Question

1. transform using the parent function (f(x) = x), create four new functions based on the parent function, one for each of the transformations below.

a. vertical shift, such as (f(x) + k), for some constant, k

b. horizontal shift, such as (f(x - h)), for some constant, h

c. vertical stretch/compression, such as (a cdot f(x)), for some constant, a

d. reflection, such as (-f(x))

Explanation:

Apply vertical shift

$$ g(x) = f(x) + 3 = x + 3 $$

Apply horizontal shift

$$ h(x) = f(x - 2) = x - 2 $$

Apply vertical stretch

$$ p(x) = 2 \cdot f(x) = 2x $$

Apply reflection

$$ q(x) = -f(x) = -x $$

Answer:

a. \(g(x) = x + 3\) (vertical shift up by 3 units)

b. \(h(x) = x - 2\) (horizontal shift right by 2 units)

c. \(p(x) = 2x\) (vertical stretch by a factor of 2)

d. \(q(x) = -x\) (reflection across the x-axis)