Answer:
Part 1) Option 3 could be the quadratic equation shown in the figure
Part 2) Option 4 [tex]y\leq 11[/tex]
Step-by-step explanation:
Part 1) we know that
The quadratic equation shown in the graph represent a vertical parabola open downward
The vertex represent a maximum
The coordinates of the vertex are positive
The y-intercept is positive
Has two real solutions (x-intercepts) one positive and one negative
In this problem, the options 2 and 4 represent a vertical parabola open upward (because the leading coefficient is positive)
so
Options 2 and 4 could not be the quadratic equation shown in the figure
Verify option 1 and 3
Option 1
[tex]y=-3x^{2}+8x-5[/tex]
Find the y-intercept
The y-intercept is the value of y when the value of x is equal to zero
so
For x=0
[tex]y=-3(0)^{2}+8(0)-5[/tex]
[tex]y=-5[/tex]
The y-intercept is negative
therefore
Option 1 could not be the quadratic equation shown in the figure
Option 3
[tex]y=-2x^{2}+12x+11[/tex]
Verify the y-intercept
Find the y-intercept
For x=0
[tex]y=-2(0)^{2}+12(0)+11[/tex]
[tex]y=11[/tex]
The y-intercept is positive
Verify the vertex
Convert to vertex form
[tex]y=-2x^{2}+12x+11[/tex]
Factor -2
[tex]y=-2(x^{2}-6x)+11[/tex]
Complete the square
[tex]y=-2(x^{2}-6x+9)+11+18[/tex]
[tex]y=-2(x^{2}-6x+9)+29[/tex]
rewrite as perfect squares
[tex]y=-2(x-3)^{2}+29[/tex]
The vertex is the point (3,29)
so
Both coordinates are positive
Verify the x-intercepts
Remember that the x-intercepts are the values of x when the vakue of y is equal to zero
For y=0
[tex]-2(x-3)^{2}+29=0[/tex]
[tex]2(x-3)^{2}=29[/tex]
[tex](x-3)^{2}=14.5[/tex]
square root bot sides
[tex]x-3=(+/-)\sqrt{14.5}[/tex]
[tex]x=3(+/-)\sqrt{14.5}[/tex]
Has two real solutions (x-intercepts) one positive and one negative
therefore
Option 3 could be the quadratic equation shown in the figure
Part 2) we know that
Using a graphing tool
Plot the points
The quadratic equation represent a vertical parabola open downward
The vertex is a maximum
so
The maximum value of y is equal to 11 (based in the table)
so
[tex]y\leq 11[/tex]
see the attached figure to better understand the problem
