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Quadratic Functions (Vertex, Roots, Axis)
Standard Form, Vertex Form, Axis of Symmetry, and Discriminant
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Quadratic functions model many curved relationships in math, science, and engineering. Their graphs are parabolas, and key features like the vertex, roots, and axis of symmetry help describe the shape and location of the curve. Learning these features makes it easier to sketch graphs, solve equations, and interpret real situations such as projectile motion or area optimization.
A quadratic function is often written as y = ax^2 + bx + c, where the value of a controls whether the parabola opens up or down and how wide it appears. The vertex gives the maximum or minimum point, the roots show where the graph crosses the x-axis, and the axis of symmetry splits the parabola into two mirror-image halves. These features are connected by formulas, so once you know some of them, you can often find the others quickly.
Key Facts
- Standard form: y = ax^2 + bx + c, where a ≠ 0
- Axis of symmetry: x = -b/(2a)
- Vertex: (-b/(2a), f(-b/(2a)))
- Quadratic formula for roots: x = (-b ± sqrt(b^2 - 4ac))/(2a)
- Discriminant: D = b^2 - 4ac; if D > 0 there are 2 real roots, if D = 0 there is 1 real repeated root, if D < 0 there are no real roots
- Vertex form: y = a(x - h)^2 + k, where the vertex is (h, k)
Vocabulary
- Quadratic function
- A function whose highest power of x is 2, usually written as y = ax^2 + bx + c.
- Vertex
- The highest or lowest point on a parabola, depending on whether it opens downward or upward.
- Root
- A value of x that makes the quadratic equal to zero, shown where the graph crosses the x-axis.
- Axis of symmetry
- The vertical line that passes through the vertex and divides the parabola into two matching halves.
- Discriminant
- The expression b^2 - 4ac, which tells how many real roots a quadratic equation has.
Common Mistakes to Avoid
- Using x = -b/a for the axis of symmetry, which is wrong because the correct formula is x = -b/(2a). Missing the factor of 2 moves the vertex and roots to the wrong locations.
- Assuming the vertex is always a minimum, which is wrong because if a < 0 the parabola opens downward and the vertex is a maximum.
- Confusing roots with the y-intercept, which is wrong because roots occur where y = 0 while the y-intercept occurs where x = 0.
- Making sign errors in the quadratic formula, which is wrong because a small mistake in -b or in b^2 - 4ac can completely change the number and value of the roots.
Practice Questions
- 1 For y = x^2 - 6x + 5, find the axis of symmetry, the vertex, and the roots.
- 2 For y = -2x^2 + 8x - 6, determine whether the parabola opens up or down, then find the vertex and the x-intercepts.
- 3 A quadratic has axis of symmetry x = 3 and one root at x = 1. Without solving a full equation, find the other root and explain your reasoning.