Graphing quadratic functions — vertex, axis, direction, roots from equation

Question

How do we graph the quadratic function $y = 2x^2 - 8x + 6$ ? Find the vertex, axis of symmetry, direction of opening, and roots. Sketch the graph.

Solution — Step by Step

The coefficient of $x^2$ is $a = 2 > 0$ , so the parabola opens upward. If $a$ were negative, it would open downward.

The x-coordinate of the vertex is:

x = \frac{-b}{2a} = \frac{-(-8)}{2(2)} = \frac{8}{4} = 2

Substitute $x = 2$ into the equation:

y = 2(4) - 8(2) + 6 = 8 - 16 + 6 = -2

Vertex = (2, -2)

The axis of symmetry is the vertical line through the vertex: $x = 2$ .

Set $y = 0$ : $2x^2 - 8x + 6 = 0$

Divide by 2: $x^2 - 4x + 3 = 0$

Factor: $(x - 1)(x - 3) = 0$

Roots: $x = 1$ and $x = 3$

Set $x = 0$ : $y = 6$ . So the y-intercept is (0, 6).

Plot: vertex at (2, -2), roots at (1, 0) and (3, 0), y-intercept at (0, 6). The parabola opens upward, symmetric about $x = 2$ .

flowchart TD
    A[Given: y = ax² + bx + c] --> B[Direction: a positive = up, a negative = down]
    B --> C[Vertex x = -b/2a]
    C --> D[Substitute x into equation for y]
    D --> E[Axis of symmetry: x = -b/2a]
    E --> F[Roots: set y = 0, solve]
    F --> G[y-intercept: set x = 0]
    G --> H[Plot points and sketch]

Why This Works

Every quadratic $y = ax^2 + bx + c$ is a parabola. The vertex formula $x = -b/(2a)$ comes from completing the square. The roots come from the quadratic formula or factoring. These five features (direction, vertex, axis, roots, y-intercept) are enough to sketch any parabola accurately.

Common Mistake

Students forget that the discriminant determines the number of roots: $D = b^2 - 4ac$ . If $D > 0$ : two real roots (parabola crosses x-axis twice). If $D = 0$ : one repeated root (vertex touches x-axis). If $D < 0$ : no real roots (parabola does not cross x-axis). In our problem, $D = 64 - 48 = 16 > 0$ , so we get two roots.

The vertex is always the minimum point if $a > 0$ (opens up) and the maximum point if $a < 0$ (opens down). This is essential for optimization problems in Class 12.

Question

Solution — Step by Step

Why This Works

Common Mistake

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