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Compare Linear Function 1 with Linear Function 2. Which one has the lesser initial value?

Linear Function 1: Lionel’s cell phone plan costs $45 per month.
Linear Function 2: The relationship between the number of months Ruby has had her cell phone (x)
and the total cost (y)
is given by y=40x+60

options:

.Linear Function 1 has the lesser initial value, with an initial value equal to $0 while Linear Function 2 has an initial value equal to $40.
Linear Function 1 has the lesser initial value, with an initial value equal to $0 while Linear Function 2 has an initial value equal to $40.

Linear Function 1 has the lesser initial value, with an initial value equal to $0 while Linear Function 2 has an initial value equal to $60.
Linear Function 1 has the lesser initial value, with an initial value equal to $0 while Linear Function 2 has an initial value equal to $60.

Linear Function 1 has the lesser initial value, with an initial value equal to $45 while Linear Function 2 has an initial value equal to $60.
Linear Function 1 has the lesser initial value, with an initial value equal to $45 while Linear Function 2 has an initial value equal to $60.

Linear Function 2 has the lesser initial value, with an initial value equal to $40 while Linear Function 1 has an initial value equal to $45.

3 answers

To compare the initial values of Linear Function 1 and Linear Function 2, we need to analyze each function:

Linear Function 1: Lionel’s cell phone plan costs $45 per month.

  • This means the total cost (y) is $45 per month. If we consider the initial value at 0 months, the initial cost is $0 (no cost before any month), and then it accumulates $45 for each month after.

Linear Function 2: The relationship between the number of months Ruby has had her cell phone (x) and the total cost (y) is given by \(y = 40x + 60\).

  • In this function, when \(x = 0\) (i.e., at the start or initial month), the total cost \(y\) is \(40(0) + 60 = 60\). So, the initial value of her cell phone plan is $60.

Now we will compare the initial values:

  • Linear Function 1: initial value = $0 (before any cost is incurred).
  • Linear Function 2: initial value = $60.

Thus, Linear Function 1 has the lesser initial value.

The correct option is: Linear Function 1 has the lesser initial value, with an initial value equal to $0 while Linear Function 2 has an initial value equal to $60.

Analyze the linear function whose slope is −3
and passes through the coordinate pair (−2,9)
. Which linear function matches this line?

options:

y=3x−3
y equals 3 x minus 3

y=−3x+3
y equals negative 3 x plus 3

y=−2x−3
y equals negative 2 x minus 3

y=−3x+9
y equals negative 3 x plus 9

To find the linear function that has a slope of \(-3\) and passes through the coordinate point \((-2, 9)\), we can use the point-slope form of a linear equation, which is given by:

\[ y - y_1 = m(x - x_1) \]

Where:

  • \(m\) is the slope,
  • \((x_1, y_1)\) is the point the line passes through.

Here, the slope \(m = -3\) and the point \((-2, 9)\) gives us \(x_1 = -2\) and \(y_1 = 9\).

Substituting these values into the point-slope formula:

\[ y - 9 = -3(x - (-2)) \]

This simplifies to:

\[ y - 9 = -3(x + 2) \]

Distributing the \(-3\):

\[ y - 9 = -3x - 6 \]

Adding \(9\) to both sides gives:

\[ y = -3x - 6 + 9 \]

\[ y = -3x + 3 \]

Now, we can compare this result with the options provided:

  1. \(y = 3x - 3\)
  2. \(y = -3x + 3\)
  3. \(y = -2x - 3\)
  4. \(y = -3x + 9\)

The correct match is:

\(y = -3x + 3\).