Students can use the Spectrum Math Grade 8 Answer Key Chapter 4 Lesson 4.6 Initial Values of Linear Functions as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 4 Lesson 4.6 Initial Values of Linear Functions Answers Key
Where a linear function crosses the y-axis is considered its initial value. One way to find the initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b, where m represents the rate of change and b represents the initial value of the linear function, to solve.
\(\frac{18-6}{6-2}\) = \(\frac{12}{4}\) = 3
Step 1: Find the rate of change for the
function table.
6 = (3)(2) + b Step 2: Substitute values of x, y, and m in the linear equation.
6 – 6 = 6 – 6 + b
b = 0 Step 3: Solve for b to find the initial value of the function.
Find the initial value of each function.
Question 1.
a.
b = ____
Answer:
5
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{17 – 8}{12 – 3}\)
= \(\frac{9}{9}\)
= 1
y = mx + b,
for x = 3, y = 8 the initial value b is,
8 = 1(3) + b
8 – 3 = b
b = 5
b.
b = _____
Answer:
0
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{10 – 3}{90 – 27}\)
= \(\frac{7}{63}\)
= \(\frac{1}{9}\)
y = mx + b,
for x = 90, y = 10 the initial value b is,
10 = \(\frac{1}{9}\)(90) + b
10 – 10 = b
b = 0
c.
b = ____
Answer:
b = \(\frac{- 11}{2}\)
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{12 – 2}{14 – 6}\)
= \(\frac{10}{8}\)
= \(\frac{5}{4}\)
y = mx + b,
for x = 6, y = 2 the initial value b is,
2 = \(\frac{5}{4}\)(6) + b
b = 2 – \(\frac{15}{2}\)
b = \(\frac{2×2 – 15}{2}\)
b = \(\frac{4 – 15}{2}\)
b = \(\frac{-11}{2}\)
Question 2.
a.
b = ____
Answer:
-6
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{22 – 2}{7 – 2}\)
= \(\frac{20}{5}\)
= 4
y = mx + b,
for x = 2, y = 2 the initial value b is,
2 = 4(2) + b
2 – 8 = b
b = -6
b.
b = ___
Answer:
11
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{23 – 14}{12 – 3}\)
= \(\frac{9}{9}\)
= 1
y = mx + b,
for x = 3, y = 14 the initial value b is,
14 = 1(3) + b
14 – 3 = b
b = 11
c.
b = ____
Answer:
8
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{-4 – 0}{12 – 8}\)
= \(\frac{-4}{4}\)
= -1
y = mx + b,
for x = 8, y = 0 the initial value b is,
0 = -1(8) + b
b = 8
Question 3.
a.
b = ____
Answer:
4\(\frac{1}{2}\)
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{45 – 9}{9 – 1}\)
= \(\frac{36}{8}\)
= \(\frac{9}{2}\)
y = mx + b,
for x = 1, y = 9 the initial value b is,
9 = \(\frac{9}{2}\)(1) + b
b = 9 – \(\frac{9}{2}\)
b =\(\frac{18 -9}{2}\)
b = \(\frac{9}{2}\)
b = 4\(\frac{1}{2}\)
b.
b = ____
Answer:
8
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{14 – 10}{3 – 1}\)
= \(\frac{4}{2}\)
= 2
y = mx + b,
for x = 1, y = 10 the initial value b is,
10 = 2(1) + b
b = 8
c.
b = ____
Answer:
-3
Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = \(\frac{2 – 0}{10 – 6}\)
= \(\frac{2}{4}\)
= \(\frac{1}{2}\)
y = mx + b,
for x = 6, y = 0 the initial value b is,
0 = \(\frac{1}{2}\)(6) + b
b = – \(\frac{6}{2}\)
b = -3
Sometimes the initial value of a linear function can be immediately found when it is described in the context of a real-world problem.
Joe’s Pizza and Subs sells a 14-inch cheese pizza for $ 10.00. Toppings can be added to the pizza for $0.50 each. How much will a pizza cost if it has three toppings?
Initial Value: $ 10.00
Step 1: Identify the output variable. In this case, the price of the pizza is the output and the number of toppings is the input variable.
Step 2: Reread the problem to find the described output starting point. In this problem, it is $ 10.00.
Find the initial value in each real-world problem below.
Question 1.
Julie is trying to grow her hair out so she can donate it to be made into a wig. When she first measures it, it is six inches long. It is growing at a rate of one inch each month. How long will her hair be in 5 months?
Initial Value:
________
Answer:
Initial value is 6 inches.
11 inches after five months.
Explanation:
Julie first measures it, it is six inches long.
It is growing at a rate of one inch each month.
Total length of her hair be in 5 months,
m = 1 inch each month
b = 6 inches
x = 5 months
y = mx + b
y = 1(5) + 6
y = 11 inches.
Question 2.
Nina bought a new fish tank that holds 25 gallons of water. If she fills up the tank at a rate of 1 gallon per 30 seconds, how full will the tank be after 5 minutes?
Initial Value:
______
Answer:
Initial value is 0.
35 gallons after 5 minutes.
Explanation:
Nina bought a new fish tank that holds 25 gallons of water.
If she fills up the tank at a rate of 1 gallon per 30 seconds,
The amount of the water in the tank after 5 minutes,
m = 1 gallon for 30 seconds
m = 2 gallons for 1 minute
b = 25 gallons of water
x = 5 minutes
y = mx + b
y = 2(5) + 25
y = 35 gallons.
Question 3.
Michael has to mow his grass when it gets too long. He likes for the grass to be 3 inches long. If it grows at a rate of 1 inch every 2 days, how long will his grass be after 8 days without mowing?
Initial Value:
_________
Answer:
Initial value is 3 inches.
y = 8 inches.
Explanation:
The grass to be 3 inches long, it grows at a rate of 1 inch every 2 days,
how long will his grass be after 8 days without mowing,
m = 1 inch every 2 days
Initial value is 3 inches.
b = 3 inches
x = 8 days
y = mx + b
y = 1(5) + 3
y = 8 inches.
Question 4.
Phillip has been assigned 50 pages of reading. If he still has 35 pages of reading left after he has been reading for 10 minutes, how many pages will he have left after he has been reading for 40 minutes?
Initial Value:
________
Answer:
Initial value is 50 pages.
Explanation:
Number of assigned 50 pages of reading.
b = 50 pages.
y = mx + b
where b is the initial value 50 pages.
Question 5.
Jack pulls the plug from a bathtub that is holding 25 gallons Initial Value: of water. After 40 seconds, there are 15 gallons of water left in the tub. How much water will be left in the bathtub one minute after Jack pulls the plug?
Initial Value:
________
Answer:
Initial value is 25 gallons.
Explanation:
Bathtub is holding 25 gallons Initial Value,
b = 25 gallons.
The amount of the water in the tank after 1 minutes,
m = 45 gallons for 40 seconds.
m = 45/4 gallons for 10 seconds.
b = 25 gallons.
x = 60 seconds.
y = mx + b
Question 6.
Penelope lives 10 miles from her school. After 5 minutes of driving, she still has 7 miles left before she gets to school. How far will she have traveled after 8 minutes of driving?
Initial Value:
________
Answer:
Initial value 10 miles.
Explanation:
Penelope lives 10 miles from her school.
After 5 minutes of driving, she still has 7 miles left before she gets to school.
b = 10 miles.
m =7/5
x = 8 minutes.
y = mx + b
y = 7/5 (8) + 10
y = 24 miles.