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A short class with Sarah and her students. The other ways are under More.
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Today we are diving into linear functions! A linear function can be expressed in the form y = mx + c. Does anyone remember what each term represents?
I think m is the slope!
And c is the y-intercept!
Correct! The slope determines the steepness and direction of the line, while the y-intercept is where the line crosses the y-axis. Remember: Slope is how steep, intercept is where we meet!
Now let’s delve into slope a bit more. The slope, represented by m, is calculated as the change in y over the change in x. Can anyone give me an example of positive and negative slopes?
A line that rises from left to right has a positive slope.
And a line that falls from left to right has a negative slope!
Exactly! Just remember, if you see a line going up, you have a positive slope. If it’s going down, it’s negative. Can someone summarize?
A positive slope goes up, a negative slope goes down!
Now, let’s graph a linear function! Remember these steps: identify the slope (m) and y-intercept (c), plot the y-intercept, then use the slope to find another point. Who wants to try plotting the line for y = -2x + 4?
I can! Starting at (0,4), I move down 2 and right 1 for my next point.
Great job! Now what can you tell me about where this line goes?
It’s going down, so that means it has a negative slope!
Correct! Remember, the slope dictates the direction of the line!
Linear functions are everywhere! Can anyone think of a real-life example where we can apply linear equations?
Like when we calculate costs based on distance traveled, like in a taxi fare!
Or a plant growing at a constant rate over time!
Exactly! Both examples can be modeled with linear functions, showing how mathematics applies to our everyday lives.
A fundamental function exhibited by a straight line.
Indicates steepness and direction of the line.
The point where a line crosses the y-axis.
The point where a line crosses the x-axis.
Identify slope and intercept, plot, and draw.
Relationships between lines based on their slopes.
Linearity and slope, we can cope, intercepts found help us elope!
Imagine climbing a mountain; each step up represents the slope and where you start is the intercept. Together, they guide your journey!
To remember slope, think 'Rise over Run' – find change in y and then x!
S.I.G. = Slope, Intercept, Graph - the three key concepts to remember!
Term
What is a linear function?
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Term
What does the slope indicate?
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How do you find the y-intercept?
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What does a negative slope look like?
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A function whose graph is a straight line, typically represented in the form y = mx + c.
A measure of the steepness of a line, defined as the change in y over the change in x.
The point where the line crosses the y-axis, indicated by the coefficient c in the equation.
The point where the line crosses the x-axis, calculated by setting y = 0.
Lines that intersect at a right angle, characterized by slopes that are negative reciprocals.
Lines that run alongside each other and never intersect, having identical slopes.
A way of writing linear equations as Ax + By = C, where A, B, and C are constants.
This section introduces linear functions and their graphical representation including slope, y-intercept, and methods for graphing such functions.
Linear functions are foundational in algebra, characterized by their straight-line graphs. The section covers the mathematical representation, importance of slope and intercepts, along with practical steps for graphing and applications in real-world scenarios.
Linear functions are vital in algebra, represented by equations of the form y = mx + c, where m is the slope and c is the y-intercept. This section explains how to graph these functions including understanding the characteristics of slope (whether it is positive, negative, zero, or undefined), and the significance of intercepts. We will also discuss the various forms of linear equations, the relationship between parallel and perpendicular lines, and how to apply these concepts to real-life situations.
Slope as a ratio: Slope tells us how steep a line is. It’s the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. Formula:
That’s why slope is a ratio — it compares how much the line goes up or down for every step it goes across.
Zero slope: A horizontal line has no rise (numerator = 0), so slope = 0. Example: line through (0,2) and (3,2).
Undefined slope: A vertical line has no run (denominator = 0), so slope is undefined (you can’t divide by 0). Example: line through (4,1) and (4,5).
👉 In short: