Question

if you got:
x = 3
a
now find:
describe the transformations that produce
g(x) from the parent graph f(x):
hint: this means to identify a, h, and k, and
state what they do.
f(x) = x² & g(x) = -2(x + 3)² - 4

Explanation:

Step1: Recall Vertex Form

The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex and \( a \) determines vertical stretch/compression and reflection. For \( f(x)=x^2 \) (vertex form \( y = 1(x - 0)^2 + 0 \)) and \( g(x)=-2(x + 3)^2 - 4 \), rewrite \( g(x) \) as \( y=-2(x - (-3))^2 + (-4) \).

Step2: Identify \( a \), \( h \), \( k \)

• \( a=-2 \): Since \( |a| = 2>1 \), there's a vertical stretch by a factor of 2. The negative sign means a reflection over the \( x \)-axis.
• \( h=-3 \): In vertex form, \( h \) is the horizontal shift. \( h=-3 \) means a shift 3 units to the left (because \( x - h \) becomes \( x - (-3)=x + 3 \)).
• \( k=-4 \): \( k \) is the vertical shift. \( k=-4 \) means a shift 4 units down.

Answer:

To transform \( f(x)=x^2 \) to \( g(x)=-2(x + 3)^2 - 4 \):

1. Vertical Transformation: \( a = -2 \) causes a vertical stretch by a factor of 2 and a reflection over the \( x \)-axis.
2. Horizontal Transformation: \( h = -3 \) causes a horizontal shift 3 units to the left.
3. Vertical Shift: \( k = -4 \) causes a vertical shift 4 units down.

(In terms of identifying \( a=-2 \), \( h=-3 \), \( k=-4 \) and their effects: \( a \) stretches/reflected, \( h \) shifts left/right, \( k \) shifts up/down.)