Quadratic Functions and Equations
Quadratic Functions and Equations connects the graph of y=ax² with solving equations and finding intersections. This practice covers transformations of the parabola, factoring and formula methods, the discriminant, and the link between roots and x-intercepts.
Core rules
- The graph of y=ax² is a parabola with vertex at the origin and axis x=0. If a>0 it opens upward; if a<0 it opens downward. Larger |a| makes the parabola narrower.
- Solve ax²+bx+c=0 by factoring first, then by completing the square or the quadratic formula x=(−b±√(b²−4ac))/(2a).
- The discriminant D=b²−4ac tells you the number of real solutions: D>0 gives two, D=0 gives one repeated solution, and D<0 gives none.
- The intersection of y=ax² and y=mx+n is found by solving ax²=mx+n, which rearranges to a quadratic equation. The x-coordinates of the intersection points are its roots.
Worked example
Find the intersection points of y=x² and y=2x+3. Set x²=2x+3, so x²−2x−3=0. Factor: (x−3)(x+1)=0, giving x=3 or x=−1. Substitute into y=x²: when x=3, y=9; when x=−1, y=1. The intersection points are (3,9) and (−1,1). Checking in y=2x+3 gives 9=6+3 and 1=−2+3, so both points are correct.
Traps
- Dividing both sides by x and losing the solution x=0.
- Forgetting the ± sign when taking a square root.
- Using the vertex x-coordinate as −b/(2a) but dropping the negative sign.
- Assuming a quadratic always has two real roots without checking the discriminant.