How to Find Slope
Table of Contents
Introduction
What is Slope?
Visualizing Slope
Positive, Negative, Zero &
Undefined Slopes
Finding Slope from Two Points
Plugging into the Slope Formula
Practice Finding Slope from Two
Points
Finding Slope from a Graph
Identifying the Slope on a Linear
Graph
Using Rise Over Run
Practice Finding Slope from a
Graph
Finding Slope from an Equation
Isolating the Variables
Plugging into the Formula
Practice Finding Slope from an
Equation
Common Mistakes with Slope
Checking Your Work
Real-World Uses of Slope
Conclusion
Frequently Asked Questions
How to Find Slope
What's up folks! 😃
Today we're going to dive into the world of slope. I know, I know - slope
sounds boring and math-y. But stick with me! Finding slope is actually pretty
straightforward and understanding it will make you feel like a geometry wizard.
🧙♂️
Introduction
So what even is slope? Simply
put, slope tells us how steep a line is. When we talk about slope, we're
talking about the slantiness of a line. If you think about a hill, the
steepness of that hill is its slope!
Now in math class, we use slope
to describe the steepness of straight lines. The steeper the line, the higher
the slope. Gentler lines have a lower slope. And flat horizontal lines have a
slope of 0.
Let's dig into the nitty
gritty of calculating slope. We'll cover:
- Visualizing slope
- Finding slope from two points
- Finding slope from a graph
- Finding slope from an equation
- Common mistakes
Ready to become a slope master?
Let's do this! 😎
What is Slope?
We touched on this briefly, but
let's talk more about what slope is exactly...
Slope is the measure of how
steep a line is. Specifically, it tells us how much a line rises or falls
as you move across it.
It's easiest to visualize
slope with a diagram:
As you can see, the slope here
measures the steepness of the diagonal line as you trace your finger from left
to right.
Mathematically, we calculate
slope with the following ratio:
Slope = Rise/Run
- Rise - How much the line
rises as you move across it (the vertical change)
- Run - How much you moved
across the line horizontally
By comparing the rise and run,
you get the overall steepness - aka the slope!
Visualizing Slope
Before calculating any slopes, it
helps to understand what different slopes look like visually. This builds your
intuition!
Here's an overview:
- Positive Slope - Rises
from left to right (/)
- Negative Slope - Falls
from left to right (\)
- No Slope (Zero) -
Perfectly flat horizontal line (_)
- Undefined - Perfectly
vertical line |
Study those for a bit. Can you
imagine tracing your finger across each one? Positive slopes rise up as you
move across. Negative slopes fall. Zero slope stays flat. And undefined has no
run - it's vertical.
Keep these images in mind as we
start calculating actual slope values...
Positive, Negative, Zero &
Undefined Slopes
Now that you know how to
visualize slope, let's assign some actual values.
Here are four examples with
the rise, run, and slope worked out:
Let's break this down:
1. Positive Slope - Rises
3 units while running 2 units across.
Slope = Rise (3) / Run (2) = 3/2
2. Negative Slope - Drops
5 units while running 2 across. Slope = -5/2
3. Undefined Slope - Rises
7 with no run (it's vertical) - so the slope is undefined
4. Zero Slope - Runs 3
across while rise = 0 Slope = 0/3 = 0
See how the rise and run relate
to the visual representation? Very handy!
Finding Slope from Two Points
One of the easiest ways to find
slope is using two points on a line.
For example, say you have the
points (1, 3) and (4, 6) And you need to know the slope between them.
Here are the steps:
Step 1) Identify the rise
and run between the points
- Rise is 6 - 3 = +3
- Run is 4 - 1 = +3
Step 2) Plug the
values into the slope formula:
Slope = Rise/Run
Slope = 3/3
Slope = 1
Pretty straightforward! Now you
calculate the slope with any two points on a line using this approach.
Plugging into the Slope Formula
The slope formula looks like
this:
Slope = (y2 - y1) / (x2 - x1)
Where:
- (x1, y1) is the first point
- (x2, y2) is the second point
Plugging in our previous
example:
(x1, y1) is (1, 3)
(x2, y2) is (4, 6)
Then:
Slope = (6 - 3) / (4 - 1) =
3/3 = 1
Using the slope formula works for
any two points. Just remember that order matters! Be consistent by always
subtracting the smaller x and y values from the larger x and y values.
Let's try one more...
Practice Finding Slope from Two Points
If given the points (2, 5) and
(-3, 1), find the slope:
First let's analyze the
points:
- (2, 5) is the point with larger
x and y values
- (-3, 1) has smaller x and y
values
Then plug into the formula:
Slope = (1 - 5) / (-3 - 2)
Working through...
Slope = (-4) / (-5)
Slope = 4/5 or 0.8
Boom! You just found the slope
between those points like a pro. 😎 Give yourself a fist
bump!
Finding Slope from a Graph
Another handy way to find slope
is by reading it directly from a linear graph.
Let's see how it works...
Identifying the Slope on a Linear Graph
Check out this neat trick.
Say we have a linear graph
with a nice straight diagonal line:
To quickly get the slope, draw a
right triangle extending out from any two points along the line.
Now think back to ratios! The horizontal side of this triangle is your
"run" extending out along the x-axis.
And the vertical side is your
"rise" going up along the y-axis.
So Rise over Run gives you the
slope! By comparing the lengths, this line's slope is 2/1 = 2. Slick!
Finding Slope from an Equation
What if all you have is an
equation - can you still get the slope? Absolutely!
It's as easy as isolating the
variables and plugging into our handy formula.
For example, take the
equation:
y = 3x + 2
Let's break this down
step-by-step...
Isolating the Variables
First, we isolate y and x by
getting one side equal to 0:
y = 3x + 2
y - 3x = 2 (Subtract 3x from both sides)
Plugging into the Formula
Now we can clearly see:
- Our rise is 3 (The y coefficient)
- Our run is 1 (The x term with
no coefficient is 1)
Plugging into slope:
Slope = Rise/Run
Slope = 3/1
Slope = 3
See - slope of 3 right from the
equation!
This process works no matter what
the equation values are. Now you try one!
Practice Finding Slope from an Equation
Using the same steps, find the
slope of this equation:
4y + 2x = 8
Pause here and work through this
on your own!
Welcome back! Let's walk through it:
First, isolate y and x:
4y + 2x = 8
4y = 8 - 2x
Rise (y) is 4
Run (x) is 2
Then plug into slope formula:
Slope = Rise/Run
Slope = 4/2 = 2
Great job finding that slope from
the equation!
Common Mistakes with Slope
Now that you know the basics,
be wary of these common slope slip-ups:
- Mixing up rise and run
- Forgetting the absolute value
with negative rise/run
- Messing up order of points when
solving for slope
- Reading slopes incorrectly from
graphs
Double check your work and don't
make assumptions. Take it slow until finding slopes becomes second nature!
And when in doubt, draw a
triangle and visualize the rise and run. This foolproof way can confirm slopes
visually.
Checking Your Work
Speaking of checking work,
here are handy ways to confirm your slopes:
- Graph the line and double check
slope (rise/run)
- Find a second slope from
different points on same line. Should equal the first!
- Plug slope and a point into
slope-intercept form (y=mx+b) and ensure it fits equation of line
Verifying your slopes will
prevent frustration on tricky math tests. Don't skip this step!
Real-World Uses of Slope
Now that you're a slope master,
where in the real world might you use this?
Here are just a few examples
where slope comes in handy:
- Architecture -
Calculating ramp and roof slopes
- Geography - Determining
steepness of hills and slopes
- Physics -Finding
velocity from position graphs
- Finance - Analyzing rate
of change of financial charts
And yes...even the slope of
walking upstairs uses this rise over run concept!
So don't underestimate the power
of thoroughly understanding slope. You'll impress friends and colleagues as you
analyze slopes all around you!
Conclusion
Phew - we covered a lot of ground
on the slippery slope that is slope! 😅
You started out visualizing
different slope types. Then stepped through finding slope from points, graphs
and equations. We covered common mistakes. And saw real-world applications.
Pat yourself on the back - you
now know the ins and outs of slope like a mathematical rockstar! 🤘
But don't stop here. Keep
practicing those slope skills every chance you get. Look for slopes all around
you in the real world. And your geometry intuition will become second nature in
time.
Thanks for learning with me - now
go wow some people with your newfound slope skills! 😎👏
Frequently Asked Questions
Still have some burning
questions? Here are answers to common slope inquiries:
What are the different types of slopes?
The four main types are
positive, negative, zero, and undefined slope:
- Positive - Rises from left to
right (/)
- Negative - Falls from left to
right (\)
- Zero - Perfectly horizontal
line (_)
- Undefined - Perpendicular
vertical line (|)
What does a zero slope look like graphically?
A zero slope is any horizontal
line. As you trace along the line left-to-right, the line does not rise or fall
at all, so the slope is zero.
What does the steepness of a slope depend on?
The steepness of a sloped line
depends on how quickly the line rises or falls (the rise or fall rate). The
faster it goes up or down as you move horizontally across, the steeper the
slope.
How is slope useful in real life?
Slope is used everywhere! For
example:
- Building wheelchair ramp
slopes
- Measuring the pitch of roofs
- Determining safety on steep
hills/roads
- Analyzing graphs in physics,
finance and more.
Can a positive slope ever equal a negative
slope?
Nope! Positive slope means the
line is rising from left to right. Negative is opposite - falling from left to
right. These two slopes directions can never be equal.
The values may coincide (like +2
and -2) But the directions will still differ.
What does an undefined slope mean?
An undefined slope occurs with a
vertical line that only moves up/down without any horizontal change. Since run
= 0, you get a divide by zero error - hence the slope being undefined.
How do I verify that I have the correct slope?
Check your slope with at least
one or more of these:
- Graph points and check rise/run
visually
- Calculate slope between second
set of points
- Plug slope/point into
slope-intercept form
What causes a flat slope versus steep slope?
The relative change in rise and
run determines slope steepness. Big rise change and little run gives a steeper
slope. Small rise for a big run change means flatter slope.
Horizontal flat lines have no
rise/only run - hence zero slope!
My two slope calculations for the same line
don't match. Help!
Double check calculations for
silly errors mixing up rise/run or slopes signs. Also graph points to check
visually if slope matches assumed value. Slopes should calculate identically on
same line.
If still off - cry for help from
teacher/friend to spot potential issue!
What is one easy way to estimate slope?
Rise over run! Pick any two
points on the line, draw right triangle. Compare vertical rise to horizontal
run. No calculator needed to get approximate slope.