Graphing Quadratic Functions: notes and common mistakes

Key ideas

• A quadratic function can be written y = ax² + bx + c with a ≠ 0. Its graph is a parabola.
• If a > 0 the parabola opens up and its vertex is the lowest point. If a < 0 it opens down and its vertex is the highest point. A larger |a| makes it narrower.
• The axis of symmetry is the vertical line x = −b/(2a). The vertex is on it: its x-coordinate is −b/(2a), and its y-coordinate comes from substituting that x.
• Points the same distance from the axis, on opposite sides, have the same y-value. Use this to plot one side and mirror it.
• The y-intercept is c (the graph crosses the y-axis at (0, c)). The x-intercepts are the x-values where y = 0, and there can be two, one or none.
• With no story attached, the domain is all real numbers. The range is y ≥ k if the parabola opens up and y ≤ k if it opens down, where k is the y-value of the vertex.

Steps

1. Put the equation in the form y = ax² + bx + c and read a, b and c. The sign of a tells you which way it opens.
2. Find the axis of symmetry, x = −b/(2a).
3. Find the vertex: substitute that x into the equation to get y.
4. Plot (0, c) and its mirror point across the axis. Find the x-intercepts, if there are any, by setting y = 0.
5. Add more points in mirror pairs, equally far from the axis.
6. Draw one smooth curve with arrowheads at both ends, and label the vertex and the axis.

Formulas

• Standard form: y = ax² + bx + c, with a ≠ 0
• Axis of symmetry: x = −b/(2a)
• Vertex: (h, k) with h = −b/(2a) and k = the y-value at x = h
• y-intercept: c (the graph crosses the y-axis at (0, c))
• Mirror point: If the axis is x = h, the points at x and at 2h − x have the same y-value
• x-intercepts: Set y = 0 and solve: two, one or no values
• Range: y ≥ k if a > 0, and y ≤ k if a < 0

Watch out

• Flip the sign of b in the axis formula: for y = x² + 6x + 2 the axis is x = −3.
• Divide by 2a, the whole thing. For y = 2x² − 16x + 35 the axis is 16/4 = 4, not 16/2 = 8.
• Only a decides whether the parabola opens up or down. The sign of b or c does not.
• The maximum or minimum value is a y-value. The vertex is a point with two coordinates.
• A parabola can have two, one or no x-intercepts. Do not assume it always crosses the x-axis twice.
• Draw a smooth, rounded curve. Do not join the points with straight segments or draw a sharp V.
• The mirror trick works only for x-values that are equally far from the axis.
• In a story, use only the part of the parabola that fits. A time cannot be negative.

Why it works

• Why does the sign of a decide the direction? The square x² is never negative, so ax² has the sign of a. Far from the vertex, where |x| is large, the ax² term is much bigger than bx + c, so the graph heads the way a points.
• Why is the axis x = −b/(2a)? Let h = −b/(2a), so 2ah + b = 0. Compare the heights at h + d and h − d, two x-values equally far from h. Subtract the two heights. The a part is a((h + d)² − (h − d)²) = 4ahd and the b part is b((h + d) − (h − d)) = 2bd. Together, y(h + d) − y(h − d) = 4ahd + 2bd = 2d(2ah + b) = 0. So the heights match for every d, which means the graph is symmetric about the line x = h.
• A quick check of the formula: y = c at x = 0 and also at x = −b/a, because ax² + bx = x(ax + b) = 0 there. The axis is halfway between these two x-values: (0 + (−b/a))/2 = −b/(2a). (If b = 0 the two x-values are the same, the axis is x = 0, and the formula agrees.)
• Why can there be zero, one or two x-intercepts? A parabola changes direction only once, at the vertex. On each side of the vertex, y moves in one direction only, so each side can cross the x-axis at most once. That gives at most two x-intercepts. If the parabola opens away from the x-axis, no point reaches it (none). If the vertex is on the axis, only the vertex touches it (one). If the parabola opens toward the axis, it crosses once on each side of the vertex (two).