i am gift ask to describe in two means why is it the $y=ax^2+bx+c$ parabola opens up up if $a$ is positive and why is it the $y=ax^2+bx+c$ opens up down when $a$ is negative. One of the explanations needs to be interpreted by beginning algebra student.I to be unsure exactly how I would describe it



If $x$ is huge and positive, and also $a$ is positive, then $ax^2$ will be very large and positive, overwhelming any effect native $bx+c$.

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If $x$ is big and negative, and $a$ is positive, climate $ax^2$ will again it is in very large and positive.

So if $a$ is positive, the parabola opens up upwards.

If $a$ is an adverse then if $x$ is big (positive or negative) the contrary occurs, and $ax^2$ will certainly be very large and negative with the parabola opened downwards.


One more method:

You recognize the parabola $y = ax^2 + bx+c$ can be created as:$$y - y_0 = a(x-x_0)^2$$Since this is simply the parabola "shifted" come the axis. Now since $(x-x_0)^2$ is constantly positive, what determines what the parabola looks prefer is just $a$.


Hint: show that the parabola has a distinct minima or maxima relying on whether $a$ is hopeful or negative respectively utilizing the second derivative idea. This is the calculus way. In the algebraic way, ns guess plotting is one option.


So i am gonna try to prove native "slope" perspective. Ns hope it"s clear the if ours parabola opens upward climate slope is positive else an unfavorable for say huge positive x values.

Consider $ y = ax^2 + bx + c $ with 2 point out $(x_1,y_1)$ and also $(x_2,y_2)$. Additionally we will certainly safely i think $x_2 > x_1 > 0$ i.e. Points are on the appropriate side the Y-axis.

Now, steep = $ fracy_2 - y_1x_2-x_1 $ = $a(x_1+x_2)+b$.

It need to be clear that:

$b>0$, climate it"s quite straightforward the slope will certainly be confident if $a>0$ and hence parabola opens upward rather $a$b0$ we will gain $x_1$ and $x_2$ such that $a(x_1+x_2) > b$ and also hence slope optimistic implying parabola opened upward.

This can convince beginning algebra college student else the answer provided by Gautam Shenoy around using second-derivative is much more concrete.

answered Mar 3 "18 at 19:22
Ayush BhatnagarAyush Bhatnagar
1111 bronze argorial
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My usual approach when to teach this topic is a "transformational" approach. We very first understand basic transformations: if we recognize what the graph the $f(x)$ look at like, then

the graph the $f(x-h)$ is the very same graph, yet translated come the left through $h$ units;the graph of $f(x) + k$ is the exact same graph, yet translated up by $k$ units;for confident $A$, the graph the $A f(x)$ is the exact same graph, however scaled vertically by a element of $A$; andthe graph that $-f(x)$ is the exact same graph, reflected vertically across the $x$-axis.

We additionally discuss horizontal scaling and also reflection, however those space not pertinent here. These type of revolutions are the bread-and-butter of a start algebra curriculum, therefore I would think the they need to be pretty accessible at the level.

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Next, think around the ur-parabola, $p(x) = x^2$. The graph that this duty is a U-shaped beast which opens up up. Even better, the is possible to show that the graph the every quadratic polynomial is a revolutionized version the the ur-parabola. The is:

Proposition: If $a,b,cin smashville247.netbbR$ with $a e 0$ (that is, if $a$, $b$, and also $c$ are any kind of real numbers with $a$ nonzero), climate there are genuine numbers $A$, $h$, and $k$ such the $$ ax^2 + bx + c = A(x-h)^2 + k. $$

Proof: us will show that such actual numbers exist by clearly solving for them. Observe that$$ colorredax^2 + colorpurplebx + colorbluec= A(x-h)^2 + k= colorredAx^2 + colorpurple(-2Ah)x + colorblue(Ah^2 + k) $$if and only if the coefficients on the left and right agree. That is$$egincasescolorreda = A \colorpurpleb = -2Ah \colorbluec = Ah^2 + k \endcases$$Solving because that $A$, $h$, and also $k$, us obtain$$A = a,qquad h = -fracb2a,qquad extand\qquad k = c - fracb^24a.$$As long as $a e 0$ (which was assumed in ~ the beginning), then $A$, $h$, and $k$ can be explicitly found, which gives the desired result.$ ag*$square$$

Remark: as a young aside, notification that we have just obtained the "completing the square" formula, and also are really close to a finish derivation the the quadratic formula. Therefore this kind of the parabola has other offers (including several that i haven"t listed).

We have actually just displayed that every parabola have the right to be created in the form$$ ax^2 + bx + c = aleft(x - h ight)^2 + +k. $$In particular, keeping our transformations in mind, we can graph any parabola by translating the ur-parabola come the right $h$ units, scaling it by a factor of $a$, possibly mirroring it vertically if $a 0$ climate the parabola maintain its initial orientation and also opens up, and also if $a