Respuesta :
y = 2x2 + 15x + 18 //With 2 being greater than 1 in 2x^2
y = x2 + 13x + 5 //Not being negative
y = -x2 + 3x + 8
y = -2x2 + 4x + 8 //Being the largest negative means it decreases the most.
Answer:
3rd Parabola < 2nd parabola < 4th parabola < 1st parabola.
Step-by-step explanation:
We are given with equation of parabolas.
We need to arrange them in increasing order with respect to y-values of their directrixes.
First we convert given equations in Standard form then find their directrix.
The standard form is (x - h)² = 4a (y - k),
then directrix is y = k - a
1). y = -x² + 3x + 8
using completing the square method,
[tex]y=-(x^2-3x-8)[/tex]
[tex]y=-(x^2-3x+(\frac{3}{2})^2-8-(\frac{3}{2})^2)[/tex]
[tex]y=-((x-\frac{3}{2})^2-8-\frac{9}{4})[/tex]
[tex]y=-((x-\frac{3}{2})^2-\frac{41}{4})[/tex]
[tex]y=-(x-\frac{3}{2})^2+\frac{41}{4}[/tex]
[tex](x-\frac{3}{2})^2=-(y-\frac{41}{4})[/tex]
Now, by comparing with standard equation
[tex]4a=-1\:\:\implies\:a=\frac{-1}{4}[/tex]
[tex]k=\frac{41}{4}[/tex]
So, Directrix, [tex]y=\frac{41}{4}-\frac{-1}{4}=\frac{42}{4}=10.5[/tex]
2). y = 2x² + 15x + 18
using completing the square method,
[tex]y=2(x^2+\frac{15}{2}x+9)[/tex]
[tex]y=2(x^2+\frac{15}{2}x+(\frac{15}{4})^2+9-(\frac{15}{4})^2)[/tex]
[tex]y=2((x+\frac{15}{4})^2+9-\frac{225}{16})[/tex]
[tex]y=2((x+\frac{15}{4})^2-\frac{81}{16})[/tex]
[tex]y=2(x+\frac{15}{4})^2-\frac{81}{8}[/tex]
[tex](x+\frac{15}{4})^2=\frac{1}{2}(y+\frac{81}{8})[/tex]
Now, by comparing with standard equation
[tex]4a=\frac{1}{2}\:\:\implies\:a=\frac{1}{8}[/tex]
[tex]k=\frac{-81}{8}[/tex]
So, Directrix, [tex]y=\frac{-81}{8}-\frac{1}{8}=\frac{-82}{8}=-10.25[/tex]
3). y = x² + 13x + 5
using completing the square method,
[tex]y=x^2+13x+5[/tex]
[tex]y=x^2+13x+(\frac{13}{2})^2+5-(\frac{13}{2})^2[/tex]
[tex]y=(x+\frac{13}{2})^2+5-\frac{169}{4}[/tex]
[tex]y=(x+\frac{13}{2})^2-\frac{149}{4}[/tex]
[tex](x+\frac{13}{2})^2=(y+\frac{149}{4})[/tex]
Now, by comparing with standard equation
[tex]4a=1\:\:\implies\:a=\frac{1}{4}[/tex]
[tex]k=\frac{-149}{4}[/tex]
So, Directrix, [tex]y=\frac{-149}{4}-\frac{1}{4}=\frac{-150}{4}=-37.5[/tex]
4). y = -2x² + 4x + 8
using completing the square method,
[tex]y=-2(x^2-2x-4)[/tex]
[tex]y=-2(x^2-2x+(1)^2-4-(1)^2)[/tex]
[tex]y=-2((x-1)^2-4-1)[/tex]
[tex]y=-2((x-1)^2-5)[/tex]
[tex]y=-2(x-1})^2+10[/tex]
[tex](x-1})^2=\frac{-1}{2}(y-10)[/tex]
Now, by comparing with standard equation
[tex]4a=\frac{-1}{2}\:\:\implies\:a=\frac{-1}{8}[/tex]
[tex]k=10[/tex]
So, Directrix, [tex]y=10-\frac{-1}{8}=\frac{81}{8}=10.125[/tex]
So, from above
y = 37.5 < y = -10.25 < y = 10.125 < y = 10.5
Therefore, 3rd Parabola < 2nd parabola < 4th parabola < 1st parabola.
