Question

In two or more complete sentences, compare the number of x-intercepts in the graph of f(t) = t2 to the number of x-intercepts in the graph of g(t) = (t – 8)2. Be sure to include the transformations that occurred between the parent function f(t) and its image g(t).

Answer 1

Answer

Both graphs have the same number of x-intercepts. The graph of the function f(t) = t² has one x-intercept, which is the value of t for which f(t) = t² = 0, and that is t = 0. The graph of the function g(t) = (t - 8)² has also one x-intercept, which is the value of t for which g(t) = 0 and that is t = 8.

The function f(t) = t² is the most simple form of a parabola, so it is considered the parent function. The function g(t) = (t - 8)² is a daughter function of f(t); then, the graph of g(t) is a horizontal translation of the graph of f(t),  8 units to the right, so the number of x-intercepts (the points where the x-axis is crossed or touched by the graph) does not change, it is just their position what changes.

Explanation:

The x-intercepts are the points where the graph of the function touches or crosses the x-axis. They are found by doing the function equal to zero. In this case f(t) = 0 and g(t) = 0.

You can solve easily f(t) = t², as, just by simple inspection, the soluton is t = 0.

Then, when you realize that the function g(t) = (t - 8)² is a horizontal translation (8 units to the right) of the parent function f(t), you can conclude quickly that the number of x-intercepts of both graphs is the same. Thus, uisng the transformation of the parent function, 8 units to the right, you conclude that both the graph of f(t) and the graph of g(t) have the same number of x-intercepts: one.

Answer 2

f rests at (0/0) and grows upwards, so there is only a single x-intercept. g is f moved 3 to the left, so it also only has one intercept but at (-3,0).