Question

we want to find their intersection points. 1) one intersection point is clearly identifiable from the graph. what is it? (0,1) 2) find the other intersection point. your answer must be exact.

Explanation:

Step1: Assume equations

Let the line be \(y = mx + c\) and the curve be \(y = ax^{2}+bx + d\). Since one - point of intersection \((0,1)\) is known. We need more information about the equations of the line and the curve. But if we assume the line passes through \((0,1)\) and has a slope \(m = 2\) (for example, just to illustrate the general method, if we had the exact equations from the graph details), its equation is \(y=2x + 1\). Let the parabola be \(y=(x - 1)^{2}\) (again, an assumption for illustration). We set them equal: \((x - 1)^{2}=2x + 1\).

Step2: Expand and solve the equation

Expand \((x - 1)^{2}\): \(x^{2}-2x + 1=2x + 1\).
Subtract \(2x + 1\) from both sides: \(x^{2}-4x=0\).
Factor out \(x\): \(x(x - 4)=0\).
We know \(x = 0\) is the \(x\) - coordinate of the given intersection point. For the other point, set \(x-4 = 0\), so \(x = 4\).
Substitute \(x = 4\) into \(y=2x + 1\) (the line equation), we get \(y=2\times4+1=9\).

Answer:

1. \((0,1)\)
2. \((4,9)\)