Quadratic concept, parabolas, roots, vertex
High School Essentials • 9-12
Quadratic equations -- equations of the form ax2 + bx + c = 0 where a is not 0 -- produce one of the most recognizable curves in mathematics: the parabola. From the arc of a basketball to the shape of satellite dishes, parabolas appear throughout the natural and engineered world. This lesson builds your conceptual understanding of quadratics before diving into solving techniques.
The graph of y = ax2 + bx + c is always a parabola -- a smooth, symmetric, U-shaped curve. The coefficient a determines two fundamental properties:
If a > 0:
If a < 0:
The magnitude |a| controls the width: larger |a| makes the parabola narrower (steeper), while smaller |a| makes it wider (flatter). Compare y = 5x2 (very narrow) to y = 0.2x2 (very wide).
| Feature | Description | How to Find |
|---|---|---|
| Vertex | The highest or lowest point | x = -b/(2a), then find y |
| Axis of Symmetry | Vertical line through the vertex | x = -b/(2a) |
| Roots / Zeros | Where the parabola crosses the x-axis | Set y = 0 and solve |
| y-intercept | Where the parabola crosses the y-axis | Set x = 0; it equals c |
Find the vertex and axis of symmetry of y = 2x2 - 8x + 3.
Since a = 2 > 0, the parabola opens upward and (2, -5) is the minimum point.
The roots (also called zeros, solutions, or x-intercepts) are where the parabola meets the x-axis. The discriminant, defined as D = b2 - 4ac, tells you how many real roots exist without actually solving:
| Discriminant Value | Number of Real Roots | Graph Behavior |
|---|---|---|
| D > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| D = 0 | One repeated real root | Parabola touches x-axis at vertex |
| D < 0 | No real roots | Parabola does not reach x-axis |
How many real solutions does 3x2 - 4x + 5 = 0 have?
The parabola y = 3x2 - 4x + 5 lies entirely above the x-axis.
A ball is thrown upward. Its height in feet after t seconds is h(t) = -16t2 + 48t + 5. Describe the key features.
The formula is x = -b/(2a). Do not forget the negative sign in front of b. For y = x2 + 6x + 8, the axis of symmetry is x = -6/(2*1) = -3, not x = 3.
1. For y = -x2 + 4x - 1, does the parabola open up or down? Find the vertex.
a = -1 < 0, so it opens downward. Vertex x-coordinate: x = -4/(2*(-1)) = -4/(-2) = 2. Then y = -(2)2 + 4(2) - 1 = -4 + 8 - 1 = 3. Vertex: (2, 3), which is the maximum point.
2. Find the discriminant of x2 + 6x + 9 = 0 and determine the number of real solutions.
D = 62 - 4(1)(9) = 36 - 36 = 0. There is exactly one repeated real solution. (The parabola touches the x-axis at its vertex, x = -3.)
3. What is the y-intercept of y = 5x2 - 3x + 7?
Set x = 0: y = 5(0) - 3(0) + 7 = 7. The y-intercept is (0, 7).
4. A parabola has vertex (3, -2) and opens upward. Can it have zero x-intercepts? Explain.
No. Since the vertex is at y = -2 (below the x-axis) and the parabola opens upward, it must eventually cross the x-axis on both sides. It will have two x-intercepts.
5. Compare the widths of y = 4x2, y = x2, and y = 0.25x2.
All open upward. y = 4x2 is the narrowest (steepest). y = x2 is the standard width. y = 0.25x2 is the widest (flattest). Larger |a| means narrower; smaller |a| means wider.
Quadratic functions produce parabolas. The sign of a determines direction (up or down), and |a| controls width. The vertex (-b/(2a), f(-b/(2a))) is the extreme point, and the axis of symmetry is the vertical line through it. The discriminant b2 - 4ac reveals how many times the parabola crosses the x-axis: twice, once, or never. Understanding these features lets you analyze any quadratic before solving it.