Example  4
The graph of the quadratic function \(f(x) = x^2\) is stretched by a factor of \(-3\), then translated \(2\) units left and \(4\) units up.

  1. What happens to the vertex when the function is multiplied by \(-3\)?

    The vertex does not change when the function is multiplied by \(-3\). Originally, the vertex was \((0, 0)\). Multiplied by \(-3\), \((0, 0)\) is still \((0, 0)\).
  2. What is the vertex of the graph of the function after it is translated \(2\) units left and \(4\) units up?

    The vertex is moved to the left \(2\) units and up \(4\) units, therefore \(p = -2\) and \(q = 4\). The new vertex is \((-2, 4)\).
  3. What is the equation of the transformed function?

    The function is stretched by a factor of \(-3\), so \(a = -3\). Substitute the values for \(a\), \(p\), and \(q\) into \(f(x) = a(x - p)^2 + q \), and the new equation is:
    \(h(x) = -3(x - (-2))^2 + 4\)
    \(h(x) = -3(x + 2)^2 + 4\)
  4. How many \(x\)-intercepts does the graph of the function have?

    Because \(a\) and \(q\) are opposite signs, there will be two \(x\)-intercepts.
  5. Sketch the graph.