? Mark the residual on your scatterplot. Be sure to include units
for your residual.
6-13. The warehouse store wants to offer a super-sized 600 in2 box.
a.
The residual for this box is 1005 grams. What is the actual weight of a 600 in2 box?
b.
Why do you suppose the residual is so large?
c.
Interpret the meaning of the slope and y-intercept of your model in the context of this problem.
Does the y-intercept make sense in the context of the problem?
6-14. Armen was concerned about the amount of sugar in his diet, so he went to the store and collected data
from several cereal boxes. Armen used the data to create a model that related the sugar in cereal to calories:
s = –16.9 + 0.23c where s is the amount of sugar in grams and c is the number of calories in one cup of cereal.
a.
What does a negative residual mean in this context? Is a cereal with a
positive or negative residual better for Armen’s diet?
b.
Interpret the meaning of the slope and y-intercept in the context of
the problem. Does the y-intercept make sense in the context of the
problem?
6-15. Write an equation in slope-intercept form for each line described
SHOW SUFFICIENT WORK FOR CREDIT!
a.
Contains (0, 3) and (-2, 5)
b.
Has an x-intercept of -7 and a y-intercept of 2
𝒚 = −𝒙 + 𝟐
c.
Is parallel to the line 𝑦 = 3𝑥 + 4 and contains (11, 0)
𝒚 = 𝟑𝟑 − 𝟑𝟑
d.
Has a slope of
−1
3
and contains the point (-6, 5)
𝒚=
𝟐
𝒙+𝟐
𝟕
𝒚=
−𝟏
𝒙+𝟑
𝟑
6-16. The price of homes (in thousands of dollars) is associated with the number of square feet in a home.
Home prices in Smallville can be modeled with the equation p = 150 + 0.041a where p is the price of the home
in thousands of dollars and a is the area of the house in square feet. Home prices in Fancyville can be modeled
with the equation p = 250 + 0.198a. Ngoc saw a real estate advertisement for a 2800 square foot home that was
selling for $280,000. Which city should she predict that the home is in?
6-17. For this sequence 5, 25, 125, 625, …
a. What kind of sequence is it?
b. Find the fifth term
c. Write an explicit equation
d. Write a recursive equation
d. 𝑎1 = 5, 𝑎𝑛+1 = 𝑎𝑛 ∗ 5
6-18. Write an explicit equation for the sequence based on the graph at right.
6-19. Lisa is weighing combinations of children’s blocks. Four blue blocks and five red blocks weigh 32
ounces. One blue block and eight red blocks weigh 35 ounces. What is the combined weight of one red block
and one blue block? Write a system of equations and solve algebraically.
a.
b. Solution for System
Define Variables:
𝑏 = weight of blue block in oz.
𝑟 = weight of rblock in oz.
(show sufficient work)
𝑏 = −8𝑟 + 35
4(−8𝑟 + 35) + 5𝑟 = 32
−32𝑟 + 140 + 5𝑟 = 32
−27𝑟 = −108
𝒓=𝟒
System of Equations:
4𝑏 + 5𝑟 = 32
𝑏 + 8𝑟 = 35
c. Answer to Question.
𝑟+𝑏
= 4+3
=𝟕
The combined weight of
1 red block and 1 blue
block is 7 ounces.
𝑏 = −8(4) + 35 = 𝟑
6-20.Solve for the indicated variable.
a. V = LWH (for W)
𝑥
b. 𝑦 = 2 + 3 (for x)
c.
𝐹
𝑅
= 𝐼 (for R)
6-21. Find the point of intersection of the lines x = 3 and 𝑦 =– 2𝑥 + 1.
1
d. 2𝑥 + 𝑦 = 3 (for y)
𝑦 = −2(3) + 1
𝑦 = −6 + 1
𝑦 = −5
(𝟑, −𝟓)