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Course: Digital SAT Math > Unit 6
Lesson 4: Graphs of linear equations and functions: mediumGraphs 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 -plane. For example, the graph of is shown below.
An equation in function notation, , can also represent this line.
In this lesson, we'll learn to:
- Identify features of linear graphs from their equations
- Write linear equations based on graphical features
- Determine the equations of parallel and perpendicular lines
You can learn anything. Let's do this!
What are the features of lines in the -plane?
Intro to slope
Features of lines in the -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 -value to its change in -value. We can calculate the slope using any two points on the line, and :
Example: Line contains the points and . What is the slope of line ?
A horizontal line has a slope of since all points on the line have the same -coordinate (so the change in is ).
A vertical line has an undefined slope since all points on the line have the same -coordinate (so the change in is ).
The -intercept
The -intercept of the line is the point where the line crosses the -axis. This point always has an -coordinate of . All non-vertical lines have exactly one -intercept.
The -intercept
The -intercept of the line is the point where the line crosses the -axis. This point always has a -coordinate of . All non-horizontal lines have exactly one -intercept.
Try it!
How do I tell the features of lines from linear equations?
Converting to slope-intercept form
How do I interpret an equation in slope-intercept form?
Lines in the -plane are visual representations of linear equations. The slope-intercept form of a linear equation, , tells us both the slope and the -intercept of the line:
- The slope is equal to
. - The
-intercept is equal to .
For example, the graph of has a slope of and a -intercept of .
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 ?
Try it!
How do I write linear equations based on slopes and points?
Slope-intercept equation from two points
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 .
The slope and a point
When we're given the slope and a point, we have values for , , and in the equation , and we just need to plug in the values and solve for the -intercept .
Example: If line has a slope of and passes through the point , what is the equation of line ?
Note: if the given point is the -intercept, then we just need to plug in the slope for and the -intercept for . 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 , , and into to find .
Example: Line passes through the points and . What is the equation of line ?
Try it!
How do I write equations of parallel and perpendicular lines?
Parallel & perpendicular lines from graph
What are the features of parallel and perpendicular lines?
In the -plane, lines with different slopes will intersect exactly once.
Parallel lines in the -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 -plane have slopes that are of each other. Perpendicular lines form angles.
The graph below shows lines , , and .
- Line
has a slope of . - Line
also has a slope of . It is parallel to line . - Line
has a slope of . It is perpendicular to both lines and .
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 and are graphed in the -plane. Line is represented by the equation . If line is parallel to line and passes through the point , what is the equation of line ?
Line is represented by the equation . What is the equation of a line that is perpendicular to line and intersects line at ?
Try it!
Your turn!
Things to remember
The slope-intercept form of a linear equation, , tells us both the slope and the -intercept of the line:
- The slope is equal to
. - The
-intercept is equal to .
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 -plane have the same slope.
Perpendicular lines in the -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.(16 votes)
- 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).(72 votes)
- i am so confused(28 votes)
- u ain't making it out(7 votes)
- When two lines' slope are equal, it's said parallel. Then, Can it is said perpendicular too?(1 vote)
- 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.(36 votes)
- I certainly didn't understand a thing he said(15 votes)
- in video6:13should not it be -3+3/-6+3 i mean y2-y1/x2-x1(5 votes)
- You will get the same answer no matter how you put it. (y1 - y2)/(x1 - x2) works fine too :)(12 votes)
- Isn't the formula for slope y2-y1/x2-x1? At6:30the variable positions in the formula are switched.(8 votes)
- 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(4 votes)
- y2-y1 over x2-x1 gives the same answer as y1-y2 over x1-x2, try it with a calculator.(10 votes)
- we can do this fellow test takers(7 votes)
- Can someone please explain how to know when finding m wether to do y2 -y1 or the opposite because he keeps switching between them(4 votes)
- To find slope, m, the formula is :(y2 - y1)/(x2-x1). Or you can do (y1 - y2)/(x1 - x2)(5 votes)
- im making dumb mistakes like not seeing negative sign or something. its frustrating. dunno what to do.(6 votes)
- a small tip. When you're dealing with problems like:
-1=2+x
always rember to put the -2 on the right side of the equation like:
-1-2=2+x-2
-3=x
otherwise there could be a mistake:
2- -1=x
2+1=x
3=x(2 votes)