Question

What are the possibilities for the number of times the graphs of two different quadratic functions intersect?

Short answer

The graphs of two different quadratic functions can intersect 0, 1, or 2 times.

Step-by-step solution

- Understand the problem

We need to determine the number of points where the graphs of two different quadratic functions intersect. Quadratic functions are parabolas, and the intersections are the points where the equations of the parabolas are equal.

- Consider the quadratic equation intersection

Let the two quadratic functions be given by \( f(x) = ax^2 + bx + c \) and \( g(x) = dx^2 + ex + f \). To find the points of intersection, set the equations equal to each other: \( ax^2 + bx + c = dx^2 + ex + f \).

- Simplify the equation

Rearrange the equation to bring all terms to one side: \( (a-d)x^2 + (b-e)x + (c-f) = 0 \). This is now a quadratic equation in terms of \( x \).

- Analyze the quadratic equation

A quadratic equation \( Ax^2 + Bx + C = 0 \) can have 0, 1, or 2 real solutions depending on the discriminant \( \text{Δ} = B^2 - 4AC \). If \( \text{Δ} > 0 \), there are 2 distinct solutions. If \( \text{Δ} = 0 \), there is 1 solution. If \( \text{Δ} < 0 \), there are no real solutions.

- Interpret results in terms of intersections

Thus, the graphs of the two quadratic functions can intersect at 0, 1, or 2 points, depending on the value of the discriminant of the resultant quadratic equation.

Key concepts

quadratic equations intersection

When we deal with quadratic functions, also known as parabolas, a common problem is finding out how many times these parabolas intersect. Understanding this involves several key concepts in algebra, mainly focusing on quadratic equations. To solve the problem, start by considering the general form of two different quadratic equations:

• First quadratic function: \(f(x) = ax^2 + bx + c\)
• Second quadratic function: \(g(x) = dx^2 + ex + f\)

To find their intersection points, we set these equations equal: \(ax^2 + bx + c = dx^2 + ex + f\). By simplifying this, we obtain a single quadratic equation: \((a-d)x^2 + (b-e)x + (c-f) = 0\). Let's walk through what this means and how it helps us determine the points of intersection.

parabolas' characteristics

Quadratic functions graph as parabolas, which are U-shaped curves that can open either up or down. Here are some key features:

• The parabola touches or crosses its vertex; the highest or lowest point on the curve.
• It is symmetric around a vertical line called the axis of symmetry.
• The direction it opens (up or down) is determined by the leading coefficient \( (a) \) in the equation \( ax^2 + bx + c \).

When we consider two different parabolas, their point(s) of intersection are the points where they cross each other on the graph. Since each parabola has a specific shape and direction, these intersections rely mainly on how their equations relate.

role of the discriminant in finding intersection points

The discriminant is a crucial component in finding the number of real solutions to a quadratic equation. For the quadratic equation \( Ax^2 + Bx + C = 0 \), the discriminant \( \text{Δ} \) is given by: \[ \text{Δ} = B^2 - 4AC \] The discriminant can tell us the nature and number of solutions:

• If \( \text{Δ} > 0 \), there are 2 distinct real solutions, meaning two distinct intersection points.
• If \( \text{Δ} = 0 \), there is exactly 1 real solution, implying the parabolas touch at a single point.
• If \( \text{Δ} < 0 \), there are no real solutions, meaning the parabolas do not intersect.

By examining the discriminant of the simplified intersection equation \( (a-d)x^2 + (b-e)x + (c-f) = 0 \), we can determine the exact nature of their intersection.