How do you find the point of intersection of a parabola and a line?

Subtract mx+d from both sides. Now we have a quadratic equation in one variable, the solution of which can be found using the quadratic formula. The solutions to the equation ax2+(b−m)x+(c−d)=0 will give the x-coordinates of the points of intersection of the graphs of the line and the parabola.

How do you show that a line cuts a parabola?

A line can meet a parabola in at most two points. If the discriminant of the resulting quadratic is negative, the line does not meet the parabola at all. If the discriminant of the resulting quadratic is positive, the line meets the parabola in two distinct points.

What is the point of intersection of the parabola?

Now, the places where the two functions cross are called their points of intersection. At these points of intersection the x-coordinate for the line equals the x-coordinate for the parabola, and the y-coordinate for the line equals the y-coordinate for the parabola.

What do you call the intersection of the parabola and its axis of symmetry?

The point where the parabola intersects its axis of symmetry is called the “vertex” and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the “focal length”.

What is the fixed line in parabola?

The parabola is defined in analytic geometry as the set of all points P in a plane that are the same distance from a given line and a fixed point not on a line. The fixed point is called the focus and the fixed line is called the directrix.

How do you find the tangent line in calculus?

1) Find the first derivative of f(x). 2) Plug x value of the indicated point into f ‘(x) to find the slope at x. 3) Plug x value into f(x) to find the y coordinate of the tangent point. 4) Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.

How do you find the point of intersection of a parabola?

Find the points of intersection of the parabola with the line given respectively by their equations y = 2 x 2 + 4 x – 3. 2y + x = 4. Solution to Example 1. We first solve the linear equation for y as follows: y = – (1 / 2) x + 2. We now substitute y in the equation of the parabola by – (1 / 2) x + 2 as follows.

What is the vertex and axis of symmetry of the parabola?

Its vertex is the point (−5, 0) and the axis of symmetry is x = −5. Write down the equation of the parabola obtained when the graph of y = x2 is translated 3 units to the left. Sketch the parabola. Describe the transformation required to move the parabola y = ( x + 3) 2 to y = ( x − 2) 2 .

What is the equation of the parabola y = -x2?

When the basic parabola y = x2 is reflected in the x-axis, the point (3, 9), for example, is reflected to the point (3, 9). Indeed the general point (a, a2) becomes (a, −a2) and so the equation satisfied by these points is y = −x2.

How to change the general point of a parabola?

When we translate the parabola vertically upwards or downwards, the y -value of each point on the basic parabola is increased or decreased. Thus, for example, translating the parabola upwards by 9 units, shifts the general point ( a, a2) to ( a, a2 + 9).