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### Course: Digital SAT Math>Unit 2

Lesson 4: Graphs of linear equations and functions: foundations

# Graphs of linear equations and functions | Lesson

A guide to graphs of linear equations and functions on the digital SAT

## What are graphs of linear equations and functions questions?

Graphs of linear equations and functions questions deal with linear equations and functions and their graphs in the $xy$-plane. For example, the graph of $y=2x-1$ is shown below.
An equation in function notation, $f\left(x\right)=2x-1$, can also represent this line.
In this lesson, we'll learn to:
1. Identify features of linear graphs from their equations
2. Write linear equations based on graphical features
3. Determine the equations of parallel and perpendicular lines
You can learn anything. Let's do this!

## What are the features of lines in the $xy$‍ -plane?

### Intro to slope

Intro to slopeSee video transcript

### Features of lines in the $xy$‍ -plane

#### The slope

The slope of a line describes its direction and steepness.
• A line that trends upward from left to right has a positive slope.
• A line that trends downward from left to right has a negative slope.
• The steeper the line is, the larger the
of its slope is.
The slope is equal to the ratio of a line's change in $y$-value to its change in $x$-value. We can calculate the slope using any two points on the line, $\left({x}_{1},{y}_{1}\right)$ and $\left({x}_{2},{y}_{2}\right)$:
Example: Line $\ell$ contains the points $\left(-1,2\right)$ and $\left(4,12\right)$. What is the slope of line $\ell$ ?
A horizontal line has a slope of $0$ since all points on the line have the same $y$-coordinate (so the change in $y$ is $0$).
A vertical line has an undefined slope since all points on the line have the same $x$-coordinate (so the change in $x$ is $0$).

#### The $y$‍ -intercept

The $y$-intercept of the line is the point where the line crosses the $y$-axis. This point always has an $x$-coordinate of $0$. All non-vertical lines have exactly one $y$-intercept.

#### The $x$‍ -intercept

The $x$-intercept of the line is the point where the line crosses the $x$-axis. This point always has a $y$-coordinate of $0$. All non-horizontal lines have exactly one $x$-intercept.

### Try it!

Try: identify the features of a graph
The graph of line $m$ is shown above.
The $y$-intercept of the line is located at $\left(0,\phantom{\rule{0.167em}{0ex}}$
$\right)$.
The $x$-intercept of the line is located at $\left($
$,0\right)$.
We can use the intercepts to calculate the slope of the line. The slope of the line is
.

## How do I tell the features of lines from linear equations?

### Converting to slope-intercept form

Converting to slope-intercept formSee video transcript

### How do I interpret an equation in slope-intercept form?

Lines in the $xy$-plane are visual representations of linear equations. The slope-intercept form of a linear equation, $y=mx+b$, tells us both the slope and the $y$-intercept of the line:
• The slope is equal to $m$.
• The $y$-intercept is equal to $b$.
For example, the graph of $y=3x-7$ has a slope of $3$ and a $y$-intercept of $-7$.
Because the slope-intercept form shows us the features of the line outright, it's useful to rewrite any linear equation representing a line in slope-intercept form.
Example: What is the slope of the graph of $3x+4y=12$ ?

### Try it!

Try: identify slope and intercept from a linear equation
$2x+y=3$
To rewrite the above equation in slope-intercept form, we can isolate $y$ by
both sides of the equation.
When $2x+y=3$ is graphed in the $xy$-plane:
• The slope of the line is
.
• The $y$-intercept of the line is
.

## How do I write linear equations based on slopes and points?

### Slope-intercept equation from two points

Slope-intercept equation from two pointsSee video transcript

### What information do I need to write a linear equation?

We can write the equation of a line as long as we know either of the following:
• The slope of the line and a point on the line
• Two points on the line
In both cases, we'll be using the information provided to find the missing values in $y=mx+b$.

#### The slope and a point

When we're given the slope and a point, we have values for $x$, $y$, and $m$ in the equation $y=mx+b$, and we just need to plug in the values and solve for the $y$-intercept $b$.
Example: If line $a$ has a slope of $2$ and passes through the point $\left(1,3\right)$, what is the equation of line $a$ ?
Note: if the given point is the $y$-intercept, then we just need to plug in the slope for $m$ and the $y$-intercept for $b$. No calculation needed!

#### Two points

When we're given two points, we must first calculate the slope using the two points, then plug in the values of $x$, $y$, and $m$ into $y=mx+b$ to find $b$.
Example: Line $b$ passes through the points $\left(-2,4\right)$ and $\left(1,-5\right)$. What is the equation of line $b$ ?

### Try it!

TRY: WRITE the equation of a line
The line shown above passes through the points $\left(0,\phantom{\rule{0.167em}{0ex}}$
$\right)$ and $\left(2,2\right)$.
The slope of the line is
.
The equation of the line in slope-intercept form is:
$y=\phantom{\rule{0.167em}{0ex}}$

## How do I write equations of parallel and perpendicular lines?

### Parallel & perpendicular lines from graph

Parallel & perpendicular lines from graphSee video transcript

### What are the features of parallel and perpendicular lines?

In the $xy$-plane, lines with different slopes will intersect exactly once.
Parallel lines in the $xy$-plane have the same slope. Parallel lines do not intersect unless they also completely overlap (i.e., are the same line).
Perpendicular lines in the $xy$-plane have slopes that are
of each other. Perpendicular lines form ${90}^{\circ }$ angles.
The graph below shows lines $\ell$, $m$, and $n$.
• Line $\ell$ has a slope of $2$.
• Line $m$ also has a slope of $2$. It is parallel to line $\ell$.
• Line $n$ has a slope of $-\frac{1}{2}$. It is perpendicular to both lines $\ell$ and $m$.
This means we can write the equation of a parallel or perpendicular line based on a slope relationship and a point on the line.

#### Let's look at some examples!

Lines $p$ and $q$ are graphed in the $xy$-plane. Line $p$ is represented by the equation $y=-2x-5$. If line $q$ is parallel to line $p$ and passes through the point $\left(0,4\right)$, what is the equation of line $q$ ?
Line $\ell$ is represented by the equation $y=-3x+2$. What is the equation of a line that is perpendicular to line $\ell$ and intersects line $\ell$ at $\left(-3,11\right)$ ?

### Try it!

TRY: identify the features of parallel and perpendicular lines
The line shown in the graph above has a slope of
and a $y$-intercept of
.
The slope of a line parallel to the line above must have a slope of
.
The slope of a line perpendicular to the line above must have a slope of
.
The slope-intercept form equation of a perpendicular line that intersects the line shown above at $\left(0,3\right)$ is:
$y=\phantom{\rule{0.167em}{0ex}}$

Practice: match an equation and its graph
Which of the following is the graph of the equation $y=-\frac{1}{2}x+3$ in the $xy$-plane?

Practice: write an equation based two points
Line $\ell$ in the $xy$-plane passes through the points $\left(-1,-2\right)$ and $\left(2,7\right)$. Which of the following equations describes line $\ell$ ?

Practice: find a point on a perpendicular line
In the $xy$-plane, line $\ell$ has a $y$-intercept of $-7$ and is perpendicular to the line with equation $y=-\frac{1}{4}x$. If the point $\left(5,c\right)$ is on line $\ell$, what is the value of $c$ ?

## Things to remember

The slope-intercept form of a linear equation, $y=mx+b$, tells us both the slope and the $y$-intercept of the line:
• The slope is equal to $m$.
• The $y$-intercept is equal to $b$.
We can write the equation of a line as long as we know either of the following:
• The slope of the line and a point on the line
• Two points on the line
Parallel lines in the $xy$-plane have the same slope.
Perpendicular lines in the $xy$-plane have slopes that are negative reciprocals of each other.

## Want to join the conversation?

• By the way, when solving the first equation in the video, Sal actually misuses the formula for a slope; instead of substituting y2-y1/x2-x1, he performs y1-y2/x1-x2. Please see to it that this is resolved for it could be misleading.
• It doesn't matter. The two versions are interchangeable, as long as you don't confuse the order(e.g. y1-y2/x2-x1 is wrong).
• i am so confused
• u ain't making it out
• When two lines' slope are equal, it's said parallel. Then, Can it is said perpendicular too?
• No. Parallel and perpendicular are two different things. And two lines cannot be both at the same time, either.
Parallel means that they have the same slope so they never intersect.
Perpendicular means that they intersect at a right angle, meaning that their slopes are vastly different(specifically, are negative reciprocals of each other).
Therefore, two lines can never be both parallel and perpendicular at the same time.

You seem to be Korean, so I'll just add a few translations below.

Parallel -> 평행(기울기가 같고 교점이 없다.)
Perpendicular -> 수직(교점이 1개 있고 기울기가 서로 곱하면 -1이 됨, 즉 역수*-1)

결론
평행이면서 동시에 수직인 두 선은 존재할 수 없음.
• I certainly didn't understand a thing he said
• me neither
• we can do this fellow test takers
• in video should not it be -3+3/-6+3 i mean y2-y1/x2-x1
• You will get the same answer no matter how you put it. (y1 - y2)/(x1 - x2) works fine too :)
• Isn't the formula for slope y2-y1/x2-x1? At the variable positions in the formula are switched.
• Yes, it is.
• Please i have a question from the parallel and pependicular line video.we were taught that slope is y2-y1 why was it done in y1-y2
• y2-y1 over x2-x1 gives the same answer as y1-y2 over x1-x2, try it with a calculator.
• Can someone please explain how to know when finding m wether to do y2 -y1 or the opposite because he keeps switching between them
• To find slope, m, the formula is :(y2 - y1)/(x2-x1). Or you can do (y1 - y2)/(x1 - x2)
• im making dumb mistakes like not seeing negative sign or something. its frustrating. dunno what to do.