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A short class with Sarah and her students. The other ways are under More.
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Today we will explore what slope, or gradient, means in our linear functions. The formula for slope is m = (Δy) / (Δx), which is the rise over the run. Can anyone tell me what we mean by 'rise' and 'run'?
Does 'rise' mean how much the line goes up?
Exactly! And 'run' represents how far it goes horizontally. If we have two points A and B, the slope can tell us if the line moves upwards or downwards. Now, let's think of a line that moves uphill from left to right. What type of slope is that?
That would be a positive slope!
Correct! Now, what about a line that goes downwards?
That one would have a negative slope.
Great! Remember: + for rising, - for falling. Let’s summarize: positive and negative slopes indicate the direction of the line.
Now, let's dive deeper into calculating the slope. If we have point A at (1,3) and point B at (3,7), how would we find the slope?
We subtract the y-values, right? So it’s 7 - 3.
Exactly! And what do we do with the x-values?
We do 3 - 1.
Well done! So, putting that together, what's the slope?
The slope m = (7 - 3) / (3 - 1) = 4 / 2 = 2.
Fantastic! So the slope is 2, which means this line rises steeply.
Let’s categorize the slopes we discussed — can anyone give me an example of something we might call zero slope?
A flat line? Like the horizon?
Correct! A horizontal line has a zero slope. And how about undefined slope?
That’s like a vertical line, which doesn’t move left or right!
Exactly! So to summarize: Positive slope rises, negative slope falls, zero is flat, and undefined is vertical. Drawing a line at home as practice can help.
Slope goes high when the line goes spry, down means fall, zero is all!
Imagine hiking a hill: The steeper the hill, the more effort to climb, resembling a positive slope. A flat road means no effort — zero slope. A vertical cliff? That's an undefined slope!
P for Positive, N for Negative, Z for Zero, U for Undefined — remember the slopes!
SPNZU (Slope: Positive, Negative, Zero, Undefined) helps recall the types of slopes.
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How is slope calculated?
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What is a positive slope?
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What is an undefined slope?
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The measure of the steepness of a line, calculated as the ratio of the vertical change to the horizontal change between two points.
Indicates that a line rises from left to right.
Indicates that a line falls from left to right.
Describes a horizontal line where there is no vertical change.
Describes a vertical line where there is no horizontal change.
This section explains the concept of slope (gradient) in linear functions, detailing how it reflects a line's steepness and direction.
Slope (gradient) is crucial in understanding linear functions as it indicates how steep a line is and whether it rises or falls. The section defines positive, negative, zero, and undefined slopes and illustrates them through examples.
The slope, represented by the letter m, indicates how steep a line is and the direction it takes. Mathematically, it is defined as the change in y over the change in x (rise over run).
For example, given two points A(1,3) and B(3,7), we calculate the slope as follows:
Here we find that the slope m = (7 - 3) / (3 - 1) = 4/2 = 2, indicating a positive slope. Understanding slopes is essential for graphing linear functions and solving problems related to rates of change, making it foundational in algebra.