a. the x - value when y = 1
b. the y - intercept
c. the y - value when x = 1
d. the x - intercept
9 answers
In a linear function, the part of the function that corresponds to the initial value is:
b. the y - intercept
The y-intercept represents the value of the function when the independent variable (x) is equal to zero, which is typically interpreted as the initial value in real-world situations.
In the given function \( f(x) = -350x + 1,400 \), the initial value is represented by the constant term when \( x = 0 \).
When \( x = 0 \):
\[ f(0) = -350(0) + 1,400 = 1,400 \]
This initial value of 1,400 represents the distance from your starting point to your final destination at the beginning of your trip (when no days have passed).
Therefore, the correct interpretation is:
a. the initial value of 1,400 represents the distance from your starting point to your final destination.
To model the relationship between the number of miles and the total cost of the shuttle, we can represent the total cost as a linear function. The cost consists of a fixed fee of $15 plus an additional charge of $0.50 per mile.
Let \( x \) be the number of miles and \( y \) be the total cost of the shuttle. The equation can be written as:
\[ y = 0.50x + 15 \]
This means for every mile traveled (represented by \( x \)), you add $0.50 to the fixed cost of $15.
Thus, the correct option is:
c. \( y = 0.5x + 15 \), where \( x \) is the number of miles and \( y \) is the total cost of the shuttle.
To find the rate of change for each linear function, we'll calculate the slope (rate of change) using the slope formula:
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \]
The points are \( (2, 12) \) and \( (1, 9) \).
Using the slope formula: \[ \text{slope} = \frac{12 - 9}{2 - 1} = \frac{3}{1} = 3 \]
The points are \( (0, 2) \) and \( (5, 42) \).
Using the slope formula: \[ \text{slope} = \frac{42 - 2}{5 - 0} = \frac{40}{5} = 8 \]
Therefore, the statement that is correct is:
c. Linear Function 2 has the greater rate of change, with a rate of change equal to 8 while Linear Function 1 has a rate of change equal to 3.
The given equation is in the slope-intercept form, which is typically expressed as:
\[ y = mx + b \]
where:
For the equation \( y = -\frac{1}{2}x + 2 \):
Thus, the correct identification is:
Slope: -1/2 Y-Intercept: 2
