graph f(x) and g(x). then describe the transformation(s) from the graph of f(x) to the graph of g(x).
17. f(x) = 8x, g(x) = 4x
18. f(x) = -x + 2, g(x) = -x - 1
19. f(x) = 3x, g(x) = 6x - 1
20. an airport parking lot charges an entry fee of $2.00 plus $2.50 for every hour that your car is parked. the total charge for parking x hours is f(x) = 2.5x + 2. how will the graph of this function change if the entry fee is increased to $3.50? if the hourly rate is reduced to $2.25?

Question

Accepted Answer

### Problem 17:
# Explanation:
## Step1: Identify Function Types
Both $ f(x) = 8x $ and $ g(x) = 4x $ are linear functions (in the form $ y = mx + b $, here $ b = 0 $).
## Step2: Analyze Slope and Transformation
The slope of $ f(x) $ is $ 8 $, and the slope of $ g(x) $ is $ 4 $. Since $ 4=\frac{8}{2} $, the graph of $ g(x) $ is a vertical compression of the graph of $ f(x) $ by a factor of $ \frac{1}{2} $ (because the slope is halved, which corresponds to compressing the graph vertically towards the x - axis).
# Answer:
The graph of $ g(x) $ is a vertical compression of the graph of $ f(x) $ by a factor of $ \frac{1}{2} $.

### Problem 18:
# Explanation:
## Step1: Identify Function Types
Both $ f(x)=-x + 2 $ and $ g(x)=-x-1 $ are linear functions with the same slope ($ m=-1 $).
## Step2: Analyze Vertical Shift
The y - intercept of $ f(x) $ is $ 2 $, and the y - intercept of $ g(x) $ is $ - 1 $. To get from $ 2 $ to $ -1 $, we subtract $ 3 $ (or add $ - 3 $). So the graph of $ g(x) $ is the graph of $ f(x) $ shifted down 3 units.
# Answer:
The graph of $ g(x) $ is the graph of $ f(x) $ shifted down 3 units.

### Problem 19:
# Explanation:
## Step1: Analyze Slope Change
The slope of $ f(x)=3x $ is $ 3 $, and the slope of $ g(x)=6x - 1 $ is $ 6 $. Since $ 6 = 2\times3 $, the first transformation related to the slope is a vertical stretch of the graph of $ f(x) $ by a factor of $ 2 $ (because the slope is doubled, which corresponds to stretching the graph vertically away from the x - axis).
## Step2: Analyze Vertical Shift
The graph of $ f(x) $ has a y - intercept of $ 0 $, and the graph of $ g(x) $ has a y - intercept of $ - 1 $. So after the vertical stretch, the graph is shifted down 1 unit.
# Answer:
The graph of $ g(x) $ is a vertical stretch of the graph of $ f(x) $ by a factor of $ 2 $ followed by a vertical shift down 1 unit.

### Problem 20:
# Explanation:
## Case 1: Entry fee increased to $3.50
- The original function is $ f(x)=2.5x + 2 $. The new function with an entry fee of $ \$3.50 $ is $ f_1(x)=2.5x+3.5 $.
- Since the slope ($ m = 2.5 $) remains the same and the y - intercept changes from $ 2 $ to $ 3.5 $, the graph of the new function is a vertical shift up of $ 3.5 - 2=1.5 $ units of the graph of the original function $ f(x) $.

## Case 2: Hourly rate reduced to $2.25
- The original function is $ f(x)=2.5x + 2 $. The new function with an hourly rate of $ \$2.25 $ is $ f_2(x)=2.25x + 2 $.
- The slope of the original function is $ 2.5 $, and the slope of the new function is $ 2.25 $. Since $ 2.25=\frac{2.25}{2.5}\times2.5=\frac{9}{10}\times2.5 $, the graph of the new function is a vertical compression of the graph of the original function $ f(x) $ by a factor of $ \frac{9}{10} $ (because the slope is multiplied by $ \frac{9}{10} $, which corresponds to compressing the graph vertically towards the x - axis) with the same y - intercept.
# Answer:
- If the entry fee is increased to $3.50, the graph shifts up 1.5 units.
- If the hourly rate is reduced to $2.25, the graph is vertically compressed by a factor of $ \frac{9}{10} $.

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QUESTION IMAGE

Question

Explanation:

Problem 17:

Step1: Identify Function Types

Step2: Analyze Slope and Transformation

Step1: Identify Function Types

Step2: Analyze Vertical Shift

Step1: Analyze Slope Change

Step2: Analyze Vertical Shift

Answer:

Problem 18: