Intersection of a straight line and a parabola
Introduction to Intersection of a Straight Line and a Parabola
- The intersection of a straight line and a parabola refers to the points where a linear equation and a quadratic equation meet on a graph.
- These points of intersection are known as the solutions to the system of equations represented by the line and the parabola.
Basic Concepts
- A straight line can intersect a parabola at zero points ( no intersection), one point (tangent line), or two points.
- The intersection points or solutions can be figured out by setting the quadratic equation (parabola) and the linear equation (straight line) equal to each other.
Process of Finding Intersection Points
- Rearrange the equations to set them equal to each other - this step can create a new quadratic equation.
- Solve the newly formed quadratic equation. Use methods such as factorising or the quadratic formula if necessary.
- Results will give the x-coordinates of the intersection points.
- Substitute these x-values into the original linear equation to find the corresponding y-values.
Example
- Suppose we have a line,
y = 2x + 3
, and a parabola,y = x^2 - 2x - 3
. - Setting them equal gives a quadratic equation
x^2 - 2x - 3 = 2x + 3
. Solving this givesx = -3, 2
. - These ‘x’ values, when substituted into the line equation yields the points
(-3, -3)
and(2, 7)
, which are the intersection points.
Extra Notes
- Finding the intersection of a straight line and a parabola is an integral concept in Algebra as it helps in solving simultaneous equations and graphing functions.
- These principles prove useful in higher mathematics such as calculus, as well as real-world applications like physics and engineering.
- Understanding these also builds a drift towards learning how to extrapolate and interpolate data.