Use the table to answer the question.

Question

x
y

1 4
−3
8
Find the rate of change and the initial value of the linear function given as a table of values.

(1 point)

Answer 1

To find the rate of change and the initial value of the linear function from the given table of values, we will identify the coordinates provided and calculate the slope (rate of change).

The table gives us the points:

• (1, 4)
• (-3, 8)

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

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

Taking the points (1, 4) and (-3, 8):

• \( (x_1, y_1) = (1, 4) \)
• \( (x_2, y_2) = (-3, 8) \)

Now, substituting into the formula:

\[ m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1 \]

Thus, the rate of change is \(-1\).

To find the initial value (the y-intercept \( b \)), we can express the linear function in the slope-intercept form \( y = mx + b \).

Using one of the points, let's use (1, 4):

\[ 4 = (-1)(1) + b \] \[ 4 = -1 + b \] \[ b = 4 + 1 = 5 \]

Thus, the initial value (y-intercept) is \( 5 \).

To summarize:

• Rate of change: \(-1\)
• Initial value: \(5\)

Answer 2

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 on both ends passes through the following closed points: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3 right parenthesis.
Determine the initial value and the rate of change of the linear function given in the graph.

Answer 3

To determine the initial value and the rate of change of the linear function depicted in the graph, we will analyze the two points provided: (0, 0) and (2, 3).

Step 1: Determine the Rate of Change (Slope)

The rate of change (or slope \( m \)) can be calculated using the formula:

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

Using the points:

• \( (x_1, y_1) = (0, 0) \)
• \( (x_2, y_2) = (2, 3) \)

Substituting these coordinates into the slope formula:

\[ m = \frac{3 - 0}{2 - 0} = \frac{3}{2} \]

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

Step 2: Determine the Initial Value (Y-Intercept)

The initial value (or y-intercept \( b \)) is the value of \( y \) when \( x = 0 \). From the point (0, 0):

When \( x = 0 \), \( y = 0 \), which means the initial value \( b = 0 \).

Summary

• Rate of change: \(\frac{3}{2}\)
• Initial value: \(0\)

Answer 4

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. A downward slanting line is plotted on the coordinate plane. The downward slanting line with arrows on both ends passes through the following closed points: left parenthesis negative 3 comma 2.5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.
Determine the initial value and the rate of change of the linear function as given in the graph.

Answer 5

To determine the initial value and the rate of change of the linear function from the points provided in the graph, we can use the coordinates of the points given: \((-3, 2.5)\), \((0, 1)\), and \((2, 0)\).

Step 1: Determine the Rate of Change (Slope)

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

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

Let's use the points \((0, 1)\) and \((2, 0)\) for our calculation:

• \( (x_1, y_1) = (0, 1) \)
• \( (x_2, y_2) = (2, 0) \)

Substituting these coordinates into the slope formula:

\[ m = \frac{0 - 1}{2 - 0} = \frac{-1}{2} \]

Thus, the rate of change (slope) is \(-\frac{1}{2}\).

Step 2: Determine the Initial Value (Y-Intercept)

The initial value (or y-intercept \( b \)) is the value of \( y \) when \( x = 0 \). From the point \((0, 1)\):

When \( x = 0 \), \( y = 1 \), which means the initial value \( b = 1 \).

Summary

• Rate of change: \(-\frac{1}{2}\)
• Initial value: \(1\)

Answer 6

Determine the rate of change in the following description. In Ms. Katy’s math class, a homework assignment is worth 100 points. If students turn the homework in late, there will be a deduction of 5 points per day. (1 point)

Answer 7

The rate of change in this scenario refers to the decrease in points for the homework assignment as the number of days it is turned in late increases.

Since the homework assignment starts at a total value of 100 points and there is a deduction of 5 points for each day it is late, the rate of change is:

• Rate of Change: \(-5\) points per day

This means for each day the homework is late, the score decreases by 5 points. Thus, the rate of change is \(-5\) points/day.

Answer 8

Use the table to answer the question.

x
y

2.6 −6

3.2 3
Determine the rate of change of the linear function given as a table of values.

Answer 9

To determine the rate of change (slope) of the linear function from the given points in the table, we can use the formula for slope:

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

The points provided are:

• \( (x_1, y_1) = (2.6, -6) \)
• \( (x_2, y_2) = (3.2, 3) \)

Now we can substitute the values into the slope formula:

\[ m = \frac{3 - (-6)}{3.2 - 2.6} \]

This simplifies to:

\[ m = \frac{3 + 6}{3.2 - 2.6} = \frac{9}{0.6} \]

Now, performing the division:

\[ m = 15 \]

Thus, the rate of change of the linear function is 15.

Answer 10

Find the initial value of the linear function, given that the rate of change is m=−4/7
, and (14,3)
is an (x,y)
value of the linear function.(1 point)

Answer 11

To find the initial value (y-intercept, \( b \)) of the linear function, we can use the slope-intercept form of a linear equation, which is:

\[ y = mx + b \]

Where:

• \( m \) is the rate of change (slope),
• \( (x, y) \) is a point on the line,
• \( b \) is the y-intercept (initial value).

Given:

• The slope \( m = -\frac{4}{7} \)
• The point \( (14, 3) \)

We can substitute these values into the equation to solve for \( b \):

\[ 3 = -\frac{4}{7}(14) + b \]

Calculating the term involving the slope:

\[ 3 = -\frac{4 \times 14}{7} + b \] \[ 3 = -\frac{56}{7} + b \] \[ 3 = -8 + b \]

Now, add 8 to both sides to solve for \( b \):

\[ b = 3 + 8 \] \[ b = 11 \]

Therefore, the initial value (y-intercept) of the linear function is 11.

Answer 12

alt="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

Answer 13

To determine the initial value and the rate of change of the linear function based on the introduced points, we will first identify the coordinates of the points given and then calculate the slope (rate of change) and the y-intercept (initial value).

Step 1: Identify the Points

The coordinates given are:

• \( (-4, -\frac{10}{3}) \)
• \( (-1, -\frac{4}{3}) \)

Step 2: Determine the Rate of Change (Slope)

The formula for calculating the slope \( m \) is:

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

Using the points:

• \( (x_1, y_1) = (-4, -\frac{10}{3}) \)
• \( (x_2, y_2) = (-1, -\frac{4}{3}) \)

Substituting into the formula:

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

This simplifies to:

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

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

Step 3: Determine the Initial Value (Y-Intercept)

To find the initial value (y-intercept \( b \)), we can use the slope-intercept form of the line \( y = mx + b \). We can use one of the points to solve for \( b \). Let's use the point \((-1, -\frac{4}{3})\):

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

Calculating the right side:

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

Now, add \(\frac{2}{3}\) to both sides to solve for \( b \):

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

Summary

• Rate of Change: \(\frac{2}{3}\)
• Initial Value (Y-Intercept): \(-\frac{2}{3}\)

Answer 14

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 1.5 comma negative 3 right parenthesis, left parenthesis 0 comma 0 right parenthesis, and left parenthesis 0.5 comma 1 right parenthesis.
Determine the initial value and the rate of change of the linear function as given in the graph.

Answer 15

To determine the initial value and the rate of change of the linear function based on the points provided, we'll identify the coordinates, calculate the slope (rate of change), and identify the y-intercept (initial value).

Step 1: Identify the Points

The points given are:

• \( (-1.5, -3) \)
• \( (0, 0) \)
• \( (0.5, 1) \)

Step 2: Determine the Rate of Change (Slope)

We can use any two points to calculate the slope. Let's use the points \( (0, 0) \) and \( (0.5, 1) \).

Using the slope formula:

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

Taking the points:

• \( (x_1, y_1) = (0, 0) \)
• \( (x_2, y_2) = (0.5, 1) \)

Substituting into the formula provides:

\[ m = \frac{1 - 0}{0.5 - 0} = \frac{1}{0.5} = 2 \]

Thus, the rate of change (slope) is 2.

Step 3: Determine the Initial Value (Y-Intercept)

The initial value (or y-intercept \( b \)) is the value of \( y \) when \( x = 0 \). From the point \( (0, 0) \), we can see that when \( x = 0 \), \( y = 0 \). Therefore, the initial value \( b = 0 \).

Summary

• Rate of Change: \( 2 \)
• Initial Value: \( 0 \)

So, the linear function has a rate of change of 2 and an initial value of 0.