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Chapter 3: Problem 95

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)$$

Short Answer

Expert verified
The axis of symmetry for the parabola \(f(x)=3(x+2)^{2}-5\) is \(x=-2\). Using this, a second point on the parabola with the same \(y\)-coordinate as the given point (-1,-2) is (-3, -2).

Step by step solution

01

Identify the form of the equation

The given equation \(f(x)=3(x+2)^{2}-5\) can be identified as a transformation of a standard quadratic function \(f(x)=a(x-h)^2+k\) where \(h\) gives us the axis of symmetry, \(x=h\). Our given equation has \(h=-2\).
02

Determine the axis of symmetry

The axis of symmetry for the equation given is \(x=h\). Substituting the value of \(h\) from our equation, we find the axis of symmetry to be \(x=-2\).
03

Find the mirrored point

A point \(p\) on a parabola is mirrored across the axis of symmetry by changing the \(x\)-value of \(p\) to the opposite side of the axis, while keeping the same \(y\)-value. Here, the given point is (-1,-2) and the distance from this point to the axis of symmetry is \(|-1-(-2)|=1\). We use this distance from the axis of symmetry \(x=-2\) to form another point that is 1 unit away on the opposite side. This gives us a second point \((-2-1,-2)=(-3,-2)\).

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