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Use the graphing calculator to graph these functions:
[tex]\[ y_1 = 2x \][/tex]
[tex]\[ y_2 = 2x^2 \][/tex]
[tex]\[ y_3 = 2^x \][/tex]

1. After approximately what y-value does the exponential function appear to surpass the linear function?

A. [tex]\( y \approx 4 \)[/tex]

2. After approximately what y-value does the exponential function surpass the quadratic function?

A. [tex]\( y \approx 80 \)[/tex]

3. Which function grows the fastest?

A. The exponential function

4. Which function grows the slowest?

A. The linear function
B. The quadratic function
C. The exponential function

Sagot :

GinnyAnsweridn
Certainly! Let's break down the process in a detailed, step-by-step manner using the correct functions and relationships between them:

### Step 1: Define the functions
We have three functions to consider:

1. Exponential function: [tex]\( y_1 = 2^x \)[/tex]
2. Quadratic function: [tex]\( y_2 = 2x^2 \)[/tex]
3. Linear function: [tex]\( y_3 = 2x \)[/tex]

### Step 2: Graphing the functions
When you graph these functions on a graphing calculator, you will notice how each function grows as [tex]\( x \)[/tex] increases.

### Step 3: Finding where the exponential function surpasses the linear function
Observe the graph to see where the exponential function [tex]\( y_1 = 2^x \)[/tex] overtakes the linear function [tex]\( y_3 = 2x \)[/tex]. From the graph:
- The exponential function surpasses the linear function once the y-value reaches 2. This means that for [tex]\( y_1 = 2^x > y_3 = 2x \)[/tex], [tex]\( y \)[/tex] must be greater than or equal to 2.

### Step 4: Finding where the exponential function surpasses the quadratic function
Next, observe the point where the exponential function [tex]\( y_1 = 2^x \)[/tex] overtakes the quadratic function [tex]\( y_2 = 2x^2 \)[/tex]. From the graph:
- The exponential function surpasses the quadratic function once the y-value reaches 4. This means that for [tex]\( y_1 = 2^x > y_2 = 2x^2 \)[/tex], [tex]\( y \)[/tex] must be greater than or equal to 4.

### Step 5: Determining which function grows the fastest
By the nature of exponential growth, the exponential function [tex]\( y_1 = 2^x \)[/tex] grows much faster compared to the linear and quadratic functions:
- The exponential function grows the fastest.

### Step 6: Determining which function grows the slowest
Comparing the growth rates of the linear and quadratic functions and knowing that linear growth is the slowest type of growth:
- The linear function [tex]\( y_3 = 2x \)[/tex] grows the slowest.

Therefore, putting it all together:

1. The exponential function appears to surpass the linear function at y-value = 2.
2. The exponential function appears to surpass the quadratic function at y-value = 4.
3. The exponential function grows the fastest.
4. The linear function grows the slowest.

I hope this detailed explanation helps you understand the relationships and growth rates of these functions!
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