Question

a computer game designer uses the function $f(x)=4(x - 2)^2 + 6$ to model the path of the fish. the horizontal path of the squid intersects the path of the fish at what other point does the squid’s path intersect the path of the fish? (type an ordered pair )

Explanation:

Step1: Determine the y - value of the squid's path

The squid's path is horizontal and intersects the fish's path at \((5,42)\), so the equation of the squid's path is \(y = 42\) (since a horizontal line has the equation \(y=k\), where \(k\) is the y - coordinate of any point on the line).

Step2: Set the fish's function equal to the squid's path equation

We have the fish's path function \(f(x)=4(x - 2)^{2}+6\) and we set it equal to \(y = 42\):
\[4(x - 2)^{2}+6=42\]
Subtract 6 from both sides:
\[4(x - 2)^{2}=42 - 6=36\]
Divide both sides by 4:
\[(x - 2)^{2}=\frac{36}{4} = 9\]
Take the square root of both sides:
\(x-2=\pm\sqrt{9}=\pm3\)

Step3: Solve for x

Case 1: \(x - 2=3\)
Then \(x=3 + 2=5\) (this is the x - coordinate of the given intersection point \((5,42)\))
Case 2: \(x - 2=- 3\)
Then \(x=-3 + 2=-1\)

Step4: Find the other intersection point

We know that when \(x=-1\), \(y = 42\) (because the squid's path is \(y = 42\)). So the other intersection point is \((-1,42)\)

Answer:

\((-1,42)\)