Construct and Find the Rate of Change of a Function Model: CCSS.Math.Content.8.F.B.4 - Common Core: 8th Grade Math

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Question

When Megan was younger, her parents started a saving account for her and deposited . Each month since then, her parents have deposited into the savings account.

What is the rate of change for this situation?

Answer

Remember, the rate of change is a rate that describes how one quantity changes in relation to another. If we were given a table, the rate of change would be described as followed:

$\frac{\textup {change in}\ y}{\textup {change in}\ x}$

If we were given a graph, then the rate of change would be the slope of the line in the graph. It could be calculated by using the formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

For this situation, we have an initial value and a rate of change.

The initial value is what Megan started with in her savings account:

The rate of change is how her savings account changes each month. Each month her parents add ; thus, her savings account changes by each month.

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Question

When Megan was younger, her parents started a saving account for her and deposited . Each month since then, her parents have deposited into the savings account.

What is the initial value for this situation?

Answer

The initial value is what Megan started with in her savings account:

The rate of change is how her savings account changes each month. Each month her parents add ; thus, her savings account changes by each month.

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Question

When Megan was younger, her parents started a saving account for her and deposited . Each month since then, her parents have deposited into the savings account.

What is the function that represents this situation?

Answer

A function is represented by an equation. When making our equation, we need to know the rate of change, or the slope, and the y-intercept, which is the initial value or starting point.

Remember, the rate of change is a rate that describes how one quantity changes in relation to another. If we were given a table, the rate of change would be described as followed:

$\frac{\textup {change in}\ y}{\textup {change in}\ x}$

If we were given a graph, then the rate of change would be the slope of the line in the graph. It could be calculated by using the formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

For this situation, we have an initial value and a rate of change.

The initial value is what Megan started with in her savings account:

The rate of change is how her savings account changes each month. Each month her parents add ; thus, her savings account changes by each month.

Our equation will be in slope intercept form:

We know that the slope is the rate of change, which in this case is and the y-intercept is the initial value, which in this case is We can plug in our known values to create our equation as follows:

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Question

When Tim was born he weighed $7\textup{ pounds}$ . Each month he has gained $2\textup{ pounds}$ . What is the rate of change for this situation?

Answer

Remember, the rate of change is a rate that describes how one quantity changes in relation to another. If we were given a table, the rate of change would be described as followed:

$\frac{\textup {change in}\ y}{\textup {change in}\ x}$

If we were given a graph, then the rate of change would be the slope of the line in the graph. It could be calculated by using the formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

For this situation, we have an initial value and a rate of change.

The initial value is the weight that Tim started at:

$7\textup{ pounds}$

The rate of change is how much he gained each month. Each month he gained $2\textup{ pounds}$ ; thus, his weight changes by $2\textup{ pounds}$ each month.

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Question

When Tim was born he weighed $7\textup{ pounds}$ . Each month he has gained $2\textup{ pounds}$ .

What is the initial value for this situation?

Answer

The initial value is the weight that Tim started at:

$7\textup{ pounds}$

The rate of change is how much he gained each month. Each month he gained $2\textup{ pounds}$ ; thus, his weight changes by $2\textup{ pounds}$ each month.

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Question

When Tim was born he weighed $7\textup{ pounds}$ . Each month he has gained $2\textup{ pounds}$ .

What is the function that represents this situation?

Answer

A function is represented by an equation. When making our equation, we need to know the rate of change, or the slope, and the y-intercept, which is the initial value or starting point.

Remember, the rate of change is a rate that describes how one quantity changes in relation to another. If we were given a table, the rate of change would be described as followed:

$\frac{\textup {change in}\ y}{\textup {change in}\ x}$

If we were given a graph, then the rate of change would be the slope of the line in the graph. It could be calculated by using the formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

For this situation, we have an initial value and a rate of change.

The initial value is the weight that Tim started at:

$7\textup{ pounds}$

The rate of change is how much he gained each month. Each month he gained $2\textup{ pounds}$ ; thus, his weight changes by $2\textup{ pounds}$ each month.

Our equation will be in slope intercept form:

We know that the slope is the rate of change, which in this case is $2\textup{ pounds}$ and the y-intercept is the initial value, which in this case is $2\textup{ pounds}$ We can plug in our known values to create our equation as follows:

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Question

What is the rate of change for the provided table?

Answer

Remember, the rate of change is a rate that describes how one quantity changes in relation to another. If we were given a table, the rate of change would be described as followed:

$\frac{\textup {change in}\ y}{\textup {change in}\ x}\textup{ or} \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

An input/output table displays sets of ordered pairs. The input column represents the x-values and the output column represents the y-values. We can select two sets of ordered pairs from the table to solve for the rate of change, or slope:

$\frac{12-3}{4-1}=\frac{9}{3}=3$

The the rate of change for our table is

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Question

What is the initial value for the provided table?

Answer

The initial value is the starting place, where the input, or x-value, is . This is also known as the y-intercept. In order to solve for the initial value, we first need to know the rate of change, or the slope.

An input/output table displays sets of ordered pairs. The input column represents the x-values and the output column represents the y-values. We can select two sets of ordered pairs from the table to solve for the rate of change, or slope:

$\frac{12-3}{4-1}=\frac{9}{3}=3$

The the rate of change for our table is .

Now that we have our slope, we can use our known values and solve for the y-intercept, or the initial value.

Remember, the equation of our line will be in slope-intercept form:

Plug in the slope and a set of coordinate points from the table provided in the question:

To solve for we can subtract from both sides:

$\frac{\begin{array}[b]{r}3=3+b\ -3-3\ \ \ \ \ \end{array}}{\\0=b}$

represents the y-intercept, or initial value; thus, the correct answer is .

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Question

What is the function that represents the table provided?

Answer

A function is represented by an equation. When making our equation, we need to know the rate of change, or the slope, and the y-intercept, which is the initial value or starting point.

An input/output table displays sets of ordered pairs. The input column represents the x-values and the output column represents the y-values. We can select two sets of ordered pairs from the table to solve for the rate of change, or slope:

$\frac{12-3}{4-1}=\frac{9}{3}=3$

The the rate of change for our table is .

Now that we have our slope, we can use our known values and solve for the y-intercept, or the initial value.

Remember, the equation of our line will be in slope-intercept form:

Plug in the slope and a set of coordinate points from the table provided in the question:

To solve for we can subtract from both sides:

$\frac{\begin{array}[b]{r}3=3+b\ -3-3\ \ \ \ \ \end{array}}{\\0=b}$

Now that we have the slope and the y-intercept, we can plug those values in for and $b\textup:$

$y=3x+0\textup{ or} \ y=3x$

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Question

What is the rate of change for the provided graph?

Answer

Remember, the rate of change, or slope, is a rate that describes how one quantity changes in relation to another. To find the rate of change, or slope, we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

In a graph, we need to find two sets of coordinate points in order to solve for the rate of change, or slope. Looking at our graph, let's pick the following points:

$(1,6)\textup{ and}\ (2,9)$

We can plug these points into the formula for slope, and solve for the rate of change:

$\frac{9-6}{2-1}=\frac{3}{1}=3$

The rate of change, or slope, for the provided graph is .

*Important note: Any two points selected from the graph of the line will equal a slope of . If you selected two points and did get a value of for the slope, make sure you select two points that are on the line.

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Question

What is the initial value for the provided graph?

Answer

The initial value is the starting place, where the input, or x-value, is . This is also known as the y-intercept. In order to solve for the initial value, we first need to know the rate of change, or the slope.

To find the rate of change, or slope, we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

In a graph, we need to find two sets of coordinate points in order to solve for the rate of change, or slope. Looking at our graph, let's pick the following points:

$(1,6)\textup{ and}\ (2,9)$

We can plug these points into the formula for slope, and solve for the rate of change:

$\frac{9-6}{2-1}=\frac{3}{1}=3$

The rate of change, or slope, for the provided graph is .

*Important note: Any two points selected from the graph of the line will equal a slope of . If you selected two points and did get a value of for the slope, make sure you select two points that are on the line.

Now that we have our slope, we can use a set of coordinate points from the line and solve for the y-intercept, or the initial value.

Remember, the equation of our line will be in slope-intercept form:

Plug in the slope and a set of coordinate points from the graph provided in the question:

To solve for we can subtract from both sides:

$\frac{\begin{array}[b]{r}6=3+b\ -3-3\ \ \ \ \ \end{array}}{\\3=b}$

represents the y-intercept, or initial value; thus, the correct answer is .

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Question

What is the function that represents the graph provided?

Answer

A function is represented by an equation. When making our equation, we need to know the rate of change, or the slope, and the y-intercept, which is the initial value or starting point.

Remember, the rate of change, or slope, is a rate that describes how one quantity changes in relation to another. To find the rate of change, or slope, we can use the following formula:

$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

In a graph, we need to find two sets of coordinate points in order to solve for the rate of change, or slope. Looking at our graph, let's pick the following points:

$(1,6)\textup{ and}\ (2,9)$

We can plug these points into the formula for slope, and solve for the rate of change:

$\frac{9-6}{2-1}=\frac{3}{1}=3$

The rate of change, or slope, for the provided graph is .

*Important note: Any two points selected from the graph of the line will equal a slope of . If you selected two points and did get a value of for the slope, make sure you select two points that are on the line.

Now that we have our slope, we can use our known values and solve for the y-intercept, or the initial value.

Remember, the equation of our line will be in slope-intercept form:

Plug in the slope and a set of coordinate points from the graph provided in the question:

To solve for we can subtract from both sides:

$\frac{\begin{array}[b]{r}6=3+b\ -3-3\ \ \ \ \ \end{array}}{\\3=b}$

Now that we have the slope and the y-intercept, we can plug those values in for and $b\textup:$

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