The Linear Function y=mx+b

🏆Practice linear function y=mx+b

The linear function y=mx+by=mx+b actually represents a graph of a straight line that has a point of intersection with the vertical Y Y axis.

m m represents the slope.
When m m is positive, the slope is positive: the line goes upwards.
When m m is negative, the slope is negative: the line goes downwards.
When m=0 m = 0 , the slope is zero: the line is parallel to the X X axis.

b b represents the point where the line intersects the Y Y axis.
If b=0 b=0 , then the line will pass through the origin of the coordinates, that is, the point (0,0) \left(0,0\right)

A - The Linear Function y=mx+b

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Test yourself on linear function y=mx+b!

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For the function in front of you, the slope is?

XY

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How do we know if a point lies on a function?

If we are given a point, we can place it into the equation of a line to see if the equation holds true.
If we are given just one part of the point: X X or Y Y , we will put the given value into the equation correctly and find the second part of the point.


How do we graph the function?

If we want a precise drawing, we'll build a table of values with 3 3 or fewer values.
We replace X X each time and obtain the value of Y Y .
We consider the slope of the function to be increasing, decreasing, or equal to 0 0 , and then we graph it.


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What do we do if the slope is undefined?

To calculate the slope, we can use a formula that finds it using two given points that the line passes through:

m=(Y2Y1)(X2X1) m=\frac{\left(Y2-Y1\right)}{(X2-X1)}


A Lesson on Linear Functions

We are given a linear function y=3x+4 y=3x+4 .

We are asked to interpret the values 3 3 and 4 4 and plot the graph of the function.

First, it appears that m=3 m=3 , meaning 3 3 represents the slope of the line (or function).

b=4 b=4 means that the line intersects the vertical axis (the y-axis) at 4 4 .

To plot the graph, all we need are 2 2 points.
We substitute values and obtain:

1.a - An exercise on the linear function

Now we will mark the two points on the coordinate system and connect them.
Looking at the graph, we can confirm that the plot intersects the y-axis at the value of 4 4 .


Examples and exercises with solutions for the linear function

Exercise #1

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

For this problem, we need to determine the nature of the slope for a given straight line on a graph.

Based on the graph provided, the red line starts at a higher point on the left (Y-axis) and moves downward toward a lower point on the right (X-axis). This indicates that as one moves from left to right across the graph, the function decreases in value. Consequently, this is typical of a line that has a negative slope.

The slope of a line is typically defined as the "rise over the run," or the ratio of the change in the vertical direction to the change in the horizontal direction. Here, as we proceed from left to right, the line goes "downwards" (negative rise), establishing a negative slope.

Thus, we can conclude that the slope of the line is negative.

Therefore, the solution to the problem is Negative slope.

Answer

Negative slope

Exercise #2

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line segment shown in the graph, follow these steps:

  • Identify the line segment on the graph; it's shown as a red line from one point to another.
  • Examine the direction the line segment travels from the leftmost point to the rightmost point.
  • Visually analyze whether the line segment is rising or falling as it moves from left to right.

Here is the detailed analysis:
- The red line segment starts lower on the left side and ends higher on the right side.
- This suggests that as we move from left to right, the line is rising.
- In terms of slope, a line that rises as it moves from left to right has a positive slope.

Therefore, the slope of the line segment is positive.

Thus, the correct answer is Positive slope.

Answer

Positive slope

Exercise #3

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line shown on the graph, we perform a visual analysis:

  • We examine the orientation of the line from left to right.
  • The red line starts at a higher point on the left and descends to a lower point on the right.
  • This indicates a downward movement, which corresponds to a negative slope.

Therefore, by observing the direction of the line, we conclude that the slope of the function is negative. This positional evaluation confirms that the correct answer is negative slope.

Answer

Negative slope

Exercise #4

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To determine the slope of the line, we'll examine the direction of the line segment on the graph:

  • The line depicted moves from the top left, passing through a point with higher y y -coordinate values, to the bottom right, ending at a point with lower y y -coordinate values.
  • This movement indicates that as x x increases (the direction to the right along the x x -axis), the y y -coordinate decreases.
  • When the y y -value reduces as the x x -value grows, the slope m m is negative.

Since the line descends from left to right, the slope of the line is negative.

Therefore, the slope of the function is a negative slope.

Answer

Negative slope

Exercise #5

For the function in front of you, the slope is?

XY

Video Solution

Step-by-Step Solution

To solve this problem, follow these steps:

  • Step 1: Observe the given graph and the plotted line.
  • Step 2: Determine the direction of the line as it moves from left to right across the graph.
  • Step 3: Understand that a line moving downwards from left to right represents a negative slope.

Now, let's work through these steps:

Step 1: The graph shows a straight line that starts higher on the left side and descends towards the right side.

Step 2: As the line moves from left to right, it descends. This is a key indicator of the slope type.

Step 3: A line that moves downward from the left side to the right side of the graph (decreasing in height as it proceeds to the right) is characteristic of a negative slope. Conversely, a positive slope would show a line ascending as it moves rightward.

Therefore, the solution to the problem is the line has a negative slope.

Answer

Negative slope

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