Question

mark is solving an equation where one side is a quadratic expression and the other side is a linear expression. he sets the expressions equal to y and graphs the equations. what is the greatest possible number of intersections for these graphs?
none
one
two
infinitely many

Explanation:

Step1: Recall Graph Types

A quadratic expression \( y = ax^2 + bx + c \) (where \( a
eq 0 \)) graphs as a parabola (a U - shaped or inverted U - shaped curve). A linear expression \( y=mx + b\) (where \( m \) is the slope and \( b \) is the y - intercept) graphs as a straight line.

Step2: Analyze Intersection Possibilities

To find the number of intersections between the parabola \( y = ax^2+bx + c\) and the line \( y=mx + b\), we set the two equations equal to each other: \( ax^2+bx + c=mx + b\). Rearranging this equation gives us a quadratic equation of the form \( ax^2+(b - m)x+(c - b)=0\) (where \( a
eq0\)). The number of solutions to a quadratic equation \( Ax^2 + Bx + C = 0\) (where \( A
eq0\)) is determined by the discriminant \( D=B^2 - 4AC\).

• If \( D>0\), the quadratic equation has two distinct real solutions. This means the line and the parabola intersect at two distinct points.
• If \( D = 0\), the quadratic equation has one real solution (a repeated root), which means the line is tangent to the parabola and they intersect at one point.
• If \( D<0\), the quadratic equation has no real solutions, which means the line and the parabola do not intersect.

Since a quadratic equation (in the variable \( x\)) can have at most two real solutions (when the discriminant is positive), the maximum number of intersection points between a parabola (graph of a quadratic function) and a straight line (graph of a linear function) is 2.

Answer:

two