Vertex form is a way to write a quadratic function so its graph is easy to understand at a glance. The form y = a(x - h)^2 + k shows the vertex directly as (h, k), which is the highest or lowest point of the parabola. This matters because the vertex tells you where the graph turns around and helps you sketch the parabola quickly.
It also connects algebra to transformations of the parent function y = x^2.
Key Facts
- Vertex form: y = a(x - h)^2 + k
- The vertex of y = a(x - h)^2 + k is (h, k).
- If a > 0, the parabola opens upward; if a < 0, it opens downward.
- The axis of symmetry is x = h.
- The value |a| controls vertical stretch or compression: larger |a| makes the parabola narrower, smaller |a| makes it wider.
- Completing the square converts standard form y = ax^2 + bx + c into vertex form.
Vocabulary
- Quadratic function
- A function whose highest power of x is 2 and whose graph is a parabola.
- Vertex form
- The form y = a(x - h)^2 + k, which shows the vertex and transformations of a quadratic function.
- Vertex
- The turning point of a parabola, located at (h, k) in vertex form.
- Axis of symmetry
- The vertical line x = h that divides a parabola into two mirror-image halves.
- Completing the square
- An algebra method that rewrites a quadratic expression as a squared binomial plus or minus a constant.
Common Mistakes to Avoid
- Using (−h, k) as the vertex, which is wrong because y = a(x - h)^2 + k has vertex (h, k). The sign inside the parentheses is opposite of how the x-coordinate appears.
- Forgetting that a negative a opens the parabola downward, which changes the vertex from a minimum to a maximum. Always check the sign of a before describing the graph.
- Treating k as a horizontal shift, which is wrong because k moves the graph up or down. The horizontal shift comes from h inside the parentheses.
- Changing only the constant when converting to vertex form, which can break equivalence. When completing the square, whatever is added inside the expression must be balanced correctly.
Practice Questions
- 1 For y = 2(x - 3)^2 - 5, identify the vertex, axis of symmetry, direction of opening, and whether the parabola is narrower or wider than y = x^2.
- 2 Convert y = x^2 - 6x + 11 to vertex form by completing the square, then identify the vertex.
- 3 A parabola has vertex (−2, 4) and opens downward with the same width as y = x^2. Write its equation in vertex form and explain how it is transformed from y = x^2.