1. A ball thrown high and follows a parabolic path. Given the equation of the
parabola 𝑦 = −2𝑥^2 + 4𝑥 + 1. Write the equation in standard form.
2. A cucumber cut at an angle to its main axis results in an elliptical shape. Suppose the ellipse is given by the equation 25𝑥^2 + 𝑦^2 = 25. Write the equation in standard form.
y = -2x^2 + 4x + 1
= -2(x^2 - 2x + 1 - 1) + 1
= -2( (x-1)^2 - 1) + 1
= -2(x-1)^2 + 2 + 1
= -2(x-1)^2 + 3
25𝑥^2 + 𝑦^2 = 25 , assuming by standard form you mean
x^2 / a^2 + y^2 / b^2 = 1
divide each side by 25
x^2 + y^2 / 25 = 1
x^2 + y^2 / 5^2 = 1