Math

Question Find the transformation properties of the quadratic function g(x)=3(x2)2g(x) = 3(x-2)^{2}, including its vertex, domain, range, and axis of symmetry.

Studdy Solution

STEP 1

Assumptions
1. The function given is g(x)=3(x2)2g(x) = 3(x-2)^2.
2. We need to find the transformations applied to the parent function f(x)=x2f(x) = x^2.
3. We will identify the vertex, domain, range, and axis of symmetry for the function g(x)g(x).

STEP 2

Identify the transformations applied to the parent function f(x)=x2f(x) = x^2 to obtain g(x)g(x).
The function g(x)g(x) can be obtained from f(x)f(x) by: - Horizontal shift 2 units to the right - Vertical stretch by a factor of 3 - No vertical shift, as there is no constant added or subtracted

STEP 3

Determine the vertex of the function g(x)g(x).
The vertex form of a quadratic function is a(xh)2+ka(x-h)^2 + k, where (h,k)(h, k) is the vertex. In g(x)=3(x2)2g(x) = 3(x-2)^2, h=2h = 2 and k=0k = 0, so the vertex is at (2,0)(2, 0).

STEP 4

Determine the domain of the function g(x)g(x).
The domain of any quadratic function is all real numbers, because there is no restriction on the values that xx can take.
Domain:(,)Domain: (-\infty, \infty)

STEP 5

Determine the range of the function g(x)g(x).
Since the coefficient of (x2)2(x-2)^2 is positive (3), the parabola opens upwards. The lowest point on the graph is the vertex. Therefore, the range is all values greater than or equal to the y-coordinate of the vertex.
Range:[0,)Range: [0, \infty)

STEP 6

Determine the axis of symmetry for the function g(x)g(x).
The axis of symmetry for a parabola in vertex form a(xh)2+ka(x-h)^2 + k is the vertical line x=hx = h. For g(x)=3(x2)2g(x) = 3(x-2)^2, the axis of symmetry is x=2x = 2.

STEP 7

Summarize the transformations, vertex, domain, range, and axis of symmetry for the function g(x)=3(x2)2g(x) = 3(x-2)^2.
Transformations: - Horizontal shift 2 units to the right - Vertical stretch by a factor of 3
Vertex: (2,0)(2, 0)
Domain: (,)(-\infty, \infty)
Range: [0,)[0, \infty)
Axis of Symmetry: x=2x = 2

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord