To solve the system of equations by graphing:

The system is:

1. $y = -x^2 + 2x + 7$ (a quadratic equation, representing a parabola).
2. $y = 2x + 7$ (a linear equation, representing a straight line).

Steps:

1. Graph the quadratic equation:

   • $y = -x^2 + 2x + 7$.
   • This is a parabola that opens downward (due to the negative coefficient of $x^2$).
   • Find the vertex:
     • The x-coordinate of the vertex is $x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1$.
     • Substitute $x = 1$ into the equation to find $y$: $y = -(1)^2 + 2(1) + 7 = 8$.
     • Vertex: $(1, 8)$.
   • Additional points:
     • Substitute $x = 0$: $y = 7$ (point $(0, 7)$).
     • Substitute $x = 2$: $y = -4 + 4 + 7 = 7$ (point $(2, 7)$).

2. Graph the linear equation:

   • $y = 2x + 7$.
   • This is a straight line with slope 2 and y-intercept 7.
   • Points to graph:
     • When $x = 0$: $y = 7$ (point $(0, 7)$).
     • When $x = -1$: $y = 5$ (point $(-1, 5)$).
     • When $x = 1$: $y = 9$ (point $(1, 9)$).

3. Find points of intersection:

   • Solve $-x^2 + 2x + 7 = 2x + 7$ algebraically:
     • $-x^2 + 2x + 7 - 2x - 7 = 0$,
     • $-x^2 = 0$,
     • $x = 0$.
   • Substitute $x = 0$ into $y = 2x + 7$: $y = 7$.
     • Point of intersection: $(0, 7)$.

Final Answer:

The parabola and line intersect at one point: $(0, 7)$.

If you have further questions or want a graph of the solution, let me know!

Here are related questions for deeper understanding:

1. How do you determine the vertex of a parabola from its equation?
2. Why does the parabola open downward in this case?
3. How can you algebraically confirm the number of intersections in a quadratic-linear system?
4. What is the significance of the slope in the linear equation?
5. How does the discriminant help in identifying the number of solutions?

Tip: Always check for symmetry when graphing a parabola for quicker plotting.