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The task is to determine in how many points a given parabola intersects with a given straight line. The problem presents three pairs of a parabola and a straight line (labeled as p and g, respectively). To solve this, we need to set the equations of the parabola and the line equal to each other and solve for the values of \( x \). The number of real solutions tells us the number of intersection points.

Let's solve each case one by one.

### a) 
Parabola: \( p: y = \frac{1}{2}x^2 - 2 \)  
Line: \( g: y = x + 4 \)

Set the two equations equal to find the intersection points:
\[
\frac{1}{2}x^2 - 2 = x + 4
\]

### b)
Parabola: \( p: y = -2x^2 + 7 \)  
Line: \( g: y = -2x + 3 \)

Set the two equations equal to find the intersection points:
\[
-2x^2 + 7 = -2x + 3
\]

### c)
Parabola: \( p: y = -\frac{1}{4}x^2 + 3 \)  
Line: \( g: y = -\frac{1}{2}x + 6 \)

Set the two equations equal to find the intersection points:
\[
-\frac{1}{4}x^2 + 3 = -\frac{1}{2}x + 6
\]

I'll now solve each of these equations to find the number of intersection points.The results show the number of intersection points for each case:

- **a)** The parabola \( p: y = \frac{1}{2}x^2 - 2 \) and the line \( g: y = x + 4 \) intersect at **2 points** (approximate solutions: \( x = -2.61 \) and \( x = 4.61 \)).
- **b)** The parabola \( p: y = -2x^2 + 7 \) and the line \( g: y = -2x + 3 \) intersect at **2 points** (exact solutions: \( x = -1 \) and \( x = 2 \)).
- **c)** The parabola \( p: y = -\frac{1}{4}x^2 + 3 \) and the line \( g: y = -\frac{1}{2}x + 6 \) have **0 real intersection points** (the solutions are complex numbers, indicating no real intersection).

### Summary:
- Case a: 2 intersection points
- Case b: 2 intersection points
- Case c: 0 real intersection points

Would you like further details on these solutions?

Here are some related questions to expand on this topic:
1. What happens when a parabola and a line intersect at exactly one point?
2. How can we visualize the intersection of two functions graphically?
3. What determines whether a quadratic equation has real or complex solutions?
4. Can the line ever lie entirely within or outside the parabola without intersection?
5. How does the discriminant of a quadratic equation relate to the number of intersection points?

**Tip:** When solving quadratic equations, the discriminant \( \Delta = b^2 - 4ac \) is key to determining whether the solutions are real (if \( \Delta \geq 0 \)) or complex (if \( \Delta < 0 \)).

In wie vielen Punkten schneiden sich die Parabel und die Gerade? Eine Skizze hilft. (How many points do the parabola and the line intersect? A sketch helps.)

Learn how to find the intersection points of a parabola and a line through step-by-step analysis using quadratic equations and linear functions. This problem involves solving quadratic equations and understanding the discriminant. Suitable for students in Grades 9-12.

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