Let's solve the problem step by step.

Step 1: Start with the given equation.

The equation is: $xy = 4 + 2x - x^2$

Step 2: Rearrange the right side to prepare for completing the square.

First, write the quadratic expression $-x^2 + 2x$ in a standard form suitable for completing the square: $xy = -x^2 + 2x + 4$

Step 3: Complete the square.

To complete the square for the quadratic expression $-x^2 + 2x$, we can factor out $-1$ from the first two terms: $xy = -(x^2 - 2x) + 4$ Now, complete the square inside the parentheses. To complete the square for $x^2 - 2x$, add and subtract $\left(\frac{2}{2}\right)^2 = 1$: $xy = -\left(x^2 - 2x + 1 - 1\right) + 4$ This simplifies to: $xy = -\left((x - 1)^2 - 1\right) + 4$ Now, distribute the negative sign: $xy = -(x - 1)^2 + 1 + 4$ Simplify further: $xy = -(x - 1)^2 + 5$

Step 4: Express the equation in the required form $xy = a(x + b)^2 + c$.

The equation is now in the form: $xy = -1 \cdot (x - 1)^2 + 5$ This can be rewritten as: $xy = -1 \cdot \left(x - (-1)\right)^2 + 5$ Therefore, comparing with $xy = a(x + b)^2 + c$, we have:

• $a = -1$
• $b = -1$
• $c = 5$

Step 5: Find the coordinates of the maximum point of the curve.

The maximum point occurs where the squared term $(x - 1)^2$ is zero, because the negative sign in front of the square means that the expression is maximized when the square is minimized (which is zero).

Set $(x - 1)^2 = 0$: $x - 1 = 0 \quad \Rightarrow \quad x = 1$

Substitute $x = 1$ back into the equation to find $y$: $xy = -1 \cdot (1 - 1)^2 + 5 = 5$ Since $x = 1$, we have: $y = \frac{5}{1} = 5$

So, the coordinates of the maximum point are $(1, 5)$.

Final Answer:

• a: $-1$
• b: $-1$
• c: $5$
• Maximum point: $(1, 5)$

Would you like more details on this method or have any other questions?

Related Questions:

1. What is the process of completing the square for a general quadratic expression?
2. How does the sign of $a$ in $y = a(x + b)^2 + c$ affect the direction of the parabola?
3. Can you express the maximum or minimum point of a parabola using the vertex form equation?
4. How do you derive the vertex form of a quadratic function from its standard form?
5. How does the transformation $(x - h, y - k)$ shift the graph of a function?

Tip:

When completing the square, always ensure that the coefficient of $x^2$ is 1. If not, factor it out first before completing the square.