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The slope of a function y (x) is the change in y divided by the change in x: (1) m = Δy/Δx. Linear graphs are graphs of straight lines, and can be defined by their slope, m, and their y-intercept, b: (2) y (x) = m*x + b. In the figure below, there is a linear function plotted on a graph. If we want to find the slope of this function, we must ...
- Aidan Philbin
- 2020
Nov 21, 2023 · Slope is used to describe the steepness of a line. The definition of slope is the rise of a line over the run of a line, or the change in the vertical direction ( y) over the change in the ...
- How do I find the slope?You can find the slope by plugging in 2 points from a line on the graph and using the slope formula, (y2 - y1) / (x2 - x1). Then, solve for the slope.
- What does a slope represent?A slope represents the steepness of a line. A slope can be on an incline or decline or it can be flat (horizontal line). The slope of a vertical li...
- Which definition best describes the word slope?The best way to describe slope is the rise over the run of a line. As the y-axis increases or decreases, the x-axis will increase.
People also ask
What does slope mean in physics?
What is the slope of a line on a graph?
Does the slope of a position graph have to equal the velocity?
What is the slope of a position versus time graph?
That is, the line on the position vs. time graph has a slope of +5 meters/1 second for the first five seconds. Thus, the slope of the line on the graph equals the velocity of the car. During the last 5 seconds (5 to 10 seconds), the line slopes up 0 meters. That is, the slope of the line is 0 m/s - the same as the velocity during this time ...
Determining the Slope on a p-t Graph. It was learned earlier in Lesson 3 that the slope of the line on a position versus time graph is equal to the velocity of the object. If the object is moving with a velocity of +4 m/s, then the slope of the line will be +4 m/s. If the object is moving with a velocity of -8 m/s, then the slope of the line ...
- Overview
- How are position vs. time graphs useful?
- What does the vertical axis represent on a position graph?
- What does the slope represent on a position graph?
- What does the curvature on a position graph mean?
- Example 1: Hungry walrus
- Example 2: Happy bird
See what we can learn from graphs that relate position and time.
How are position vs. time graphs useful?
Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a conveniently small space.
What does the vertical axis represent on a position graph?
The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.
Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a convenient...
The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.
Concept check: What is the position of the object at time t=5 seconds according to the graph above?
The slope of a position graph represents the velocity of the object. So the value of the slope at a particular time represents the velocity of the object at that instant.
To see why, consider the slope of the position vs. time graph shown below:
[Wait, why is the vertical axis called x?]
The slope of this position graph is slope=riserun=x2−x1t2−t1 .
This expression for slope is the same as the definition of velocity: v=ΔxΔt=x2−x1t2−t1 . So the slope of a position graph has to equal the velocity.
This is also true for a position graph where the slope is changing. For the example graph of position vs. time below, the red line shows you the slope at a particular time. Try sliding the dot below horizontally to see what the slope of the graph looks like for particular moments in time.
Look at the graph below. It looks curvy since it's not just made out of straight line segments. If a position graph is curved, the slope will be changing, which also means the velocity is changing. Changing velocity implies acceleration. So, curvature in a graph means the object is accelerating, changing velocity/slope.
On the graph below, try sliding the dot horizontally to watch the slope change. The first hump between 1 s and 5 s represents negative acceleration since the slope goes from positive to negative. For the second hump between 7 s and 11 s , the acceleration is positive since the slope goes from negative to positive.
Concept check: What is the acceleration of the object at t=6 s according to the graph above?
[Show me the answer.]
To summarize, if the curvature of the position graph looks like an upside down bowl, the acceleration will be negative. If the curvature looks like a right side up bowl, the acceleration will be positive. Here's a way to remember it: if your bowl is upside down all your food will fall out and that is negative. If your bowl is right side up, all your food will stay in it and that is positive.
Finding the velocity at 2 s :
We can find the velocity of the walrus at t=2 s by finding the slope of the graph at t=2 s : slope=x2−x1t2−t1(use the formula for slope) Now we will pick two points along the line we are considering that conveniently lie at a hashmark so we can determine the value of the graph at those points. We'll choose the points (0 s,1 m) and (4 s,3 m) , but we could pick any two points between 0 s and 4 s . We must plug in the later point in time as point 2, and the earlier point in time as point 1. slope=3 m−1 m4 s−0 s(Pick two points and plug the x values into the numerator and the t values into the denominator.) slope=2 m4 s=12 m/s(Calculate and celebrate.) So, the velocity of the walrus at 2 s was 0.5 m/s .
Finding the velocity at 5 s :
To find the velocity at 5 s , we just have to note that the graph is horizontal there. Since the graph is horizontal, the slope is equal to zero, which means that the velocity of the walrus at 5 s was 0 m/s .
Finding the velocity at 8 s :
slope=x2−x1t2−t1(Use the formula for slope.) We'll pick the points at the beginning and end of the final line segment, which are (6 s,3 m) and (9 s,0 m) . slope=0 m−3 m9 s−6 s(Pick two points and plug the x values into the numerator and the t values into the denominator.) slope=−3 m3 s=−1 m/s(Calculate and celebrate.) So, the velocity of the walrus at 8 s was −1 m/s .
The motion of an extraordinarily jubilant bird flying straight up and down is given by the graph below, which shows the vertical position y as a function of time t . Answer the following questions about the motion of the bird.
What was the average velocity of the bird between t=0 s and t=10 s ?
Slope = Δ y Δ x = 2 − 5 4 − 0 = − 3 4. In other words, for every three units we move vertically down the line, we move four units horizontally to the right. A coordinate plane. The x- and y-axes each scale by one. A graph of a line intersects the points zero, five and four, two. Both of these points are plotted on the graph.
Intro to slope. Slope tells us how steep a line is. It's like measuring how quickly a hill goes up or down. We find the slope by seeing how much we go up or down (vertical change) for each step to the right (horizontal change). If a line goes up 2 steps for every 1 step to the right, its slope is 2.
- 7 min
