Question

The graph shows g(x), which is a translation of

f(x) = x². Write the function rule for g(x).
-10 -8 -6 -4 -2
104
-8-
-6-
-4-
2-
0
-2-
-4-
-6-
-8-
-1.0
у
2
4
6
-CO
8
Write your answer in the form a(x − h)² + k,
where a, h, and k are integers or simplified
fractions.
g(x) =

Answer 1

Since g(x) is a translation of f(x) = x², we can write the function rule for g(x) in terms of f(x) as:

g(x) = f(x - h) + k

where h and k are the horizontal and vertical translations, respectively. To determine the values of h and k, we need to look at the graph of g(x).

From the graph, we can see that the vertex of g(x) is at the point (-2, -8). This means that the horizontal translation is 2 units to the right (since the vertex of f(x) = x² is at the origin), and the vertical translation is 8 units downward.

Therefore, we have:

h = 2

k = -8

Substituting these values into the general form of g(x), we get:

g(x) = a(x - 2)² - 8

To determine the value of a, we can use another point on the graph. For example, we can use the point (-6, 2), which is 4 units to the left and 10 units upward from the vertex (-2, -8).

Substituting x = -6 and y = 2 into the equation for g(x), we get:

2 = a(-6 - 2)² - 8

Simplifying and solving for a, we get:

a = 1/8

Therefore, the function rule for g(x) is:

g(x) = (1/8)(x - 2)² - 8

Written in the form a(x - h)² + k, where a = 1/8, h = 2, and k = -8.