Do Parabolas and Other Non-Linear Graphs Have Slope?

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Discussion Overview

The discussion revolves around the concept of slope in relation to parabolas and other non-linear graphs, exploring whether these graphs possess a slope and how it can be determined. The scope includes mathematical reasoning and conceptual clarification regarding derivatives and tangent lines.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant notes that slope is defined as the ratio of change in y-coordinates to change in x-coordinates, questioning how this applies to non-linear equations like parabolas.
  • Another participant explains that the slope at a specific x-value can be determined using the derivative, providing the derivative of a quadratic function as an example.
  • A later reply clarifies that the slope refers to the slope of a tangent line at a given point on the curve, indicating that it varies from point to point depending on the function.

Areas of Agreement / Disagreement

Participants generally agree that the concept of slope can be applied to non-linear graphs through the use of derivatives, but the discussion remains open regarding the interpretation and visualization of slope in these contexts.

Contextual Notes

The discussion does not resolve how to visualize the slope of a parabola or other non-linear graphs, and it relies on the understanding of derivatives without delving into specific examples beyond quadratics.

DS2C
So the slope is of course a ratio of the change in y-coordinates to the change in x-coordinates. This is easy to see with a linear equation.
I just came across a cool math simulator ( https://phet.colorado.edu/sims/equation-grapher/equation-grapher_en.html), and I left the first value (ax^2) alone and messed with the other two, which acted as a normal linear equation because the x^2 value was 0.
But when the x^2 term is a non-zero value, and you change the values of b, the graph gets weird and starts tilting. Is this still the slope changing? Does every graph of every power have slope? I'm having a hard time picturing the slope of a parabola.
 
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You can determine the slope at a specific x-value. This is possible for every polynomial and for many other functions as well. It is called the derivative of the function. The derivative of ax2+bx+c is 2ax+b. As you can see, it depends on x. As an example, the slope is zero at x=-b/(2a). At this point the parabola has its minimum or maximum.
 
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Ah ok that makes more sense. Thank you.
 
What mfb didn't mention is that the slope is that of a tangent line to function at that point. Thus varies point to point depending on the functionl
 
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