tiffany is solving an equation where both sides are quadratic expressions. she sets each quadratic equation…
tiffany is solving an equation where both sides are quadratic expressions. she sets each quadratic equation equal to y and graphs the resulting system. if the graph of one quadratic opens upward and the other opens downward, what is the greatest possible number of intersections for these graphs? none one two infinitely many
Answer
Explanation:
Step1: Recall quadratic - graph properties
A quadratic function has the form $y = ax^{2}+bx + c$. If $a>0$, the parabola opens upward; if $a < 0$, it opens downward.
Step2: Consider intersection of two parabolas
The intersection points of two graphs $y = f(x)$ and $y = g(x)$ are found by solving $f(x)=g(x)$. A quadratic - quadratic system of equations can be written as $ax^{2}+bx + c=dx^{2}+ex + f$, which simplifies to a quadratic equation $(a - d)x^{2}+(b - e)x+(c - f)=0$.
Step3: Use the discriminant of the quadratic equation
The number of solutions of a quadratic equation $Ax^{2}+Bx + C = 0$ is determined by the discriminant $\Delta=B^{2}-4AC$. The maximum number of solutions of a quadratic equation is 2. So, the maximum number of intersection points of two parabolas (one opening upward and one opening downward) is 2.
Answer:
C. two