The phrase absolute lowest point refers to the point on the graph where the function attains its smallest possible \(y\)-value over its domain (a global minimum). For a quadratic function with a positive leading coefficient, that point is the vertex of the parabola.
\[ f(x)=x^2-4x+1. \]
Vertex form and the absolute lowest point
Completing the square rewrites the quadratic so the minimum is visible directly:
\[ f(x)=x^2-4x+1 =\bigl(x^2-4x+4\bigr)-4+1 =(x-2)^2-3. \]
Since \((x-2)^2\ge 0\) for every real \(x\), the smallest possible value of \((x-2)^2-3\) occurs when \((x-2)^2=0\), which happens at \(x=2\).
Minimum value and range
\[ f(2)=(2-2)^2-3=-3. \]
The absolute lowest point is therefore \(\,(2,-3)\,\). The minimum value is \(-3\), and the range over all real numbers is
\[ y\ge -3. \]
Summary table
| Item | Result | Meaning |
|---|---|---|
| Vertex form | \(f(x)=(x-2)^2-3\) | Shows the minimum shift from a perfect square |
| Axis of symmetry | \(x=2\) | Vertical line through the vertex |
| Absolute lowest point | \((2,-3)\) | Global minimum point of the parabola |
| Minimum value | \(-3\) | Smallest possible output |
| Range | \(y\ge -3\) | All attainable outputs |
Visualization
Common pitfalls
The y-intercept is not generally the absolute lowest point. The sign of the leading coefficient controls the opening direction: a positive coefficient produces a global minimum (absolute lowest point), while a negative coefficient produces a global maximum.