You are training your dog to catch a frisbee. You are playing in a large field, and you are standing next to your dog when you throw the frisbee. If the path of the frisbee is y = –x2 + 7x + 1 and the path of the dog is modeled by y = 2x + 5, will the dog catch the frisbee? If so, what are the coordinates of the point or points where they meet? 




To determine whether the dog catches the frisbee, we need to find the points of intersection between the paths of the frisbee and the dog. This involves solving the system of equations:

1. The path of the frisbee: \( y = -x^2 + 7x + 1 \)
2. The path of the dog: \( y = 2x + 5 \)

To find the points of intersection, set the two equations equal to each other:

\[ -x^2 + 7x + 1 = 2x + 5 \]

Next, simplify and solve for \( x \):

\[ -x^2 + 7x + 1 - 2x - 5 = 0 \]

Combine like terms:

\[ -x^2 + 5x - 4 = 0 \]

This is a quadratic equation. To solve it, we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, \( a = -1 \), \( b = 5 \), and \( c = -4 \). Plugging in these values:

\[ x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(-4)}}{2(-1)} \]

\[ x = \frac{-5 \pm \sqrt{25 - 16}}{-2} \]

\[ x = \frac{-5 \pm \sqrt{9}}{-2} \]

\[ x = \frac{-5 \pm 3}{-2} \]

This gives us two solutions for \( x \):

\[ x = \frac{-5 + 3}{-2} = \frac{-2}{-2} = 1 \]

\[ x = \frac{-5 - 3}{-2} = \frac{-8}{-2} = 4 \]

So the points of intersection occur at \( x = 1 \) and \( x = 4 \).

Now, we need to find the corresponding \( y \) values for each \( x \) to get the coordinates of the points of intersection. We can use the equation \( y = 2x + 5 \) (the path of the dog):

For \( x = 1 \):

\[ y = 2(1) + 5 = 2 + 5 = 7 \]

For \( x = 4 \):

\[ y = 2(4) + 5 = 8 + 5 = 13 \]

Thus, the coordinates of the points where the dog catches the frisbee are:

\[ (1, 7) \] and \[ (4, 13) \]

### Conclusion
The dog will catch the frisbee at the points \((1, 7)\) and \((4, 13)\).

Would you like more details or have any questions?

### Relative Questions:
1. How can we verify the solutions for the points of intersection?
2. What does the path of the frisbee look like in terms of its graph?
3. How does the slope of the dog's path compare to the frisbee's path?
4. What are some other methods to solve quadratic equations?
5. Can the quadratic formula be used for all types of quadratic equations?
6. How would the paths change if the initial velocities of the frisbee and dog were different?
7. What real-world factors might affect whether the dog catches the frisbee?
8. How can you use graphing to visually confirm the intersection points?

### Tip:
When solving quadratic equations, always double-check your work by plugging the solutions back into the original equations to ensure they satisfy both equations.

Explore how to solve the problem of determining if a dog catches a frisbee using quadratic equations. Learn step-by-step how to find intersection points between the frisbee's path y = -x^2 + 7x + 1 and the dog's path y = 2x + 5. Suitable for advanced high school students.

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