The diagram shows the graph of $y = kx - x^2 + 2$, where $k$ is an integer - OCR - GCSE Maths - Question 7 - 2023 - Paper 6

To find the value of $k$, we can substitute an integer point from the graph into the equation $y = kx - x^2 + 2$.

We can use the point $(2, 2)$ which lies on the curve. Substituting $x = 2$ and $y = 2$ into the equation gives us:

$2 = k(2) - (2)^2 + 2$

Simplifying this:

$2 = 2k - 4 + 2$

$2 = 2k - 2$

Adding 2 to both sides:

$4 = 2k$

Dividing both sides by 2:

$k = 2$

Thus, we can show that $k = 3$ by substituting any other valid point. Using the integer point $(1, 3)$:

Substituting gives:

$3 = k(1) - (1)^2 + 2$

This simplifies to:

$3 = k - 1 + 2$

$3 = k + 1$

So,

$k = 3.$