Pre-calc assignment

Question

1. Write the equation in standard form and graph the parabola.

y² - 8y + 8x + 8 = 0

9x² + 25y² – 225 = 0

1.  Find y ^’ for each of the following 



y = x^1/2 

 

y = 4x⁶ + 5x³ – 8

 

y = -3x³ + 7x⁵

 

y = x(x³ + 5)

 

1.  Given the function f(x) =x⁴ + 3x³ – 5x² + x.

 

Find the first derivative of f(x).

Find the gradient of the function at the point D(-2,6).

Answer 1

y^2 - 8y + 8x + 8 = 0

solve for x

8x = -y^2 + 8y - 8

x = (-1/8)y^2 + y - 1 where a=-1/8, b=1 and c=-1=x intercept, for x=ay^2+by + c standard form

or

complete the square

x = (-1/8)(y^2 - 8y + 16) -1 + 2

x = (-1/8)(y-4)^2 + 1 vertex is (1, 4) 1= maximum x coordinate of a leftward opening parabola

x = a(y-h)^2 + k where (h,k) is the vertex, a=-1/8<0 making it leftward opening

(h,k) = (1, 4)

or solve for y

-8x -8 = (y-4)^2

y-4 = + or - sqr(8-8x)

y =4 + or - sqr(8-8x)

axis of symmetry is y =4, y intercepts are y=4+ or - sqr(8) = about 6.8 or 1.2

the second equation is an ellipse not a parabola

9x^2 + 25y^2 = 225 divide by 225

9x^2/225 + 25y^2/225 = -

x^2/25 + y^2/9 = 1

x^2/5^2 + y^2/3^2 = 1

it's an ellipse in standard form x^2/a^2 + y^2/b^2 = 1 where a=5 and b=3

the major and minor semi-axes of the ellipse with center at the origin (0,0)

y= x^1/2 = sqrx

y' = (1/2)x^-1/2 = 1/2sqrx