help quad functions

Question

1. Describe how each of a, h, and k in a quadratic function of the form
   f(x) = a(x − h)2 + k affects the graph of f(x) = x2 (how is the graph of f(x) = x2 transformed?). Provide examples.

2. Graph a quadratic function that includes multiple transformations
   (a ≠ ±1, h ≠ 0, k ≠ 0). You must use the step pattern or the mapping rule.
   You can graph by hand or using Geogebra Notes (if you use Geogebra Notes, you must use the Point tool to plot the points yourself).
   Explain each step involved in graphing the function in detail.

3. State the domain and range of the function from part b. using proper set notation.

4. Determine f(−10) for the function from part b. using proper function notation.
Show all steps.

Answer 1

(1) The basic parabolic shape is a U-shaped curve y=x² which sits on the origin (0,0), the vertex of the U. The arms of the U spread with height of the U.

h and k move the vertex to a different point. h shifts the vertex to right when positive and to the left when negative, while k shifts the vertex up when positive and down when negative. So the vertex is at (h,k) and the equation takes the general form y-k=a(x-h)² which when expanded becomes a quadratic expression of the form y=ax²+bx+c, where b=-2ah and c=ah²+k.

The value of a affects the spread of the arms of the U. The larger the magnitude of a the narrower is the shape. When the magnitude of a is between  0 and 1 the U becomes wider. Positive values of a produce an upright U (vertex is a minimum), while negative values produce an inverted U (vertex is a maximum).

(2)

The blue parabola is y=x² and the red parabola is y+2=-3(x-1)² so h=1, k=-2, a=-3. In quadratic form this is y=-3x²+6x-5. The vertex is shifted to (1,-2) and the spread of the parabolic arms is reduced from the standard (blue curve).

(3) Domain is D(x)=(−∞,∞); Range is R(y)=(−∞,-2].

(4) y=f(x)=-3x²+6x-5; f(-10)=-300-60-5=-365.