# Lesson 4

• Let’s compare quadratic and exponential changes and see which one grows faster.

### Problem 1

The table shows values of the expressions $$10x^2$$ and $$2^x$$

1. Describe how the values of each expression change as $$x$$ increases.
2. Predict which expression will have a greater value when:

1. $$x$$ is 8
2. $$x$$ is 10
3. $$x$$ is 12
3. Find the value of each expression when $$x$$ is 8, 10, and 12.

4. Make an observation about how the values of the two expressions change as $$x$$ becomes greater and greater.

$$x$$ $$10x^2$$ $$2^x$$
1 10 2
2 40 4
3 90 8
4 160 16
8
10
12

### Problem 2

Function $$f$$ is defined by $$f(x)=1.5^x$$. Function $$g$$ is defined by $$g(x)=500x^2 + 345x$$.

1. Which function is quadratic? Which one is exponential?
2. The values of which function will eventually be greater for larger and larger values of $$x$$?

### Problem 3

Create a table of values to show that the exponential expression $$3(2)^x$$ eventually overtakes the quadratic expression $$3x^2+2x$$.

### Problem 4

The table shows the values of $$4^x$$ and $$100x^2$$ for some values of $$x$$.

Use the patterns in the table to explain why eventually the values of the exponential expression $$4^x$$ will overtake the values of the quadratic expression $$100x^2$$.

$$x$$ $$4^x$$ $$100x^2$$
1 4 100
2 16 400
3 64 900
4 256 1600
5 1024 2500

### Problem 5

Here is a pattern of shapes. The area of each small square is 1 sq cm.

1. What is the area of the shape in Step 10?
2. What is the area of the shape in Step $$n$$?
3. Explain how you see the pattern growing.
(From Unit 6, Lesson 2.)

### Problem 6

A bicycle costs $240 and it loses $$\frac{3}{5}$$ of its value each year. 1. Write expressions for the value of the bicycle, in dollars, after 1, 2, and 3 years. 2. When will the bike be worth less than$1?
3. Will the value of the bike ever be 0? Explain your reasoning.
(From Unit 5, Lesson 4.)

### Problem 7

A farmer plants wheat and corn. It costs about \$150 per acre to plant wheat and about \$350 per acre to plant corn. The farmer plans to spend no more than \\$250,000 planting wheat and corn. The total area of corn and wheat that the farmer plans to plant is less than 1200 acres.

This graph represents the inequality, $$150w + 350c \leq 250,\!000$$, which describes the cost constraint in this situation. Let $$w$$ represent the number of acres of wheat and $$c$$ represent the number of acres of corn.
1. The inequality, $$w + c < 1,\!200$$ represents the total area constraint in this situation. On the same coordinate plane, graph the solution to this inequality.