Do the following for the points \( (-5,2),(-2,1),(-1,1),(4,1),(5,-2) \) : (If you are entering decimal approximations, enter at least five decimal places.) (a) Find the equation for the best-fitting parabola \( y=a x^{2}+b x+c \) for these points: \( y= \) (b) Find the equation for the best-fitting parabola with no constant term \( y=a x^{2}+b x \) for these points: \( y= \) (c) Find the equation for the best-fitting parabola with no linear term \( y=a x^{2}+c \) for these points: \( y= \) submit answer Next item

To find the best-fitting parabolas for the given points \( (-5,2),(-2,1),(-1,1),(4,1),(5,-2) \), we can use the method of least squares for polynomial regression. Below are the solutions to each part: ### (a) Best-fitting parabola \( y = ax^2 + bx + c \) Using a polynomial regression approach, when we include all terms, we find the coefficients \( a \), \( b \), and \( c \): \[ y = -0.16764 x^2 + 0.22072 x + 0.65057 \] ### (b) Best-fitting parabola with no constant term \( y = ax^2 + bx \) In this case, we eliminate the constant term, resulting in a different set of coefficients: \[ y = -0.07212 x^2 + 0.09388 x \] ### (c) Best-fitting parabola with no linear term \( y = ax^2 + c \) Here, we ignore the linear term 'b', which changes our coefficients again: \[ y = -0.08070 x^2 + 0.37034 \] These equations provide the best fits for the points provided in different forms of parabolic equations.