Consider the following graph of a quadratic function. A quadratic function has the form \(y=ax^2+bx+c\) with \(a\neq 0\), and its graph is a parabola whose opening and symmetry are controlled by \(a\) and whose key features include a single vertex and a vertical axis of symmetry.
Assumed plotted features for the shown parabola
The graph below is treated as an upward-opening parabola with x-intercepts at \((-1,0)\) and \((3,0)\) and a lowest point at the vertex \((1,-4)\). These values define a unique quadratic function consistent with the drawing.
Quadratic equation consistent with the graph
The x-intercepts \((-1,0)\) and \((3,0)\) imply a factored form \(y=a(x+1)(x-3)\) for some nonzero constant \(a\). The axis of symmetry for a parabola lies halfway between the roots, giving \(x=\frac{-1+3}{2}=1\), which matches the drawn symmetry line. Substituting the vertex point \((1,-4)\) determines \(a\):
\[ -4 \;=\; a(1+1)(1-3)\;=\;a(2)(-2)\;=\;-4a \quad\Rightarrow\quad a=1. \]
The quadratic function is therefore
\[ y=(x+1)(x-3)=x^2-2x-3. \]
Vertex form and transformation meaning
Completing the square converts the standard form into vertex form, which displays the horizontal and vertical shifts relative to \(y=x^2\):
\[ y=x^2-2x-3 =\bigl(x^2-2x+1\bigr)-1-3 =(x-1)^2-4. \]
The expression \((x-1)^2-4\) indicates a shift right by 1 and down by 4, with no vertical stretch because the leading coefficient is 1.
Domain, range, and monotonic behavior
A quadratic function is defined for every real input, so the domain is all real numbers. The opening direction is upward because \(a=1>0\), so the vertex gives a minimum value. With vertex \(y=-4\), the range is \(y\ge -4\):
\[ \mathrm{Domain}=(-\infty,\infty), \qquad \mathrm{Range}=[-4,\infty). \]
The function decreases on \((-\infty,1]\) and increases on \([1,\infty)\), reflecting symmetry about \(x=1\).
Key features (summary table)
| Feature | Value | Graph interpretation |
|---|---|---|
| Equation (vertex form) | \(y=(x-1)^2-4\) | Vertex at \((1,-4)\) with upward opening |
| Equation (standard form) | \(y=x^2-2x-3\) | Expanded from \((x+1)(x-3)\) |
| Vertex | \((1,-4)\) | Lowest point of the parabola |
| Axis of symmetry | \(x=1\) | Vertical line through the vertex |
| x-intercepts (zeros) | \((-1,0)\), \((3,0)\) | Where the graph crosses the x-axis |
| y-intercept | \((0,-3)\) | Where the graph crosses the y-axis |
| Domain | \((-\infty,\infty)\) | All real inputs are allowed |
| Range | \([-4,\infty)\) | Minimum at the vertex with upward opening |
Common pitfalls
Range direction errors. An upward-opening parabola has a minimum at the vertex, while a downward-opening parabola has a maximum at the vertex.
Axis placement errors. The symmetry line passes through the vertex and lies midway between the x-intercepts when two real roots exist.
Intercept confusion. The y-intercept corresponds to \(x=0\), while x-intercepts correspond to \(y=0\).