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The problem appears to involve a curve defined by an equation \( y = -x^2 + 8x + c \), where \( c \) is an integer, and the curve passes through a specific point \( (1,5) \). We are tasked with finding the value of \( c \).

The general approach to solving this type of problem involves substituting the point \( (1,5) \) into the equation to find \( c \).

### Steps:

1. Substitute \( x = 1 \) and \( y = 5 \) into the equation \( y = -x^2 + 8x + c \).
   
   \[
   5 = -(1)^2 + 8(1) + c
   \]
   
2. Simplify:

   \[
   5 = -1 + 8 + c
   \]
   
   \[
   5 = 7 + c
   \]
   
3. Solve for \( c \):

   \[
   c = 5 - 7 = -2
   \]

Thus, the value of \( c \) is \( -2 \).

Would you like further details or explanations on this, or have any other questions related to this problem?

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Here are 5 related questions for practice:

1. How do you determine the vertex of a parabola from a quadratic equation?
2. What is the significance of the value of \( c \) in a quadratic equation?
3. How would you find the roots of the equation \( y = -x^2 + 8x - 2 \)?
4. How does the point \( (1, 5) \) affect the shape or position of the parabola?
5. Can you find the maximum value of the function \( y = -x^2 + 8x - 2 \)?

### Tip:
When analyzing parabolic equations, always check if the equation is in standard form to quickly identify properties like the vertex, axis of symmetry, and direction of the parabola.

Bentuk persamaan kurva di bawah yaitu y = -x^2 + 8x + c, dengan x ∈ ℝ. C merupakan bilangan bulat. Salah satu titik yang melalui kurva tersebut adalah (1,5). Tentukan nilai c = ...

Learn how to solve the quadratic equation y = -x^2 + 8x + c by substituting the point (1,5) into the equation. This problem covers algebraic concepts and solving quadratic equations using substitution. Suitable for students in Grades 8-10.

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