Can someone please answer these?

1.The amount of profit Bill makes per toy when he increases or decreases the price of his handmade toys can be modeled by the function f(x) = –x2 – 2x + 3. What price change gives him the highest profit? What is the highest profit per toy?

2.A pebble is tossed into the air from the top of a cliff. The height, in feet, of the pebble over time is modeled by the equation y = −16x2 + 32x + 80. What is the maximum height, in feet, reached by the pebble?

3.Function g is a transformation of the parent function f(x) = x2. The graph of f is reflected across the x-axis, and then translated left 4 units and down 2 units to form the graph of g.
Write the equation for g in the form y = ax2 + bx + c.

4.The shape of the inside of a glass follows a parabola with the function f(x) = x2 + 6x + 9. What point represents the bottom of the inside of the glass?

5.The equation y = −4.9x2 + 14 represents the height y in meters of a rock dropped off a bridge over time x in seconds. Which of the following is a graph of this equation from the time the rock dropped to when it reached the water?

-Thank You

6.Function g is a transformation of the parent function f(x) = x2. The graph of f is reflected across the x-axis, and then translated right 5 units and up 1 unit to form the graph of g. Write the equation for g in the form y = ax2 + bx + c.

1. The function [tex]f(x) = -x^2-2x + 3[/tex] represents the amount of profit Bill makes per toy when he increases or decreases the price of his handmade toys.

The graph of this function is a parabola. The maximum profit is at the vertex of the parabola.

Find the vertex:

[tex]x_v=\dfrac{-b}{2a}=\dfrac{-(-2)}{2\cdot (-1)}=-1\\ \\y_v=f(x_v)=f(1)=-(-1)^2-2\cdot (-1)+3=-1+2+3=-1+5=4[/tex]

This means that decrease of 1 money unit (dollar, cent, euro,...) will give Bill the maximum profit of 4 money units (dollars, cents, euros,...)

2. The height, in feet, of the pebble over time is modeled by the equation [tex]y = -16x^2 + 32x + 80.[/tex]

The maximum height of the pebble is at parabola's vertex.

Find the vertex:

[tex]x_v=\dfrac{-b}{2a}=\dfrac{-32}{2\cdot (-16)}=1\\ \\y_v=-16\cdot 1^2+32\cdot 1+80=-16+32+80=96[/tex]

Thus, the maximum height the pebble can reach is 96 feet.

3. First, reflect across the x-axis the graph of the parent function [tex]f(x)=x^2.[/tex] The reflection across the x-axis will give us the function [tex]h(x)=-x^2[/tex]

Now, translate this function left 4 units and down 2 units to form the graph of g(x):

[tex]g(x)=-(x+4)^2-2[/tex]

Open the brackets:

[tex]g(x)=-(x^2+8x+16)-2=-x^2-8x-16-2=-x^2-8x-18[/tex]

Hence,

[tex]\bf{g(x)=-x^2-8x-18}[/tex]

4. The shape of the inside of a glass follows a parabola with the function [tex]f(x) = x^2 + 6x + 9.[/tex]

The vertex of the parabola represents the bottom of the inside of the glass. Find the vertex of the parabola:

[tex]x_v=\dfrac{-b}{2a}=\dfrac{-6}{2\cdot 1}=-3\\ \\y_v=(-3)^2+6\cdot (-3)+9=9-18+9=0[/tex]

So, point (-3,0) represents the bottom of the inside of the glass.

5. The equation y = −4.9x2 + 14 represents the height y in meters of a rock dropped off a bridge over time x in seconds.

The graph of the function is attached.

6. First, reflect across the x-axis the graph of the parent function [tex]f(x)=x^2.[/tex] The reflection across the x-axis will give us the function [tex]h(x)=-x^2[/tex]

Now, translate this function right 5 units and up 1 unit to form the graph of g(x):

[tex]g(x)=-(x-5)^2+1[/tex]

Open the brackets:

[tex]g(x)=-(x^2-10x+25)+1=-x^2+10x-25+1=-x^2+10x-24[/tex]

Hence,

[tex]\bf{g(x)=-x^2+10x-24}[/tex]