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I cannot dicepher the difference between a quadratic equation and a quadratic function. I read the following "A quadratic equation can tell us a lot about the graph of a quadratic function." I see the following equation:

f(x) = 10x^2 - 8x

That to me is a quadratic equation, because the x term is squared. And the x squared is the highest power on x. This quadratic equation can be broken down into a linear equation by factoring.

How is this different from a quadratic function?

JohnMerlino
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  • A function is a triplet $(f, A, B)$, where $f\subseteq A\times B$ satisfies certain properties, does this help? – user2345215 Nov 08 '14 at 17:20
  • @user2345215 No, can you show a quadratic equation and then a quadratic function and show how they are different – JohnMerlino Nov 08 '14 at 17:21
  • Wait until you get to the quadratic formula, quadratic forms, quadratic residues, quadratic integrals, quadratic means, the space of quadratic polynomials, and quadratic fields. There are a lot of different things with degree $2$. :) –  Nov 08 '14 at 18:22

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My explanation is that a quadratic equation is a set of terms of the form (in general): $ax^2+bx+c=0$. A quadratic function is one where the right-hand constant (call it $f$) is allowed to vary with $x$, thus giving: $f(x)=ax^2+bc+c$.

FundThmCalculus
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$$y=f(x)=10x^2-8x$$

is a quadratic function: the set of all points in the plane of the form $\;\left(x\,,\,10x^2-8x\right)\;$

A quadratic equation "asks" for what value(s) of $\;x\;$ it equals some definite values, for example $\;10x^2-8x=0\;,\;\;10x^2-8x=16\;$ are quadratic equations

Timbuc
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A quadratic equation is made for the purpose of solving for a specific variable and so it will the equation will always be equal to a number. For example: 0 = 10x(squared) + 4

A quadratic function is made for the purpose of graphing and so it will either be set to be equal to f(x) or y.

For example: f(x) = 10x(squared) + 4x

Another example: y = 10x(squared) + 4x

Also, both a quadratic function and a quadratic equation can have x to the second power.

So lastly, I think the difference between a function and an equation lies in what it has been set equal to, and in the purpose (whether it be to solve, or to graph).

Harry McKenzie
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Amayah
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    "In addition, the equation does help you graph because after you solve for all of the variables, you can put them into the function and then graph it." This part is incorrect. Solving the equation $x^2 - 2x = 0$ gives $x = 0$ and $x = 2$. There is no function into which these two values are to be substituted, and no graph to construct. – Michael Albanese Jan 04 '16 at 00:38
  • thanks for your comment! that part confused me coz it seemed weird. glad you corrected this :) – Harry McKenzie Aug 13 '22 at 11:45