Question

Please help:

write a rule for g described by the transformations of the graph of f. Then identify the vertex.
1. f(x) = x²; vertical shrink by a factor of 1/2 and a reflection in the y-axis, followed by a translation 4 units left
2. f(x) = 9x²- 3; horizontal stretch by a factor of 3 and a translation 4 units up followed by a reflection in the y-axis


​

Answer 1

The transformed functions g(x) for each case are ½ (-(x + 4))² and 9(-x/3)² + 1, with vertices at (-4, 0) and (0, 1) respectively.

Step-by-step explanation:

Transformations of Quadratic Functions

Given the function f(x) = x², the rule for g after applying the described transformations would be:

1. A vertical shrink by a factor of 1/2 gives us g(x) = ½ x².
2. A reflection in the y-axis changes x to -x, resulting in g(x) = ½ (-x)².
3. Translation 4 units left turns x into x + 4, resulting in g(x) = ½ (-(x + 4))².

The vertex of g(x) is at the point (-4, 0).

For the second function f(x) = 9x² - 3, the rule for g is:

1. A horizontal stretch by a factor of 3 changes x to x/3, yielding g(x) = 9(x/3)² - 3.
2. Translation 4 units up adds 4 to the function, resulting in g(x) = 9(x/3)² - 3 + 4 or g(x) = 9(x/3)² + 1.
3. A reflection in the y-axis changes x to -x, resulting in g(x) = 9(-x/3)² + 1.

The vertex of g(x), in this case, is at the point (0, 1).