A quadratic function has the form , and its graph is always a parabola. When the parabola opens upward (minimum at the vertex); when it opens downward (maximum). The vertex represents the turning point of the parabola, and the axis of symmetry - a vertical line through the vertex - divides the parabola into two mirror-image halves.
Quadratics appear throughout physics (projectile motion), economics (profit maximization), and engineering (cable shapes in suspension bridges). The quadratic formula, the discriminant, and factoring methods all give different views of the same object: where the parabola crosses the x-axis and what shape it takes.
Understanding Quadratic Functions
One useful way to understand a quadratic is to start with the parent curve, y equals x squared. Changing the number in front of the squared term changes how the curve looks. A larger absolute value makes it narrower because the output rises or falls more quickly for each step away from the center.
A value between zero and one makes it wider. A negative leading value reflects the curve across the horizontal axis.
In vertex form, the number inside the brackets shifts the graph left or right, while the final number shifts it up or down. These changes help students sketch a graph before calculating many points.
The vertex can be found by completing the square. This method rewrites a quadratic so its turning point becomes visible. It may feel like an algebra trick at first, but it has a clear purpose.
A squared quantity can never be negative. Once the expression is written as a squared part plus a constant, the smallest or largest possible output is clear from the sign of the leading coefficient.
Completing the square is especially useful when a problem asks for a maximum height, a minimum cost, or the time when an object reaches its highest point. It connects algebraic manipulation directly to a feature on the graph.
Roots describe input values that make the output equal zero. On a graph, they are the places where the curve meets the horizontal axis. Factoring is usually quickest when the factors are obvious.
The quadratic formula works even when factoring does not. The discriminant tells you what kind of answer to expect before you finish the calculation. A positive discriminant gives two different real crossing points.
A zero discriminant means the curve just touches the axis at its vertex. A negative discriminant means the graph stays entirely above or entirely below the axis, so there are no real crossing points. This is important because not every algebraic equation has a solution that can be placed on the usual coordinate plane.
Quadratics often model situations with a changing rate. A thrown ball rises quickly, slows near its peak, then falls faster. Its height over time can be modeled by a quadratic when air resistance is ignored.
In a business model, profit may increase as production begins, reach a best value, then decrease if costs become too high. Students should keep track of what each variable means and which values are realistic. Negative time may be a valid algebraic root but not a meaningful answer in a motion problem.
Check calculated roots by substituting them into the original expression. When graphing, use symmetry to reduce work. Find the center line, plot the vertex, then place matching points at equal horizontal distances on both sides.
Key Facts
- Standard form:
- Vertex form: , vertex at
- Quadratic formula: x = (-b ± √(b² - 4ac)) / (2a)
- Axis of symmetry: x = -b/(2a)
- Discriminant : two real roots; one root; no real roots
- Sum of roots = ; product of roots = .
Vocabulary
- Vertex
- The maximum or minimum point of a parabola, located at .
- Axis of symmetry
- The vertical line x = -b/(2a) that divides the parabola into two symmetric halves.
- Discriminant
- The expression b² - 4ac under the square root in the quadratic formula; determines the number and type of roots.
- Root (zero)
- An x-value where ; the x-intercept(s) of the parabola.
- Vertex form
- , which directly shows the vertex and direction of opening.
Common Mistakes to Avoid
- Forgetting that the axis of symmetry is , not . The negative sign is easy to drop.
- Thinking a negative discriminant means 'no answer.' The equation has two complex (imaginary) roots - just no real ones.
- Expanding vertex form incorrectly. (x - h)² ≠ x² - h². The correct expansion is x² - 2hx + h².
- Misidentifying the vertex as the y-intercept. The y-intercept is the constant c in standard form; the vertex is at a different x-value.
Practice Questions
- 1 Find the vertex and roots of . Does the parabola open up or down?
- 2 A ball's height is h(t) = -5t² + 20t + 1 meters. When does it reach maximum height and how high is that?
- 3 Convert to vertex form and identify the vertex.