The Table Represents A Linear Function. The Rate Of Change Between The Points ( − 5 , 10 (-5, 10 ( − 5 , 10 ] And ( − 4 , 5 (-4, 5 ( − 4 , 5 ] Is − 5 -5 − 5 . What Is The Rate Of Change Between The Points ( − 3 , 0 (-3, 0 ( − 3 , 0 ] And $(-2,

Introduction

In mathematics, a linear function is a type of function that can be represented by a straight line on a graph. The table represents a linear function, and we are given two points on the line: $(-5, 10)$ and $(-4, 5)$. The rate of change between these two points is given as $-5$. In this article, we will explore the concept of rate of change and how it applies to linear functions. We will also use the given information to find the rate of change between the points $(-3, 0)$ and $(-2, y)$.

What is Rate of Change?

Rate of change is a measure of how much a function changes as its input changes. It is a fundamental concept in calculus and is used to describe the behavior of functions. In the context of linear functions, the rate of change is the slope of the line. The slope is a measure of how steep the line is and can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Calculating Rate of Change

Using the given points $(-5, 10)$ and $(-4, 5)$, we can calculate the rate of change as follows:

$$m = \frac{5 - 10}{-4 - (-5)} = \frac{-5}{1} = -5$$

This confirms that the rate of change between the points $(-5, 10)$ and $(-4, 5)$ is indeed $-5$.

Finding the Rate of Change between Two Points

Now that we have confirmed the rate of change between the points $(-5, 10)$ and $(-4, 5)$, we can use this information to find the rate of change between the points $(-3, 0)$ and $(-2, y)$. To do this, we need to find the slope of the line using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

We know that the rate of change between the points $(-5, 10)$ and $(-4, 5)$ is $-5$, so we can use this information to find the slope of the line.

Using the Rate of Change to Find the Slope

Since the rate of change between the points $(-5, 10)$ and $(-4, 5)$ is $-5$, we can write the equation of the line as:

$$y - 10 = -5(x + 5)$$

Simplifying the equation, we get:

$$y = -5x - 25 + 10$$

$$y = -5x - 15$$

This is the equation of the line in slope-intercept form, where the slope is $-5$.

Finding the Rate of Change between the Points $(-3, 0)$ and $(-2, y)$

Now that we have the equation of the line, we can use it to find the rate of change between the points $(-3, 0)$ and $(-2, y)$. To do this, we need to find the slope of the line using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

We know that the point $(-3, 0)$ is on the line, so we can substitute this point into the equation of the line to get:

$$0 = -5(-3) - 15$$

Simplifying the equation, we get:

$$0 = 15 - 15$$

$$0 = 0$$

This confirms that the point $(-3, 0)$ is on the line.

Finding the Value of $y$

Now that we have confirmed that the point $(-3, 0)$ is on the line, we can use the equation of the line to find the value of $y$ at the point $(-2, y)$. To do this, we need to substitute $x = -2$ into the equation of the line:

$$y = -5(-2) - 15$$

Simplifying the equation, we get:

$$y = 10 - 15$$

$$y = -5$$

This confirms that the value of $y$ at the point $(-2, y)$ is $-5$.

Conclusion

In this article, we explored the concept of rate of change and how it applies to linear functions. We used the given information to find the rate of change between the points $(-5, 10)$ and $(-4, 5)$, and then used this information to find the rate of change between the points $(-3, 0)$ and $(-2, y)$. We confirmed that the rate of change between the points $(-3, 0)$ and $(-2, y)$ is indeed $-5$.

Introduction

In our previous article, we explored the concept of rate of change and how it applies to linear functions. We used the given information to find the rate of change between the points $(-5, 10)$ and $(-4, 5)$, and then used this information to find the rate of change between the points $(-3, 0)$ and $(-2, y)$. In this article, we will answer some frequently asked questions related to the concept of rate of change and linear functions.

Q&A

Q: What is the rate of change between two points on a linear function?

A: The rate of change between two points on a linear function is the slope of the line. It can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

where $m$ is the slope, and $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

Q: How do I find the rate of change between two points on a linear function?

A: To find the rate of change between two points on a linear function, you need to use the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

You can substitute the coordinates of the two points into the formula to find the rate of change.

Q: What is the equation of a linear function in slope-intercept form?

A: The equation of a linear function in slope-intercept form is:

$$y = mx + b$$

where $m$ is the slope, and $b$ is the y-intercept.

Q: How do I find the y-intercept of a linear function?

A: To find the y-intercept of a linear function, you need to substitute $x = 0$ into the equation of the line. This will give you the value of $y$ when $x = 0$, which is the y-intercept.

Q: What is the significance of the rate of change in a linear function?

A: The rate of change in a linear function is a measure of how much the function changes as its input changes. It is a fundamental concept in calculus and is used to describe the behavior of functions.

Q: Can the rate of change be negative?

A: Yes, the rate of change can be negative. This means that the function is decreasing as its input increases.

Q: Can the rate of change be zero?

A: Yes, the rate of change can be zero. This means that the function is not changing as its input changes.

Q: How do I find the rate of change between two points on a linear function when the points are not given?

A: To find the rate of change between two points on a linear function when the points are not given, you need to use the equation of the line. You can substitute the coordinates of one point into the equation of the line to find the slope, and then use the slope to find the rate of change between the two points.

Conclusion

In this article, we answered some frequently asked questions related to the concept of rate of change and linear functions. We hope that this article has provided you with a better understanding of the concept of rate of change and how it applies to linear functions.

Additional Resources

If you are interested in learning more about the concept of rate of change and linear functions, we recommend the following resources:

• Mathway: A math problem solver that can help you solve math problems and understand the concept of rate of change and linear functions.
• Khan Academy: A free online learning platform that provides video lessons and practice exercises on math and other subjects.
• Wolfram Alpha: A computational knowledge engine that can help you solve math problems and understand the concept of rate of change and linear functions.

We hope that this article has been helpful in your understanding of the concept of rate of change and linear functions. If you have any further questions or need additional help, please don't hesitate to ask.