I really need help

Question

Write the equation for the following function that represents a horizontal shift left 4 and a vertical shrink by a factor of 1/2 on the parent function f(x) = x ^ 2 ?

Answer 1

The function is a quadratic, degree 2, highest power is a squared variable x. . That's a parabola, opening upwards, when the x² has a positive coefficient.

To move it left, move the vertex to the left. Initially the function has vertex at the origin (0,0) You want to change that to move left by 4 units to a new vertex of (-4,0)

y-k=a(x-h)² is the general form for a parabola opening up or down. with vertex (h,k)

Shifting left is y-0 = 1(x--4)² = (x+4)² or y=x²+8x+16

That's x-h where h= -2 or x - -2 = x+2

To shrink or narrow the parabola by a factor of 2 make a=2. That makes it half as wide across.

y=2(x+4)² As the "a" in y=a(x+4)² approaches infinity, gets very large, the parabola approaches a vertical line x=-4. Or two nearly vertical lines just to the left and right of x=-4, The larger "a" the coefficient of the squared term, the more narrow the parabola.

Answer 2

y = x^2

To shift up, subtract the shift from y

To shift right, subtract the shift from x

To scale vertically, multiply by the x term (which is the same as dividing by y)

SO:

y = 1/2 * (x + 4)^2