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Given the parabola $y=a x^{2}+b x+c$ (a) determine the relationship between $a, b$ and $\bar{c}$ if the parabola is to have no real $x$ -intercepts. (b) When will its graph lie above the $x$ -axis? (When will it lie below the $x$ -axis?

(a) $b^{2}-4 a c < 0$ (b) $b^{2}-4 a c < 0$ and $a > 0$ (c) $b^{2}-4 a c < 0$ and $a < 0$

Algebra

Chapter 1

Functions and their Applications

Section 4

Quadratic Functions - Parabolas

Functions

Campbell University

Baylor University

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Parabolas\begin{equati…

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Parabolasa. Find the c…

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a. Find the coordinates of…

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The $x$ -coordinate of the…

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The $y$ -coordinate of the…

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(a) graph each quadratic f…

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What is the $y$ -intercept…

03:59

All right. So the key to not having any X intercepts is is two different things. Um, but the main main thing is with the quadratic formula, because inside the quadratic formula is, uh, well, just write out negative B plus or minus the square root of B squared minus four a c all over to a, uh and so inside the quadratic formula is I'll do this right. This piece right here is the square root. And pretty much what it's saying is, uh if this is, if we're square rooting and negative only put that way if we square root of negative, we get imaginary roots. Those are non real. Uh, and so what it boils down to is what's under the radical B squared minus four. A. C must be less than zero in order to have no real X intercepts. So I'll circle that, and hopefully that made perfect sense. So then, in the second part, where they're asking for, when is it above the X axis? Well, what makes something above the X axis? Well, if we haven't, if we are above the X axis but has no X intercepts, then we need to have an upwards, perhaps. So what makes something have an upwards? Palavela, as if the A value is positive. Um, and we also want to have it satisfy that B squared minus four A. C is negative as well. And then the last thing is, Well, what if it's below the X axis? Well, if it's below the X axis, we need to have a downward problem. What makes that a downwards problem is if that a value is negative, as well as this statement has to be true. There you go. Yeah.

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