Question

Can someone please help? just need help with the possibilities...

Answer 1

There are infinite possible graphs for each of the questions.  I am assuming that the purpose of this assignment is to get you to graph different types of equations and then transform them (by shifting to the left, right, up, down and/or stretching them) in order to see how the graphs move.  I suggest you do this because it will be helpful for you to understand transformations.

Here are the answers and possible graphs.  I have attached a snapshot of the maximum number of intersections for each graph.

a) They can intersect 0, 1, or 2 times

$\begin {array}{c||l|l}\underline{Intersections}&\underline{Line}&\underline{Parabola}\\ 0&y=x&y=x^2+2\\1&y=2&y=x^2+2\\2&y=x&y=x^2-2\\\end{array}$

b) They can intersect 0, 1, or 2 times

$\begin {array}{c||l|l}\underline{Intersections}&\underline{Parabola\ 1}&\underline{Parabola\ 2}\\0&y=x^2+2&y=-x^2-2\\1&y=x^2-2&y=(x-2)^2\\2&y=x^2-2&y=\dfrac{1}{2}x^2\\\end{array}$

c) They can intersect 0, 1, 2, 3, or 4 times

$\begin {array}{c||l|l}\underline{Intersections}&\underline{Parabola}&\underline{\qquad Circle\qquad }\\0&y=x^2+1&x^2+(y-4)^2=4\\\\1&y=x^2-2&x^2+(y-4)^2=4\\\\2&y=x^2-3&x^2+(y-4)^2=4\\\\3&y=x^2&x^2+(y-\dfrac{22}{5})^2=4\\\\4&y=x^2-2&x^2+(y-2)^2=9\\\end{array}$

d) They can intersect 0, 1, 2, or 3 times

$\begin {array}{c||l|l}\underline{Intersections}&\underline{\quad Parabola\quad }&\underline{Hyperbola}\\0&y=(x+5)^2+1&y=1/x\\1&y=x^2+2&y=1/x\\2&y=(x+5)^2-2&y=1/x\\3&y=x^2-4&y=1/x\\\end{array}$

Answer 2

Not sure about the last question