Sketching quadratic graphs

Introduction to Quadratic Graphs

Quadratic equations take the form y = ax^2 + bx + c where a, b, and c are constants.
Understanding them is essential for exams as they commonly appear in Algebra materials.

Sketching the Graph

Quadratic graphs are always parabolas. Parabolas can open upwards (if ‘a’ is positive) or downwards (if ‘a’ is negative).
The graph crosses the y-axis at the y-intercept (c).
The maximum or minimum point is the vertex and its x-coordinate can be found using the formula -b/2a.

Identifying Key Points

The line of symmetry of a parabola is a vertical line that passes through the vertex.
The roots or x-intercepts of a quadratic equation are the x-values when y equals zero, i.e. the points where the curve crosses the x-axis.
Quadratic equations can have two roots, one root or no roots dependent on the value of the discriminant (b^2 - 4ac). Greater than 0 indicates two distinct roots, equal to 0 one real root, and less than 0 indicates no real roots.

Examples

For the equation y = x^2 - 2x - 3, ‘a’ is positive so the graph opens upwards.
The y-intercept is at -3.
The x-coordinate of the vertex can be found by -(-2)/2(1) = 1.
By setting the equation to zero, the roots can be found which are x = -1 and x = 3.

Conclusion

Practising sketching quadratic graphs frequently will help memorise the process and develop confidence in it. Remember to always label key parts like the vertex, the y-intercept and the roots.
Having a good grasp of quadratic graphs not only helps in maths but is also a crucial tool for a range of scientific calculations, making it a key skill worth mastering.