kramtus51

2021-12-15

Which is n\Rightarrower?
$f\left(x\right)=2{x}^{2}+3x$ or $g\left(x\right)={x}^{2}+4$

Elaine Verrett

Let us write these equations of parabolas in their vertex form i.e.
$y=a{\left(x-h\right)}^{2}+k$ where (h.k) is the vertex and a is quadratic coefficient. The greater the quadratic coefficient, the n\Rightarrower is the parabola.
$f\left(x\right)=2{x}^{2}+3x=2\left({x}^{2}+\frac{3}{2}x\right)$
$=2\left({x}^{2}+2×\frac{3}{4}x+{\left(\frac{3}{4}\right)}^{2}\right)-2×{\left(\frac{3}{4}\right)}^{2}$
$=2{\left(x+\frac{3}{4}\right)}^{2}-\frac{9}{8}$
and $g\left(x\right)={x}^{2}+4={\left(x-0\right)}^{2}+4$
To find whether a parabola is n\Rightarrow or wide, we should look at the quadratic coefficient of the parabola, which is 2 in f(x) and 1 in g(x) and hence $f\left(x\right)=2{x}^{2}+3x$ is n\Rightarrower

intacte87

Let's graph them both and then see for sure. Here is $f\left(x\right)=2{x}^{2}+3x$
And this is $g\left(x\right)={x}^{2}+4$
Why is it that g(x) is fatter than f(x)?
The answer lies in the coefficient for the ${x}^{2}$ term. When the absolute value of the coefficient gets bigger, the graph gets n\Rightarrower (positive and negative simply show the direction the parabola is pointing, with positive opening up and negative opening down).
Let's compare the graphs of $y=±{x}^{2},±5{x}^{2},±\frac{1}{3}{x}^{2}$. This is $y={x}^{2}$

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