Linear Functions

Summary

This educational video script covers fundamental concepts of linear functions. It demonstrates how to calculate the slope of a line given two points, explains the slope-intercept form of a line, and shows how to graph vertical and horizontal lines. The script also guides viewers through graphing equations using the slope-intercept method and finding x and y-intercepts. It further explains how to write equations of lines given a point and slope, and how to derive equations for lines parallel or perpendicular to a given line. Each concept is accompanied by step-by-step calculations and explanations, making it an informative resource for learners.

Takeaways

• 📏 To calculate the slope of a line given two points, use the formula \( (y_2 - y_1) / (x_2 - x_1) \).
• 📈 The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
• 📍 For graphing, a line equation \( x = a \) results in a vertical line, and \( y = b \) results in a horizontal line.
• 📉 The slope-intercept method involves using the slope and y-intercept to plot points and draw the line.
• 🔍 To find the x-intercept of a line, set \( y = 0 \) and solve for \( x \).
• 🔎 To find the y-intercept, set \( x = 0 \) and solve for \( y \).
• 🖊️ The point-slope form of a line is \( y - y_1 = m(x - x_1) \), which is useful when you know a point and the slope.
• ✂️ Parallel lines have the same slope, which can be used to find the equation of a line parallel to another.
• 🔄 Perpendicular lines have slopes that are negative reciprocals of each other.
• 📘 There are three main forms of linear equations: slope-intercept form, standard form, and point-slope form.

Q & A

What is the formula to calculate the slope of a line given two points?

The formula to calculate the slope (m) between two points (x1, y1) and (x2, y2) is m = (y2 - y1) / (x2 - x1).

How do you find the slope of the line passing through the points (2, -3) and (4, 5)?

Using the slope formula, the slope is calculated as (5 - (-3)) / (4 - 2) = 8 / 2 = 4.

What is the slope-intercept form of a linear equation and what does it represent?

The slope-intercept form of a linear equation is y = mx + b, where 'm' represents the slope and 'b' is the y-intercept.

What is the slope and y-intercept of the line y = 2x - 3?

The slope of the line y = 2x - 3 is 2, and the y-intercept is -3.

How do you graph the equation x = 2?

The equation x = 2 graphs as a vertical line at x = 2.

What type of line does the equation y = 3 represent?

The equation y = 3 represents a horizontal line at y = 3.

How do you graph the equation y = 3x - 2 using the slope-intercept method?

Start with the y-intercept at (0, -2). Using a slope of 3, move up 3 units and to the right 1 unit to get the next point (1, 1), then repeat to find (2, 4) and connect these points with a line.

How do you find the x-intercept of the equation 2x - 3y = 6?

To find the x-intercept, set y = 0 and solve for x: 2x = 6, so x = 3. The x-intercept is at the point (3, 0).

What is the y-intercept of the equation 2x - 3y = 6?

To find the y-intercept, set x = 0 and solve for y: -3y = 6, so y = -2. The y-intercept is at the point (0, -2).

How do you write the equation of a line given a point and a slope?

Use the point-slope form: y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point.

What is the equation of the line passing through (-3, 1) and (2, -4)?

First, calculate the slope: m = (-4 - 1) / (2 - (-3)) = -5 / 5 = -1. Then use the point-slope form with one of the points, for example, (-3, 1): y - 1 = -1(x - (-3)), which simplifies to y = -x - 2.

How do you determine if two lines are parallel and what does it mean for their slopes?

Two lines are parallel if they have the same slope. The slope of a line parallel to another line with slope 'm' is also 'm'.

What is the equation of a line parallel to the line 2x + 5y - 3 = 0 and passing through the point (3, -2)?

First, find the slope of the given line (2x + 5y - 3 = 0) which is -2/5. The parallel line will have the same slope. Using the point-slope form with (3, -2) and m = -2/5 gives y + 2 = -2/5(x - 3), which simplifies to y = -2/5x + 6/5.

How do you find the slope of a line perpendicular to another line given by its equation?

The slope of a line perpendicular to another line with slope 'm' is the negative reciprocal of 'm'.

What is the equation of a line perpendicular to the line 3x - 4y + 5 = 0 and passing through (-4, -3)?

First, find the slope of the given line (3x - 4y + 5 = 0) which is 3/4. The perpendicular line's slope is the negative reciprocal, -4/3. Using the point-slope form with (-4, -3) gives y + 3 = -4/3(x + 4), which simplifies to y = -4/3x - 20/3.

Outlines

00:00

📐 Calculating Slope and Graphing Linear Equations

This segment of the video explains how to calculate the slope of a line given two points and how to graph linear equations. The presenter uses the formula for slope (y2 - y1) / (x2 - x1) to find the slope between the points (2, -3) and (4, 5), resulting in a slope of 4. The video then moves on to discuss the slope-intercept form of a line (y = mx + b), identifying 'm' as the slope and 'b' as the y-intercept. An example is given where the slope is 2 and the y-intercept is -3. The presenter also covers graphing vertical and horizontal lines, as well as using the slope-intercept method to graph an equation like y = 3x - 2. The process involves identifying key points based on the slope and y-intercept, then connecting these points to form the line.

05:12

📘 Graphing Linear Equations Using Intercepts and Point-Slope Form

The second part of the video focuses on graphing linear equations using x and y-intercepts, as well as writing equations in point-slope form. The presenter demonstrates how to find the x-intercept by setting y to zero and solving for x, and similarly finds the y-intercept by setting x to zero. An example is given where the equation 2x - 3y = 6 is graphed by first finding the intercepts (3,0) and (0,-2) and then connecting these points. The video also explains how to write the equation of a line given a point and a slope using the point-slope formula (y - y1 = m(x - x1)). The process is illustrated by finding the equation of a line passing through (2,5) with a slope of 3, and then converting it to slope-intercept form.

10:14

🔍 Writing Linear Equations for Parallel and Perpendicular Lines

The final segment of the video script deals with writing equations for lines that are parallel or perpendicular to a given line. The presenter explains that parallel lines have the same slope and demonstrates how to find the equation of a line parallel to 2x + 5y - 3 by using the point (3, -2) and the known slope of 2. The equation is then written in slope-intercept form. For perpendicular lines, the presenter uses the concept of negative reciprocal slopes, showing that if a line has a slope of 3/4, a perpendicular line through the point (-4, -3) would have a slope of -4/3. The equation of the perpendicular line is then derived using the point-slope form and converted to slope-intercept form, resulting in the final equation -4/3x - 25/3.

Keywords

💡Slope

The slope of a line measures its steepness and is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. In the video, the slope is a key concept used to determine the direction and angle of a line, as seen in the example of calculating the slope between the points (2, -3) and (4, 5), where the slope is found to be 4.

💡Y-Intercept

The y-intercept is the point where a line crosses the y-axis, represented by the 'b' in the slope-intercept form of a linear equation (y = mx + b). It indicates the value of y when x is 0. The video discusses how to identify the y-intercept from the equation of a line, such as y = 2x - 3, where the y-intercept is -3, meaning the line crosses the y-axis at (0, -3).

💡Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line as y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This form is useful for quickly graphing a line and understanding its properties. The video demonstrates converting equations into this form, emphasizing its importance in visualizing and analyzing linear functions.

💡Graphing

Graphing involves plotting points on a coordinate plane to visualize the behavior of a function. In the video, various methods of graphing linear equations are shown, including using the slope-intercept method and finding intercepts. For example, graphing the equation y = 3x - 2 involves plotting points derived from the slope and y-intercept, resulting in a visual representation of the line.

💡Point-Slope Form

Point-slope form is another way of expressing the equation of a line, written as y - y1 = m(x - x1), where 'm' is the slope and (x1, y1) is a specific point on the line. This form is particularly useful when the slope and one point on the line are known. The video explains how to use this form to write the equation of a line, such as when given a point (2, 5) and a slope of 3.

💡Linear Equation

A linear equation is an algebraic equation in which the highest exponent of the variable is 1, producing a straight line when graphed. The video covers different forms of linear equations, including standard form, slope-intercept form, and point-slope form, and demonstrates how to graph and interpret them. For example, the equation 2x - 3y = 6 is a linear equation that can be graphed by finding its intercepts.

💡X-Intercept

The x-intercept is the point where a line crosses the x-axis, representing the value of x when y is 0. In the video, the x-intercept is used to graph linear equations by replacing y with 0 and solving for x. For instance, in the equation 2x - 3y = 6, setting y to 0 gives an x-intercept of 3, meaning the line crosses the x-axis at (3, 0).

💡Parallel Lines

Parallel lines are lines in the same plane that never intersect, having the same slope but different y-intercepts. The video discusses how to write the equation of a line parallel to a given line by ensuring the slopes are identical. For example, if a line has a slope of 2/5, a parallel line will also have a slope of 2/5 but with a different y-intercept.

💡Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees), and their slopes are negative reciprocals of each other. The video explains how to find the slope of a line perpendicular to a given line, and then use that slope to write the equation of the new line. For example, if a line has a slope of 3/4, the perpendicular line will have a slope of -4/3.

💡Negative Reciprocal

The negative reciprocal of a number is found by taking the reciprocal (flipping the numerator and denominator) and then changing the sign. It is used to find the slope of a line that is perpendicular to another. In the video, the concept is applied to determine the slope of a line perpendicular to one with a slope of 3/4, which would have a slope of -4/3.

Highlights

Introduction to basic questions about linear functions.

Explanation of how to calculate the slope between two points using the formula (y2 - y1) / (x2 - x1).

Demonstration of calculating the slope with given points (2, -3) and (4, 5).

Identification of the slope and y-intercept from the equation y = 2x - 3.

Description of how to graph the equations x = 2 and y = 3 to create vertical and horizontal lines.

Guidance on graphing the equation y = 3x - 2 using the slope-intercept method.

Tutorial on finding x and y-intercepts to graph the equation 2x - 3y = 6.

Instruction on writing the equation of a line given a point and a slope using the point-slope formula.

Conversion of the point-slope form to slope-intercept form for the equation passing through (2, 5) with a slope of 3.

Procedure to find the slope of a line given two points and then use it to write the equation of the line.

Method to write the equation of a line parallel to another given line using the same slope.

Explanation of how to convert an equation to slope-intercept form to find the slope of a given line.

Use of the point-slope formula to write the equation of a line parallel to a given line.

Calculation of the slope of a line perpendicular to another given line and writing its equation.

Conversion of the point-slope form to slope-intercept form for the equation of a line perpendicular to a given line.

Summary of the three forms of linear equations: slope-intercept, standard, and point-slope forms.

Transcripts

in this video i'm going to go over some

basic questions relating to linear

functions

so let's start with this problem what is

the slope of the line passing through

the points 2 comma negative 3

and four comma five

so to calculate the slope between two

points you need to use this formula it's

y two minus y one

divided by x two minus x one

the first number is x the second one is

y so let's call two negative three x one

y one and four comma five x two y two

so y two in this example is five

y one is negative three

x two is four

x 1 is 2.

now 5 minus negative 3 is the same as 5

plus 3

and 5 plus 3 is 8

4 minus 2 is 2

and 8 divided by 2 is 4.

so that's the slope that passes through

these two points

number two what is the slope and

y-intercept of the line

y equals two x minus three

now you need to know that this equation

is in slope intercept form which is

y equals mx plus b

so m represents the slope

and b is the y-intercept

so we could see that the slope is equal

to 2

and the y-intercept

is negative 3.

and so that's it for this question

now the y-intercept

you can write it as an ordered pair if

you want you could say it's

zero negative three

or simply the y-intercept is equal to

negative three

you can write it both ways

but that's it for this problem

number three

graph the equations x equals two and y

equals three

so let's start with x equals two

how can we graph this particular

equation

now whenever x is equal to a number

the type of graph you're going to get

it's going to be a line but specifically

a vertical line

at x equals 2.

now for the other one y equals 3

the graph is simply a horizontal line

where y is string

and that's it that's all you need to do

in order to graph

these two equations

number four

graph the equation

using the slope-intercept method

so we have the equation y is equal to

three x minus two

so we could see that the slope is equal

to three

and a y-intercept

is negative two i'm just going to write

the ordered pair so we have the point

zero negative two

so here's the first point at zero

negative two

now the slope is 3 which represents the

rise over the run

so we need to rise three units

and go over one unit to the right to get

the next point

so the next point is going to be at one

comma one

and then we could do it again

rise three over one

so the next point is going to be

two comma four

and then we can just connect these

points with a straight line

and so that's a rough estimate of

the graph y equals three x minus two

number five

graph the equation two x minus three y

is equal to six

using the x and y-intercepts

now to find the x-intercept replace y

with zero

negative three times zero is simply zero

and so zero is nothing so this just

disappears so 2x is equal to 6

and if we divide both sides by 2

we can see that x is equal to 3.

so that's the x-intercept which means we

have the point 3 comma 0.

now

to calculate the y-intercept we need to

replace x with zero

and solve for y

so two times zero is nothing

and so we have negative three y is equal

to six

and now let's divide both sides by

negative three so then positive six

excuse me divided by negative three

that's

negative two

and so that's the y intercept

so we have the point zero negative two

at this point we can make the graph

so the first point is three zero that's

the x-intercept the next one is zero

negative two that's the y-intercept

and then just connect these two points

with a straight line

and it appears i missed it

so there it is

so that's how you can graph a linear

equation

in standard form

there's three forms you need to be

familiar with

this is known as slope intercept form

this is called standard form

and then

this equation

is the point-slope form

number six

write the equation of the line passing

through the point two comma five with a

slope of three

so whenever you're given the point and a

slope

it's best to use the point-slope formula

to write the linear equation

so x one is two y one is five and m is

three so it's going to be y minus five

is equal to three times x minus two

so that's the linear equation in point

slope form

now let's convert it to slope intercept

form

so i'm going to distribute the three

three times x that's three x and then

it's three times negative two which is

negative six

so now let's add five to both sides

negative six plus five is negative one

so this is the linear equation in slope

intercept form

and that's the answer

number seven

write the equation of the line passing

through the points

negative three comma one and two comma

negative four

now we can't use the point slope formula

yet because

i don't have the slope

so we need to calculate the slope first

so it's y2 minus y1

divided by x2 minus x1

so this is going to be x1 and that's

going to be y1

and this is x2 and y2

so y2 is negative 4

y1 is positive 1.

x2 is 2 x one is negative three

so negative four minus one is negative

five

two minus negative three is the same as

two plus three that's five

negative five divided by five is

negative one

so that's the slope

it's equal to negative one

now we can use the point slope formula

so i'm going to use the first point

negative three comma one

so y minus y one is equal to m

times x minus x one

so y one is one

m is negative one and x one is negative

three

so this is y minus one equals to

negative x

negative one times x plus three

these two negative signs will become

positive

now let's distribute the negative one

so it's going to be negative x minus

three

and then let's add one to both sides

so y is equal to negative x and negative

three plus one is negative two

so this is the final answer it's

negative x minus two

number eight

write the equation of the line passing

through the point three negative two and

parallel to the line

two x plus five y minus three

so what you need to know is that

parallel lines

have the same slope

so if the slope here is two then the

slope of the other line will also be 2.

so we already have the point passing

through

or that's part of that line we need to

find the slope of this line which will

be the slope

of the line of the equation that we're

looking to find

so let's turn this equation

and change it into its slope intercept

form

so first i'm going to move the 3

to this side

so it's going to be positive 3 on the

right side

and then i'm going to take the 2x move

it to that side

where it's going to be negative 2x so i

have 5y is equal to negative 2x plus 3

and then divide every term

by

5. so y is equal to negative two over

five times x plus three over five

so the slope of this line is the number

in front of x when it's in slope

intercept form

so the slope is negative two over five

so now i can use the point slope formula

to write the equation of the line

so this is going to be x1

and y1

so y one

that's negative two

m is negative two over five

x one is three

so this is gonna be y plus 2

and then let's distribute the negative 2

over 5 to x minus 3.

now negative 2 over 5 times negative 3

negative 2 times negative 3 is 6.

so this is going to be 6 over 5. and now

let's subtract both sides by two

now two over one i'm going to multiply

this by five over five to get common

denominators

so negative 2 is the same as negative 10

over 5.

now 6 minus 10 is 4.

so this is going to be negative 4 over

5.

and this is the answer

so that's the equation of the line in

slope-intercept form

number nine

write the equation of the line passing

through the point negative four negative

three and perpendicular to the line

three x minus four y plus five

so first we need to find the slope of

this equation

so let's get y by itself

i'm going to move the 3x and the 5 to

the other side

so it's going to be negative 3x and

negative 5 on the right side

now i need to divide each term

by negative four

so y is equal to three over four times x

plus

five over four

whenever you divide two negative numbers

you're going to get a positive result

so the slope of this line is 3 over 4.

now let's say

if we have two lines

line k

and line out

and let's say that these two lines are

perpendicular which means that they meet

at right angles or at a 90 degree angle

now let's say that the slope of line k

is 2 over 5.

what do you think

is the slope of line l

perpendicular lines have a slope that is

the negative reciprocal of each other

so this slope is positive the other one

will be negative and then you need to

flip the fraction

so the reciprocal of two over five is

five over two

and so that tells us that these two

lines are perpendicular

so the slope for this equation is three

over four

so therefore the slope of the

perpendicular line that we want to find

is going to be the negative reciprocal

of that fraction

which is negative four divided by three

now let's finish this problem

so let's use the point slope formula

so x1 is going to be negative 4

y1 is negative 3.

so it's going to be y minus negative 3

and the slope is negative 4 over 3

and then x1 is negative 4.

so this becomes y plus 3

and that's equal to negative 4 divided

by 3 times x plus 4.

so that's the answer in point slope form

now let's distribute

the negative 4 over 3. let's convert it

to slope intercept form

so this is going to be negative 4 over 3

times x

and then negative 4 times 4

is negative 16.

now let's subtract both sides by three

now we need to get common denominators

so i'm going to multiply this 3 by

3 over 3

so it becomes negative 9 over 3.

negative 16 minus 9 is negative twenty

five

so this is the final answer

it's negative four over three x minus

twenty five over three

you