Question

PLEASE HELP ME
A video game producer is using the quadratic function f(x) = 4 – x^2 to model the path of an arrow shot by a game player. The path of the arrow is shown in the graph below. The point (2, 3) represents the top of the castle wall.

If the player moves so that the arrow will clear the wall at the maximum height, what is the new function that the game producer would need to use? Use complete sentences, and explain your work.

Answer 1

here, we have to traslate our parabola of 1 y-unit down, and 3 x-units right

Answer 2

Ok how do I do that mathematically because it asks what the equation would be to do that

Answer 3

yes I know. Mathematically speaking, we have to change our system of coordinates. In so doing, we get, the subsequent equation:

\[\large Y - 1 = 4 - {\left( {X + 2} \right)^2}\]

Answer 4

where Y, and X are the new coordinates. So simplifying that expression, we get:

Answer 5

\[\large \begin{gathered}
y = 4 - {x^2} - 4 - 4x \hfill \\
y = - {x^2} - 4x \hfill \\
\end{gathered} \]

Answer 6

the traslation of our parabola is 2 x-units to right and 1 y-units down

Answer 7

Ok thank you

Answer 8

please, wait I have made a typo error:
here is the right step:

Answer 9

okay lol

Answer 10

\[\large \begin{gathered}
y = 4 - {x^2} - 4 - 4x + 1 \hfill \\
y = - {x^2} - 4x + 1 \hfill \\
\end{gathered} \]

Answer 11

Okay thank you so much again I might need help with another one

Answer 12

ok! I'm ready!

Answer 13

Yeah this I don't know what to do
Create a quadratic equation in standard form that can be factored.
Write your equation in standard form. Use complete sentences to explain the benefits of writing your equation in standard form.
Write your equation in factored form. Use complete sentences to explain the benefits of writing your equation in factored form.
Write your equation in vertex form. Use complete sentences to explain the benefits of writing your equation in vertex form.
Explain how all three forms can be used together to help you graph a quadratic function. Graph your function, and label the y-intercept, the x-intercepts, and the vertex.

Answer 14

a quadratic equation, in standard form is:
\[y = a{x^2} + bx + c\]

Answer 15

or more generally, can be this:
\[\large a{x^2} + b{y^2} + cxy + dx + ey + f = 0\]
where a, b, c, d, e, f are real coefficients

Answer 16

Okay so I just make up random numbers?

Answer 17

Generally, the coefficient a, b, c for the first equation, or
a, b, c, d, e, f for the second equation, can be determined using some condition, which are given by the proble that we are trying to solve

Answer 18

they can not random values

Answer 19

oops..they can not be random values

Answer 20

Oh then can you make one up cause i'm not good at that stuff

Answer 21

for example let's consider the first equation, namely:
\[\large y = a{x^2} + bx + c\]

Answer 22

a possible exercise, which is the easier, asks to compute those coefficients a, b, c such that, the curve represented by our equation, which is a parabola, passes at point P(1,-1), Q(-2,11) and R(3,1)

Answer 23

Now, I have to substitute thos coordinates into my equation:
\[\large y = a{x^2} + bx + c\]

Answer 24

those*

Answer 25

after that, I will get the subsequent algebraic system:

Answer 26

Okay

Answer 27

\[\large \left\{ \begin{gathered}
a + b + c = - 1 \hfill \\
4a - 2b + c = 11 \hfill \\
9a + 3b + c = 1 \hfill \\
\end{gathered} \right.\]

Answer 28

the solution of that system give us three values, namely
a=1, b=-3, and c=1
so the requested parabola, or the requested equation, is:
\[\large y = {x^2} - 3x + 1\]

Answer 29

Okay thank you

Answer 30

thank you!