Example  3
Given the equations of the following functions, determine the number of \(x\)-intercepts on the graph of the function. Verify by graphing.

  1. \(f(x) = -4(x - 2)^2 + 6\)

    First, write down the values of \(a\) and \(q\).

    \(a = -4\); \(q = 6\)

    Because \(a < 0 \), the graph opens downward.

    Because \(q > 0 \), the graph will intersect the \(x\)-axis two times. There are two \(x\)-intercepts.

    Verify by graphing:
  2. \(g(x) = 2(x - 5)^2 + 3\)

    \(a = 2\); \(q = 3\)

    Because \(a > 0 \), the graph opens upward.

    Because \(q > 0 \), the graph will not intersect the \(x\)-axis. There are zero \(x\)-intercepts.

    Verify by graphing:
  3. \(h(x) = -2(x + 3)^2\)

    \(a = -2\); \(q = 0\)

    Because \(a < 0 \), the graph opens downward.

    Because \(q = 0 \), the graph will intersect the \(x\)-axis one time. There is one \(x\)-intercept.

    Verify by graphing: