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