Question

describe the possible numbers of intersections for each of the following pairs of graphs. sketch a graph for each possibility. for example, a circle could intersect a line twice, once, or not at all. your solution to each part should include all of the possibilities and a sketched example of each one. 8 - 10 hw etool. homework help a. two different lines. b. a line and a parabola. c. two different parabolas. d. a parabola and a circle.

Explanation:

Part a: Two different lines

Step1: Analyze non - parallel lines

Two non - parallel lines in a plane intersect at exactly one point.

Step2: Analyze parallel lines

Two parallel lines in a plane do not intersect.

Part b: A line and a parabola

Step1: Use the discriminant of the quadratic equation

Let the equation of the line be \(y = mx + c\) and the equation of the parabola be \(y=ax^{2}+bx + d\). Substitute \(y\) from the line into the parabola: \(ax^{2}+bx + d=mx + c\), or \(ax^{2}+(b - m)x+(d - c)=0\). The discriminant is \(\Delta=(b - m)^{2}-4a(d - c)\)

Step2: Case 1: \(\Delta>0\)

If \(\Delta>0\), the line intersects the parabola at two points.

Step3: Case 2: \(\Delta = 0\)

If \(\Delta = 0\), the line is tangent to the parabola and intersects it at one point.

Step4: Case 3: \(\Delta<0\)

If \(\Delta<0\), the line and the parabola do not intersect.

Part c: Two different parabolas

Step1: Consider the system of equations

Let \(y = a_{1}x^{2}+b_{1}x + c_{1}\) and \(y=a_{2}x^{2}+b_{2}x + c_{2}\). Then \((a_{1}-a_{2})x^{2}+(b_{1}-b_{2})x+(c_{1}-c_{2}) = 0\)

Step2: Case 1: Quadratic equation (\(a_{1}

eq a_{2}\))
If \(a_{1}
eq a_{2}\), the discriminant \(\Delta=(b_{1}-b_{2})^{2}-4(a_{1}-a_{2})(c_{1}-c_{2})\)

• If \(\Delta>0\), the two parabolas intersect at two points.
• If \(\Delta = 0\), the two parabolas intersect at one point.
• If \(\Delta<0\), the two parabolas do not intersect.

Step3: Case 2: Linear equation (\(a_{1}=a_{2}\))

If \(a_{1}=a_{2}\) and \(b_{1}
eq b_{2}\), the equation \((b_{1}-b_{2})x+(c_{1}-c_{2}) = 0\) has exactly one solution (one intersection point). If \(a_{1}=a_{2}\), \(b_{1}=b_{2}\) and \(c_{1}
eq c_{2}\), the two parabolas are parallel (no intersection).

Part d: A parabola and a circle

Step1: Substitute the equation of the parabola into the circle

Let the parabola be \(y = ax^{2}+bx + c\) and the circle be \((x - h)^{2}+(y - k)^{2}=r^{2}\). Substitute \(y\) from the parabola into the circle: \((x - h)^{2}+(ax^{2}+bx + c - k)^{2}=r^{2}\)

Step2: Analyze the degree of the resulting equation

The resulting equation is a quartic (degree - 4) equation.

• Case 1: The quartic equation can have \(0\) intersection points (e.g., the parabola is outside the circle).
• Case 2: It can have \(1\) intersection point (e.g., the parabola is tangent to the circle).
• Case 3: It can have \(2\) intersection points (e.g., the parabola cuts the circle at two points).
• Case 4: It can have \(3\) intersection points.
• Case 5: It can have \(4\) intersection points.

Answer:

• a: Two different lines can intersect \(0\) (parallel) or \(1\) (non - parallel) times.
• b: A line and a parabola can intersect \(0\), \(1\), or \(2\) times.
• c: Two different parabolas can intersect \(0\), \(1\), or \(2\) times.
• d: A parabola and a circle can intersect \(0\), \(1\), \(2\), \(3\), or \(4\) times.