Algebra — Equations
GradesAlgebra IAlgebra — EquationsGraphing Quadratic Functions

Graphing Quadratic Functions

⚖️ Algebra — Equations
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Graphing Quadratic Functions

Sketch parabolas using vertex, axis of symmetry, and intercepts.

Every quadratic function y = ax² + bx + c graphs as a parabola. The sign of a tells you which way it opens: if a > 0 the parabola opens upward (happy face ☺); if a < 0 it opens downward (sad face ☹). The vertex is the highest or lowest point.

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Axis of symmetry: x = −b / (2a). The vertex x-coordinate is always this value.

Graphing y = ax² + bx + c

1Find the axis of symmetry: x = −b/(2a).
2Substitute back to find the vertex y-coordinate: y = f(−b/(2a)).
3Find x-intercepts by setting y = 0 and solving (factor or use quadratic formula).
4Find the y-intercept: set x = 0, so y = c.
5Plot vertex, intercepts, and one or two symmetric points; draw a smooth curve.
✏️Graph y = x² − 4x + 3
a=1, b=−4, c=3. Axis: x = −(−4)/(2·1) = 2. Vertex: y = 4−8+3 = −1, so vertex is (2,−1). y-intercept: (0, 3). x-intercepts: x²−4x+3 = (x−1)(x−3) = 0 → x=1 and x=3.

Standard Form

y = ax² + bx + cVertex x = −b/(2a)Easy to find y-intercept (c)

Vertex Form

y = a(x − h)² + kVertex is (h, k) directlyEasy to read min/max value
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Remember This!

The vertex gives the minimum value of the function when a > 0, or the maximum value when a < 0. Real-world problems like "maximum height" of a ball always look for the vertex.

✏️ Try It!

For y = 2x² − 8x + 6, what is the x-coordinate of the vertex?

Practice with CalcVerse
🌊Quadratic Formula Solver📈Line Grapher (y = mx + b)📊Function Grapher
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