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Find the initial value of the linear function, given that the rate of change is m=−47, and (14,3) is an (x,y) value of the linear function.(1 point)

Responses

b=15.71

b=12.29

b=−5

b=11

5 answers

To find the initial value (y-intercept) \( b \) of the linear function given the rate of change \( m \) and a point \( (x, y) = (14, 3) \), we can use the point-slope form of a linear equation, which is:

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

where \( (x_1, y_1) \) is the point on the line and \( m \) is the slope. Here, \( m = -47 \), \( x_1 = 14 \), and \( y_1 = 3 \).

Substituting these values into the equation, we get:

\[ y - 3 = -47(x - 14) \]

Now let's rearrange this equation to find the y-intercept. First, distribute \(-47\):

\[ y - 3 = -47x + 658 \]

Next, add 3 to both sides to solve for \( y \):

\[ y = -47x + 661 \]

In this form, the y-intercept \( b \) (the initial value) is the constant term, which is 661.

None of the provided options (15.71, 12.29, -5, 11) are correct since the calculation shows that the y-intercept is \( 661 \). However, if you're looking for the initial \( b \) with the slope provided, the values or context might need clarification, or it could hint at a misunderstanding about the specific task or function.

If the context requires a different interpretation or setup, please clarify!

m isn't -47, it's -4 over 7. Its a fraction

Thank you for clarifying that the rate of change \( m \) is \( -\frac{4}{7} \).

We can use the point-slope form of a linear equation again:

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

Given:

  • \( (x_1, y_1) = (14, 3) \)
  • \( m = -\frac{4}{7} \)

Substituting these values into the equation:

\[ y - 3 = -\frac{4}{7}(x - 14) \]

Now let's rearrange this equation to find the y-intercept \( b \).

First, distribute \( -\frac{4}{7} \):

\[ y - 3 = -\frac{4}{7}x + \frac{56}{7} \]

Simplifying \( \frac{56}{7} \):

\[ y - 3 = -\frac{4}{7}x + 8 \]

Now, add 3 to both sides:

\[ y = -\frac{4}{7}x + 11 \]

So, the y-intercept \( b \) (the initial value) is 11.

Thus, the correct response is:

b = 11

n.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. An upward slanting line is plotted on the coordinate plane. The upward slanting line with arrows at both ends passes through the following closed points: left parenthesis negative 4 comma start fraction negative 10 over 3 end fraction right parenthesis, and left parenthesis negative 1 comma start fraction negative 4 over 3 end fraction right parenthesis.

Determine the initial value and the rate of change of the linear function as given in the graph.

(1 point)
Responses

The initial value is 16
1
6
, and the rate of change is 32
3
2
.
The initial value is Start Fraction 1 over 6 End Fraction , and the rate of change is Start Fraction 3 over 2 End Fraction .

The initial value is 23
2
3
, and the rate of change is −23
−
2
3
.
The initial value is Start Fraction 2 over 3 End Fraction , and the rate of change is negative Start Fraction 2 over 3 End Fraction .

The initial value is −2
−
2
, and the rate of change is 23
2
3
.
The initial value is negative 2 , and the rate of change is Start Fraction 2 over 3 End Fraction .

The initial value is −23
−
2
3
, and the rate of change is 23
2
3
.

To determine the initial value and the rate of change of the linear function based on the provided points \( \left(-4, -\frac{10}{3}\right) \) and \( \left(-1, -\frac{4}{3}\right) \), we can follow these steps:

Step 1: Find the rate of change (slope)

The slope \( m \) of the line can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

where \( (x_1, y_1) = \left(-4, -\frac{10}{3}\right) \) and \( (x_2, y_2) = \left(-1, -\frac{4}{3}\right) \).

Plugging in the values:

\[ m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)} \]

This simplifies to:

\[ m = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]

So, the rate of change is \( \frac{2}{3} \).

Step 2: Find the initial value (y-intercept)

To find the y-intercept \( b \), we can use the slope-intercept form of the line:

\[ y = mx + b \]

We already know \( m \) and can use one of the points to find \( b \). We'll use the point \( (-1, -\frac{4}{3}) \):

\[ -\frac{4}{3} = \frac{2}{3}(-1) + b \]

Solving for \( b \):

\[ -\frac{4}{3} = -\frac{2}{3} + b \]

Adding \( \frac{2}{3} \) to both sides:

\[ b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3} \]

Summary:

Thus, we find:

  • The initial value (y-intercept) is \( -\frac{2}{3} \)
  • The rate of change (slope) is \( \frac{2}{3} \)

Correct Response:

The correct response is:

The initial value is negative \( \frac{2}{3} \), and the rate of change is \( \frac{2}{3} \).