How to Graph Lines in Slope Intercept Form (y=mx+b)

Summary

This tutorial offers an insightful look into graphing linear functions with a focus on the slope-intercept form. The instructor clarifies that 'f(x)' and 'y' represent the same function output, simplifying the process. By identifying the slope as -3/4 and the y-intercept at +2, the lesson demonstrates how to plot the intercept and use the slope to create a descending line from left to right. The methodical approach of plotting points, moving down 3 units and to the right 4 units, builds a consistent pattern, ultimately resulting in a complete graph of the function y = -3/4x + 2. The video concludes with a reminder that 'y =' and 'f(x) =' are interchangeable, emphasizing the simplicity of graphing linear functions.

Takeaways

• 📚 The lesson focuses on graphing a linear function, specifically in slope-intercept form (y = mx + b).
• 🔄 It's important to understand that 'f(x)' and 'y =' represent the same concept, the output of a function.
• ✍️ The function is rewritten in terms of 'y =' to simplify the process of graphing.
• 📉 The script explains that a negative slope (-3/4) indicates a line that descends from left to right.
• 📍 The y-intercept (b) is the point where the graph crosses the y-axis, in this case, at positive 2.
• 📈 The first step in graphing is to plot the y-intercept on the y-axis.
• 🔢 The slope is interpreted as 'rise over run', which helps in plotting additional points on the graph.
• 📊 Plotting points involves moving down 3 units and to the right 4 units from the y-intercept to create a 'staircase' effect.
• 🔄 To graph on the left side of the y-intercept, the process is reversed: moving left 4 units and up 3 units to plot a point.
• 📝 The script emphasizes that all plotted points should align to form a consistent line, confirming the correct application of the slope.
• 🎓 The lesson concludes by reiterating that the graphed function y = -3/4x + 2 is equivalent to f(x) = -3/4x + 2, and encourages subscribing to the channel for more lessons.

Q & A

What is the main topic of this lesson?

The main topic of this lesson is graphing a linear function.

Why does the instructor suggest rewriting the function in terms of 'y equals'?

The instructor suggests rewriting the function in terms of 'y equals' to make it easier to work with when graphing.

What form of the equation is the instructor referring to when they mention 'MX plus B form'?

The 'MX plus B form' refers to the slope-intercept form of a linear equation, where M represents the slope and B represents the y-intercept.

What does a negative slope indicate about the graph of a linear function?

A negative slope indicates that the line will be descending from left to right on the graph.

How does the instructor determine the y-intercept from the given equation?

The instructor determines the y-intercept by identifying the B value in the slope-intercept form of the equation, which is positive 2 in this case.

What is the first step in graphing a linear function according to the instructor?

The first step is to plot the y-intercept on the y-axis.

What does the instructor mean by 'rise over run' in the context of graphing a line?

'Rise over run' refers to the concept of slope, where 'rise' is the change in y (vertical change) and 'run' is the change in x (horizontal change).

How does the instructor apply the slope to find additional points on the graph?

The instructor applies the slope by moving down 3 units on the y-axis (the rise) and then moving to the right 4 units on the x-axis (the run), plotting a new point with each repetition.

What does the instructor suggest doing if you want to plot points on the left side of the y-intercept?

The instructor suggests reversing the process by moving to the left 4 units and then up 3 units to plot points on the left side of the y-intercept.

How does the instructor confirm that the points align correctly on the graph?

The instructor confirms alignment by seeing that all plotted points form a consistent 'staircase' pattern, indicating that they can construct a line that passes through all points.

What is the final step in graphing the linear function according to the lesson?

The final step is to construct the line that passes through all the plotted points, completing the graph of the linear function.

Why does the instructor emphasize that 'y equals' and 'f of x equals' mean the same thing?

The instructor emphasizes this to clarify that both expressions represent the output of the function and can be used interchangeably when graphing.

How can viewers stay updated with new lessons from the instructor?

Viewers can subscribe to the instructor's YouTube channel, where new lessons are added every week.

What does the instructor promise regarding viewer comments on the YouTube channel?

The instructor promises to respond to every single comment, including the mean ones, but encourages keeping the discussion nice.

Outlines

00:00

📚 Introduction to Graphing Linear Functions

This paragraph introduces the topic of the video, which is graphing linear functions. The instructor emphasizes the equivalence of 'f of X' and 'y equals' in representing the output of a function. The function is rewritten in the form of 'y equals' to facilitate the graphing process. The instructor then identifies the equation in the slope-intercept form, highlighting the negative slope (-3/4) and the y-intercept (+2), which are crucial for visualizing the line's trajectory and its intersection with the y-axis.

📈 Plotting the Y-Intercept and Applying the Slope

The instructor outlines the first step in graphing a linear function: plotting the y-intercept. In this case, the point (0, +2) is plotted on the y-axis. Following this, the negative slope of -3/4 is used to determine the direction of the line's descent from left to right. The concept of 'rise over run' is introduced to explain how to move along the line, with the instructor demonstrating how to plot additional points by descending 3 units and running 4 units to the right, creating a consistent pattern.

🔍 Constructing the Line and Verifying with Additional Points

After plotting the initial points using the slope, the instructor proceeds to construct the line by continuing this pattern, thus forming a 'staircase' effect. To ensure the accuracy of the graph, additional points are plotted on the opposite side of the y-intercept by reversing the process, moving to the left 4 units and up 3 units. The alignment of these points confirms the correct application of the slope, allowing the instructor to draw the line that passes through all plotted points, completing the graph of the linear function y = -3/4x + 2.

📝 Recap and Encouragement for Further Engagement

The instructor concludes by reiterating the equivalence of the graphed function to the original function, emphasizing the importance of understanding 'y equals' and 'f of x' as representing the same concept. The video ends with an invitation for viewers to subscribe to the YouTube channel for weekly lessons and an open invitation for questions and comments, promising a response to every comment, even the less pleasant ones, with a light-hearted tone.

Keywords

💡Graphing

Graphing is the process of visually representing data or mathematical functions on a coordinate plane. In the context of the video, graphing is used to visually represent a linear function, which is a fundamental concept in mathematics. The script describes the steps to graph a linear function, emphasizing the importance of understanding the slope and y-intercept for accurate representation.

💡Linear Function

A linear function is a mathematical function that represents a straight line when graphed on a coordinate plane. The video script discusses the process of graphing a linear function, specifically one in slope-intercept form, which is a standard way to express such functions. The example given in the script is y = -3/4x + 2, illustrating the linear relationship between x and y.

💡Slope-Intercept Form

The slope-intercept form is a way of writing a linear equation in the format y = mx + b, where 'm' is the slope and 'b' is the y-intercept. The script explains that the given function is already in this form, making it easier to graph. The slope indicates the steepness of the line, while the y-intercept is the point where the line crosses the y-axis.

💡Slope

Slope is a measure of the steepness of a line, indicating the rate of change between the dependent and independent variables in a linear function. In the script, the slope of the function is -3/4, which means for every 4 units the line moves to the right, it moves down 3 units on the y-axis, creating a descending line from left to right.

💡Y-Intercept

The y-intercept is the point where a linear function crosses the y-axis on a graph. It is represented by the 'b' in the slope-intercept form of a linear equation. In the video, the y-intercept is given as +2, which means the line crosses the y-axis at the point (0, 2).

💡Rise Over Run

Rise over run is a term used to describe the slope of a line in terms of the vertical change (rise) and horizontal change (run) between two points on the line. The script uses this term to explain how to determine the slope of the line, with a negative rise indicating a downward movement on the y-axis for every positive run to the right.

💡Coordinate Plane

A coordinate plane is a two-dimensional plane that uses a grid of perpendicular and equally spaced lines called axes, typically the x-axis (horizontal) and y-axis (vertical). The script refers to the coordinate plane as the space where the linear function is graphed, with points plotted according to their x and y coordinates.

💡Plotting Points

Plotting points is the action of placing specific coordinates on a coordinate plane to represent data or the values of a function. In the script, the process of graphing involves plotting the y-intercept and additional points derived from the slope to create the line of the linear function.

💡Staircase Pattern

The staircase pattern is a visual description of how points are plotted when graphing a line with a negative slope. The script mentions this pattern to illustrate the process of plotting points by moving down three units and then to the right four units, creating a descending line that resembles a staircase.

💡Visual Representation

Visual representation refers to the use of graphical or pictorial methods to convey information or concepts. The video's main theme revolves around the visual representation of a linear function through graphing, which helps in understanding the function's behavior and characteristics.

💡YouTube Channel

A YouTube channel is a platform where content creators share videos on various topics. In the script, the speaker encourages viewers to subscribe to their YouTube channel for more lessons, indicating the channel's role in providing educational content on a regular basis.

Highlights

Introduction to the lesson on graphing a linear function.

Understanding that 'f(x)' and 'y=' represent the output of a function.

Rewriting the function in terms of 'y=' for ease of graphing.

Identifying the equation in slope-intercept form (Mx + B).

Recognizing a negative slope of -3/4 and its implications for the graph.

Understanding that the line will descend from left to right with a negative slope.

Plotting the y-intercept at positive 2 on the y-axis.

Building the line from the y-intercept point using the slope.

Using the slope as 'rise over run' to determine the direction and distance to move.

Plotting new points by applying the slope method consistently.

Constructing a staircase pattern to visualize the line's path.

Plotting points on the left side of the y-intercept using the reverse process.

Ensuring all plotted points align to confirm the accuracy of the slope application.

Constructing the final line that passes through all plotted points.

Reiterating that the graphed function y = -3/4x + 2 is equivalent to f(x) = -3/4x + 2.

Emphasizing the importance of understanding 'y=' and 'f(x)' as interchangeable.

Concluding the lesson on graphing a linear function.

Invitation to subscribe to the YouTube channel for weekly lessons.

Encouragement to comment with questions or concerns for response.

Promise to respond to every comment, maintaining a positive community.

Transcripts

[Music]

hey what's up everyone thank you again

for stopping by on this lesson it's kind

of a short one where we are going to

visually explore the concepts and

procedures behind graphing a linear

function now the first thing that we

want to do is apply our understanding

that f of X equals and y equals are the

same thing they both represent the

output of the function so we're actually

going to rewrite this function in terms

of y equals it makes it a little bit

easier to work with and now we should

see that this y equals equation is in MX

plus B form that slope-intercept form

and when it's in this form it's actually

pretty easy to graph so we can see that

our slope is going to be that negative 3

over 4 okay when we have a negative

slope we know that our line is going to

be descending from left to right so we

should have somewhat of an idea of what

our line is gonna look like when we do

graph it and the next thing that we see

is that our y-intercept that B value is

at positive 2 so we know that the graph

is going to hit that y axis at positive

2 so the first step to graphing a linear

function in slope intercept form is to

plot that y-intercept in this case I'm

putting a point at positive 2 on the y

axis and now we're ready to build that

line from that point now that slope of

negative 3 over 4 I'm going to take the

negative sign and push it up to the

value in the numerator ok so I put it

with the 3 to make it a negative 3

and if we think of slope in terms of

rise over run as in changing Y over

change in X so when our rise is negative

3 think about as rising down we're going

down 3 units on the y axis and then our

run our change in X is a positive 4 so

we move to the right 4 units and then we

plot a new point and we can continue to

build off of each point in this case our

new point by repeating that slope again

we rise down three units and then run to

the right four and again we plot another

point and if we are applying the slope

correctly we should see that we're

building a pretty consistent staircase

here and if we want to plot some points

on the other side on the left side of

that y-intercept that to just repeat

this process in Reverse in this case

going to the left 4 units and then up 3

and plotting another point and we should

still see that our points are all

aligned which means that we can

construct the line that passes through

all the points and now we have

successfully graphed our linear function

y equals negative 3 over 4x plus 2 and

just to reiterate again the graph that

we just constructed is the same thing as

the function f of x equals negative 3

over 4x plus 2 so remember that y equals

and f of X equals mean the same thing

and that's all there is to it when it

comes to graphing a linear function and

we'll catch you guys next time haha cool

all right so that's it for that lesson

hope you found it helpful and if you did

please click that link below and

subscribe to our youtube channel we add

new lessons every week we don't want you

to miss out and also if you have any

questions or concerns feel free to

comment down in the comment section

below we respond to every single comment

I promise you will respond even the mean

ones ok but let's just try to keep it

nice those ones are always a lot more

fun to read and we'll catch you guys

next time